The Theoretical Minimum  Lecture 5
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Formal Metadata
Title 
The Theoretical Minimum  Lecture 5

Title of Series  
Part Number 
5

Number of Parts 
10

Author 

License 
CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Stanford University

Release Date 
2012

Language 
English

Content Metadata
Subject Area  
Abstract 
(February 6, 2012) Leonard Susskind discusses an array of topics including uncertainty, the Schroedinger equation, and how things evolve with time.

00:00
Typesetting
Ruler
Juli Fernández
Schubvektorsteuerung
Mechanical fan
Day
Measurement
FACTS (newspaper)
Kopfstütze
Basis (linear algebra)
Night
Focus (optics)
Excited state
Permittivity
Fieldeffect transistor
Meeting/Interview
Cartridge (firearms)
Canadair CL44
Tool
Rail transport operations
08:31
Matrix (printing)
April
Schubvektorsteuerung
Measurement
Year
Basis (linear algebra)
Avro Canada CF105 Arrow
Image projector
Roll forming
Comet
Meeting/Interview
Barrel
Cardinal direction
Tool
Tape recorder
Rail transport operations
Mail (armour)
Magnetic moment
Quantum
Progressive lens
Typesetting
Pooltype reactor
Catadioptric system
Refractive index
Air compressor
Pendulum
Barque
Paul <Firma>
FACTS (newspaper)
Classical mechanics
Finger protocol
Concentrator
Excited state
Permittivity
Pager
Rolling (metalworking)
Orbit
Sheet metal
Microwave
18:53
Bomb
Mechanical fan
Short circuit
Year
Foot (unit)
Mitsubishi A6M Zero
Watch
Continuous track
Automobile
Tool
Rail transport operations
Hyperbelnavigation
Water vapor
Magnetic moment
Progressive lens
Dolch
Automobile platform
White
Map
FACTS (newspaper)
Excited state
Permittivity
Sizing
Noble gas
Cartridge (firearms)
Plant (control theory)
Measurement
23:09
Ruler
Prozessleittechnik
Hot working
Quality (business)
Bomb
Schubvektorsteuerung
Year
Measurement
Summer (George Winston album)
Bill of materials
Energy level
Particle
Rocket engine
Basis (linear algebra)
Fieldeffect transistor
Roll forming
Meeting/Interview
Automobile
Cardinal direction
Spare part
Rail transport operations
Station wagon
Typesetting
Cut (gems)
Sensor
Machine
Source (album)
FACTS (newspaper)
Classical mechanics
Excited state
Cartridge (firearms)
Plant (control theory)
Mass
Food storage
Flugbahn
Gas turbine
Noise figure
Orbit
33:33
Matrix (printing)
Weapon
Theory of relativity
Analog signal
Brillouin zone
Quality (business)
Auto mechanic
Multiplizität
Variable star
Avro Canada CF105 Arrow
Bird vocalization
Comet
Turning
Roll forming
Colorfulness
Continuous track
Sunrise
Limiter
Tool
Tape recorder
Rail profile
Quantum
Water vapor
Mechanic
Combined cycle
Augustus, Count Palatine of Sulzbach
Space probe
Bracket clock
Ground station
Printing
Missile
Cartridge (firearms)
Empennage
Gas turbine
Measurement
Porcelain
Ruler
Spaceflight
Transmission line
Bomb
Year
Lunar phase
Rotating radio transient
Ship class
Automobile
Litter (vehicle)
Spare part
Steckverbinder
Rail transport operations
Hose coupling
Watercraft rowing
Catadioptric system
Transmission tower
Stagecoach
Key (engineering)
Kilogram
White
Punch (tool)
Classical mechanics
Yacht
Finger protocol
Metre
Glass
Fuel
Motion capture
Sizing
Hour
Plant (control theory)
Cab (locomotive)
Miner
Amplitudeshift keying
50:05
Matrix (printing)
Mechanic
Stagecoach
Schubvektorsteuerung
Bracket clock
Electronics
Fire apparatus
Basis (linear algebra)
Metre
Excited state
Rootstype supercharger
Spin (physics)
Direct current
Roll forming
Automobile
Cartridge (firearms)
Gas turbine
Wire
Magnetism
Porcelain
56:45
Typesetting
Excited state
Meeting/Interview
Sunrise
Rail transport operations
Star
59:13
Weapon
Matrix (printing)
Strahlungsnebel
Effects unit
Negativer Widerstand
Cosmic distance ladder
Variable star
Image projector
Bird vocalization
Phase (matter)
Separation process
Roll forming
Screw cap
Cardinal direction
Limiter
Tool
Magnetic moment
Halpha
Truck
Mechanic
Refractive index
Paper
FACTS (newspaper)
Excited state
Cartridge (firearms)
Gas turbine
Measurement
Orbit
Storm
Ruler
Jet (lignite)
Schubvektorsteuerung
Year
Printing press
Crystal structure
Basis (linear algebra)
Printer (publishing)
Fieldeffect transistor
Apparent magnitude
Potentiometer
Volumetric flow rate
Automobile
Steckverbinder
Orbital period
Spare part
Shutter (photography)
Rail transport operations
Star
Telephone
Catadioptric system
Digital electronics
Machine
AC power plugs and sockets
Cogeneration
Yacht
Classical mechanics
Finger protocol
Audio frequency
Forging
Fiat Brava
Schwache Lokalisation
Hour
Plant (control theory)
Miner
1:17:38
Matrix (printing)
Bestrahlungsstärke
Mechanical fan
Absorbance
Particle
Group delay and phase delay
Bird vocalization
Meeting/Interview
Tool
Magnetism
Magnetic moment
Truck
Mechanic
Combined cycle
Refractive index
Bracket clock
Single (music)
FACTS (newspaper)
Missile
Cartridge (firearms)
Signal (electrical engineering)
Gas turbine
Book design
Noise figure
Ruler
Spindle (textiles)
Hot working
Transmission line
Schubvektorsteuerung
Year
Measurement
Force
Access network
Fire apparatus
Mitsubishi A6M Zero
Automobile
String theory
Institut für Raumfahrtsysteme
Musical ensemble
Steckverbinder
Spare part
Electric charge
Rail transport operations
Stock (firearms)
Sigma
Electronic component
Magnetspule
Bending (metalworking)
White
Magnet
Classical mechanics
Yacht
Kopfstütze
Spin (physics)
Sizing
Transistor
Wire
Miner
1:36:04
Matrix (printing)
Prozessleittechnik
Magnetic core
Mechanical fan
Spring (device)
Tool
Magnetism
Linear motor
Magnetic moment
Quantum
Water vapor
Striking clock
Mechanic
Combined cycle
Sensor
Photon
Single (music)
Electronics
Paper
Source (album)
Focus (optics)
Excited state
Quantum fluctuation
Direct current
Cartridge (firearms)
Signal (electrical engineering)
Mintmade errors
Measurement
Book design
Volkswagen Golf
Ruler
Hot working
Spaceflight
Schubvektorsteuerung
Gyroscope
Year
Summer (George Winston album)
Access network
Fire apparatus
Energy level
Towing
Basis (linear algebra)
Apparent magnitude
Rootstype supercharger
Ship class
Wednesday
Spare part
Rail transport operations
Hose coupling
Cylinder head
Stock (firearms)
Electronic component
White
Magnet
Chemical substance
Classical mechanics
Fulling
Fuel
Permittivity
Angle of attack
Rotor (electric)
Spin (physics)
Hour
Dashboard
Semitrailer truck
Miner
Last
1:54:29
Matrix (printing)
Prozessleittechnik
Hot working
Submarine
Schubvektorsteuerung
Year
Summer (George Winston album)
Fire apparatus
Energy level
Apparent magnitude
Roll forming
Barrel
Cardinal direction
Magnetism
Progressive lens
Speckle imaging
Electronic component
Gate (airport)
Bending (metalworking)
Photon
Car seat
White
Twoway radio
Hilt
Suitcase
Map
Kopfstütze
Audio frequency
Direct current
Spin (physics)
Computer animation
Photonics
Hour
Flugbahn
Satellite
Measurement
Miner
Orbit
00:05
Stecher University poultry
00:09
tonight going to do a number of things were to talk about uncertainty we're going there are tools talk more about the Schroedinger equation the various versions of the Schroedinger equation where talk about how things evolve with time we talked about that some already are worth see what else we have him Now we're applying the Schroedinger equation idea for the evolution of systems so our favorite little system once banned but better we have time and our hope we will bar our right to move past the single we have always a general theory of quantum mechanics for which I announced the basic rules Warren basic principles permission operators are observables vectors our state and so forth and so on North Fargo vectors means distinguishable very very general principles and so far we've applied it to exactly 1 simple system the simple as possible system in the world when not finished with it yet but hopefully the nite for next time we will move on to the next simple it's the next next course is a simple 1 in between but gasping for Spain's is next the next course system why do you think that to spins on next next simplest system as opposed to the next the simplest Espoo has 2 states states the beans have tour Wafa growth states to 4 so I have to this day with restates the system with 3 states you could add to the system of only 3 states that's the next course of toast the next nexus of course brought but let's skipped on 1st of all the the concept of concerns are not going to do the mathematical Uncertainty principle married but we should just talk a little bit about were fact that there are observables perils of observables which cannot simultaneously be measured on a basic principle is that if you measure something than by the definition of a good measure you will leave the system with the same value but you measure otherwise you couldn't confirm the next round you could confirm what you have measured source important that you leave the system at least momentarily or instantaneously having the same value that you measure Arthur D. measurable quantities are system are the Eigen values of the observable operators observer operators represent observables and the states in which the observables definite which is not ambiguity about things you get AST you've done a measurement of a certain quantity is eigenvectors on eigenvectors of the observables policy we hand tool observables which could be simultaneously measures if you measure of Yukon measure want and I will leave it in the state without saying value but you could also measured 2 things simultaneously a dozen medical B 1 after the other but about what such a score simultaneously make 2 measurements simultaneously of food different things and you leave both of them In eigenvectors of the quantities measures through distinct quantities that must mean that the states that you leave things in our simultaneous eigenvectors of tool of the distinct things if you could measure things simultaneously it means basically that they have the same eigenvectors or that there exists a complete set of basis of basis in settling the terms somebody pointed out to me that the terms based our revenues using a little bit probably true I use the term basis to mean a complete set Of or the normal actors you to say the role of farright quantum physicist to use it where you are I think in mathematics the notion of a basis does not require the vectors be orthogonal it just requires them to be linearly independent snubbing the artist pointed out to you I will use the term basis to mean a complete set Of far removed normalized vectors arm the eigenvectors broth they're observable can be made to be a complete or from normal best search by silica remake the beginning you have you do have the ambiguity what happens if they are to widen values with exactly the same value then you have an ambiguity is were her choose the eigenvectors but you can choose and perpendicular as you have look distinct operators which can be simultaneously measured and therefore simultaneously left in eigenvectors of those quantities it must mean back there is a base a basis of vectors which a simultaneous eigenvectors both quantities I'm there exists a basis if that's the observable L & & user permission operators Ellen M. means that LOT and to be simultaneously measurable and must mean that complete basis basis of states which are both simultaneously eigenvectors of L M eigenvectors of him that must mean that L on i.e. equals firm let's use the for eigenvalue values hello lets users notation little at Willie Nelson stove war and also by France i.e. that's a condition that are eigenvectors of L with eigenvalue Elsa body end at the same time saying several of states must be eigenvectors of the emir observable but so this must be true for completes states now let's apply bill in in the sequence 1st L & M and then try it the other way and see what we get some rest let's start with L on argue harassed by now let's take end multiplied by and then hit the eigenvectors are well from the 1st relationship this is equal to rule and friends else abiding by July for both for 1st equation but now else body that's just number as a number maker is just rule number value that means can writers Elsa by times are but we now know that in times are a little em what does give this gives also buy and of fans are a proper focus of the surprising on May is a thing which can be measured are sorry and
08:32
then they'll our things which from both be measured simultaneously and if we multiply the 2 of them together as matrices owners operators they become another observable whose eigenvalues adjuster products but if we did exactly in the opposite order Helen and on are we would get exactly the same thing we water users numbers so the order that you write the Bendel's matter Elsa by pendulum and survives a Satanism Sobotka also right and so this is equal to a certain thing saying Nossa Paul seal have tool operators which have exactly the same action on it every member on a basis that means they will have the same action on any superposition of vectors formed out of a superposition of the base of vectors another words they will have the same action on any vector em Times l and l have exactly the same action on any vector and therefore him times is equal to plans and as operators matzo April the following year after that of things are simultaneously measurable the talks observables orbit tool operators representing those observables calm you at all times and in and out of the same form plans and my and which is the comet favorite holiday l can't come today and must be equal to 0 you control for a stronger former of this theorem which is cozy overweight but you have to operate is where are we used to you and you have to operators which Camilo remember if you're of tour operators which commute it means that you confined the complete basis of states it would both of them are simultaneously emerged Vera complete basis states simultaneous eigenvectors Of the 2 operators it goes both ways given that you have 2 things which can simultaneously be measured the follows of a commute 2 things commute it follows a bacon be simultaneously measured by or complete set of common eigenvectors ISO the condition for things to be simultaneously measurable is that they commute was they don't commute the fact we're already has some examples of things don't sit next and instead whitey and Sigma z no pool of them commute No 2 of them commute with each other and therefore they do not have simultaneous Oregon records of surprise is we know that the eigenvectors of Sigma Zia barrel the eigenvectors of Sigma extra left and right and about the same thing the different era while often are not proportional weigela so there are examples of operators which don't exactly where I work out the commutation relations for them they don't infected on you and that's an indication that you can measure them simultaneously that's the idea of uncertainty I give uncertainty is when you have 2 objects to observables each of which you could decide the measure give them and get a mail so you may like that you may not know what the answers gonna be statistical bark you can measure the FT measure them be certain you could repeat the measurement of find consistency each of them but not both of them take and classical physics is no such thing if you can measure a committed beaded you both a and B and you could be certain about both of them if you be certain about either of them quite so that so that's just and will combat tour theme of uncertainty quantified quantified on there maybe states where are although you can be sure you're not circle our but will come back toward combined with more quantitative measure of necessary degree of uncertainty In measuring 2 things simultaneously and that the uncertainty principle but for the moment we have the idea of explanation why are uncertainty uh between observables in the 1st place young why it is each year can't
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5 you measure them and you get both of them you have to leave the system in the state which is a eigenvectors what you 1 of them both of them that's a role new measure something it believes it In a state of definite back whatever happens to be which means an eigenvectors you measure things it you and me and your if you could measure toasting simultaneously it must mean the state that you leave it is an eigenvectors both that's over West comeback Meryl tour the evolution of systems some West Forum so I went back and looked at the classical mechanics are of not now and I realize OK I have something like about 50 pages of quantum notes the equivalent classically was how many pages of classical mechanics of I have about a quarter of pager just statement that states set of brings in 3rd the Greens right on a blackboard points on acts of 50 pages to get where classical mechanics we were after about 30 seconds twostory Mexico we did when we talked about classical mechanics would we talked about a state change with time that the other page the paid arrows gone from 1 from here it's not so bad like the us another 50 pages fortunately fact rural wave of our that took basically I would say something like about 5 pages of something for a fight do more the more so or were beginning a compressor for things were want you got the idea states next Zohrabi 0 OK so let's remind ourselves what we said about the time evolution slowing her equation the Schroedinger equations equations that govern the time evolution of systems we began with the idea Of the conservation of distinctions or concentration of distinguished ability of states the implication was that states which are orthogonal remain off it was easy the drove from Netanyahu you've proved it you should be able progress fresh are there you will operator sure broad rarely Cairns you operator worth operators this if you want to know what state is at a later time from the value at times you're all stands with Prime 0 then you apply certain operations it's more than just an operation it's linear operator Olivia operating you and you get sigh at time t you of course is a function of sheet later or the earlier but you want a projector vector forward to put in different here and that was a basic equations but then we require another ingredient and the other ingredient is orthogonal states say off on a roll soul for example if we have sure firing states let's call them for the moment I J instead of sorry i and j orthogonal that I and J might be members of an awful normal basis with taken and to be members of an awful normal basis and they will allow I too involved for a time your allowed J of all for a torrent the brought the cat RIA becomes you'll see you of T plans or ii Oshawa right after retirement you feel right J. gave also where the permission tragedy of U J and the assumption areas therefore I J were awful normal you won't times I and you start times Jr also wore from normal way to say this is if JRI is equal to or chronic adult and this is also we called Konica Delta another words putting you'll annually dagger sandwiching them between also normal basis he does not seem to be in a product from that you could conclude since this is true affect any members of the basis that you dagger you is
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just a year operate we do this last time I know you're operator the does nothing you dagger times you know another step said White let's consider the evolution from zeros what is the personally short time later we're interest in what's calling you of Epsilon platform a small car and we argued for 1st of all the 0 nothing happened said nothing happens means the you'll be unit operator I've your 0 is just you and operator of mopping as we they say we want what things and that was a kind of continuity and may not of spell that our kind of continuity and that was that you are a small is closed the identity number words are very small prove the state doesn't change much tho that's same as saying that our you is equal to the unit operator plus something small of water Epsilon end I used my freedom of definitions just choosing definitions of writers as minors i Epsilon times age who was no content in their mind Epsilon I can just exorbitant in age I put it there in fact have idea that I really wanted to keep track of what will be interesting constants I would have put each of the past but this is all a matter of conventional whether or not we include major follow whether or not we include high were not
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include maps his stance was simply a definition of how the late we could call this whole thing case Annan said Katie is Eagle by epsilon over a year or more progress Pickaway BH bomb and forget it anyway units units what if and then we said this tried to warn you dagger of epsilon is equal to 1 plus epsilon age bad bad remaining permission conjured really said these tools into Europe parity conditioned by what the refining we found a H minus aged dagger it must be equal to 0 as Rabin's if your feet these 2 interview Terry condition all alright Norris I had ages equal dagger or age is her nation not as big nobles White ebbing those needs at age is an observable something that we can measure and that we should be entrusted him at what is it costly it's a hammered out of course but as I said last time is essential use apart from the fact that plants constant it is a Hamiltonian watch it at the door the classical Hamiltonian we talk about a little bit or come back to it but for the moment it's just called the quantum mechanical Hamiltonian it's a which in some way or another but very clear way generates kind evolution it is a you might think of us With the score 1 we run will be another step OK let's ride the equation sign of Epsilon was pure reviews is equal the size of 0 I Epsilon a sigh of I just want for euros the France pose this tour fans rioted and
23:10
provided by epsilon Trans posing for lefthand side makes his mind this makes an equal sign here and then divide by Epsilon but the obvious thing Auriemma left we get the time derivative of state vector of the system par and the real estate sector of the system's D sigh bar a B C is able minors I H thought each is it's a rule how you will update the the state of a system knowledge and instant the kind the next since the time it's you would make an incremental change In the state of the system proportional why my side by side with total machine for tell you have a update the state of the system sound awfully determine this sounds awfully much like classical determined classical because was also a set of rules for updating R B over the state of from so what until things not because this sick if you know how to change the state from warns that the plant would not paternalistic In this sense that even if you know the state of system you simply don't know the the results of every measure the knowledge of the state of the system in quantum mechanics is not equivalent to a knowledge of how come on every experiment what is it tell you the state of the system it tells you probabilities so this is basically a rule for updating probabilities it's not a rule thought updating knowledge complete knowledge of the state of the art of set of experiments that you could do but this summer sensor the state vector revolves I don't want a quality determine strictly because determine the Stickley year carries baggage with revolves known and that way novices troll but only if you don't interfere with system in classical mechanics you could interfere with the system or arbitrarily gently and not that caused year course and changes in the way it's evolving and quantum mechanics when you make a measurement you necessarily disturbances so this to the evolution of the system of an undersea of an undisturbed system was not the process of being measured would this be it so I could sort it is not I what a special case of Fiat New right right if you look at that price and the special case they system you measure it it doesn't start half right so that Tibet is an exception of course but generally speaking the generic situations here's for new measures something you disturb something else about the source cut this equation is destroying our equations done sometimes called the generalized shoring requires the actual Schroeder equation ensuring that figure cars just emotional particle but has exactly was formed this 1st became known as the generalized frontier required people's dropped the generalized we now think of this as as the Schroeder a form of the Schroedinger equation this is an equation which tells us how States change with time it can be applied to the simplest system already singles bill they can be applied for the 2nd simplest and the 3rd simple system and the system of arbitrary complexity is the former House that's so that's where we're more or less warned last time shorter erased everything but the Shahriar equation actually this is called the kind deep pain and Schroeder equation finding and ensuring a equations is equation for how things change with time was also another equation which is called the kind independent Schroeder equation H is observable it corresponds Hamiltonian but Hamiltonian is nothing but energy released in classical mechanics so 1 might expect that a is the operator representing observable energy of a suspected rocket now where make a measurement of the energy of some
28:23
subsystem how we do it that's not worry about that was 1 way we can do it we use equals MC Square and then we weigh the system carefully really really carefully and In that way determinants mass and its energy that's not a very efficient way to to do it because the change of the energy of a system when you heated up for something so negligible there are about us 1 way many ways to measure energy bought possible outcomes of the measurement of the energy of a system that of course depends on the system that's not something you could ask general but Our depending on the system for example a few measure energies governor Adam you get 1 of a innumerable family of possible love answers namely the energy levels of the Adam if you measure the energies of a harmonic oscillator again you get the numerable collection of possible answers if you measure the energy of a particle moving in a closed orbit are then are in fact you get a continuous possible family of possible answers so the question of what the energies are dependent on the system and depends on the choices quoted our choice systems have Hamiltonian is not up to to you let the dictate to a system What's Hamiltonian those bomb may be up to you to try to discover what its Hamiltonian by works user terminology you get to choose a Hamiltonian meaning to say you get the study different possibilities and depending on the choice of operator with you put he to call a student at different energy levels I did describe how start honey of calculated Herman with a possible outcomes of the energy measurement aren't we use statement that values at the energy can take on RBI Eigen values of the Hamiltonian like any other observable so you right on other equations H on let's call it be right now I l represents argues state of these energy it doesn't represent any old basis it represents in particular Beeler that may do we should indicate that by writing in here energy we are quite possible energy level H Lisa it is equal Eigen value east of why finds the eigenvectors associated with that energy notation are putting Esau body inside the said that vector just means nothing more misses the kept vector which happens to be the eigenvectors each with I get value Lisa body and how many such he supplies or their wealth there's a complete basis of whatever the originality of the system is where the amount of space Estates's that many eigenvectors of energy and your job as a physicist is to try fight store that's the kind independent Charita equation don't work Mexican we did get Law review Mexico did was talk about evolution crime evolution of expectations about average values our complained bitterly about terminology expectation values right but many many years ago I gave up on this and source become expectation value should be average but talk am OK with expectations are your use expert but really should be average per day on and he is always found well Ali said is the thing changes with time is the state South we're interested in calculating society and the expectation value of this remind you the expectation value you get by sandwiching the observable sandwiched between the brought version of the state vector and the cadavers we progress we prove that this is the same as the the average statistical average if over back over now this is the average value of L Lee observable we can ask how it changes with time the change in part because cited changes were
33:35
part of here we Herzog changed time restore factors of year so what we go to differentiate this was respect tying it be terms 1 coming from differentiating this 1 coming from differentiating that if you remember that when you flip from brought cat you have the complex conjugate and complex conjugate changes assigned I you will come to the conclusion that this it is equal to ruin was sign wrong we try to get the sunrise of miners are car use L H minus H. L the average value of Back side so I would work that out we used to l on sigh you a chance I assist peace over here we use that L climax of a laugh gives you return with a checking the left with a minus sign OK so his always find where we find that the time derivative of the average of all shares rise persuade Debye already average of El argues that notation for the average of as a function of tar is equal come minus further use the average of the commutator carved fellow with a Hamiltonian if you know what Hamiltonian and is as an operator or a matrix and you know what failures as an operator or matrix Ukraine computer we expect the commutator when you've computed the comet tail you cool computer average value we just use is wrong the which roll called a sandwich and it tells you the left and the right hand side shows you what the left hand side tower changes were hard to find their auditors are hiding their why did the 1st Commerce August 1st time I saw this awaited a white America prime derivative of a real thing can be imaginary whether find our that calmly a real things are always imaginary of imaginative will go away really some examples and you'll see this is missus Lafarge proved although it by example sometimes the Ritner shorthand shorthand as just DLB DTD 0 from either side of the track their chief but at least at this stage of our knowledge what this means is that should always be sandwich and equation for the average values of things where we are or called a Heisenberg equations of motion a system that brings us to the study of commentators study of commentators through letters together will cunctator some wolves associated with the rows of very very reminiscent of the rules for passed on racket go back and read about if if or song rackets all right now have a Royals are general roles algebraic rules of fossil rackets commentators and I'll just check they really true for both of them were rules on records we do ask quarter or song rackets are written this way carry a beat but not right out how you express them in terms of the P of classical mechanics but it was a few simple rules associated with their 1st of all I was familiarity rules are better fuel over a few dead tool operators sorry to 2 things and then take a pass on racket with a 3rd and and so forth this is certainly true of commutator but those a couple nontrivial rules well 1st all racket is ah which means that when you interchange the 2 entries changes sigh minus be paid 1 property this is certainly shared by the commentator during the change H. U.S. details listed firms a new jails a sign of a comic starts drove a car repair or salt if a and B are now quantum variables missiles also recalled for miners beat and they really only other 1 important relationship that's a proper role the product of AT and be more Assam where see the operational costs only if you remember the past on racket was a combination of derivatives derivatives were various things ought to go in you lose that definition of course bracket rules for derivatives of prop up what you find out this is equal to or from right side are equal to 80 pounds of strong bracket of B with C plus plus bracket of B with times sorry for record of 80 with city I be now order that you'll write their AT and porcelain bracket that's important these numbers and is just ordinary functions on numbers and as an hour which order you multiply half by wrote them in order which I you know the commutator will also satisfy operators the order does matter specific water namely equip a left over he would be on the ride over here then maybe a subtle surprising a appears on the left in this uh factory and be appears on the right so I kept urine OK let's see if this is true for commutator was always say it would say whatsoever about a punch connotative BC sorry for back it was sorry Crowley Acadia Ave ADB worth seeing equal probe will grow question mark aeronautical yet Pfizer commentator B with say class a with city it was city funds be let's check prefectural is a very easy to do what we are doing the river so if door ride out the meaning of this comment there baby was City is the same a B and C minus see plans a B and
41:50
wine 1st to put a B on left arm see really puts CEO OF be on the right and to take a a different definition of the comic What's your get right hand side on the right hand side we get a color these city show but sister from here we get class ATC miners CASE times these operators miscalculation their operators the order matters OK let's see what we have we have a ABC Here is ABC that's good we have our services this is our friend we have my ears see a B as this 1 over here CAB across friends our enemies are a CBE but their car both at the same time the same way a B C not they me with a bad 1 but here we have also a C B 1 or assigned form of across ACB minus a CBE affairs be commutator satisfies the same algebraic ruled as a plus right this is an accident no of course is not an accident but will get through I will get a deeper reason for eventually which comes 1st them with which is more basic certainly the commentators more basic quantum mechanics of more basic and classical mechanics classical mechanics is an approximation requires so it stands to reason there ought to be a classical approximation of emotions of of commutator litters mixer that leads us speculate if white but a halt take it is now speculation or procure something that will discover all approve later and we go from Kerik's classical mechanics across all roster With the across all bracket becomes I think Martinez all right Travis a quality doesn't quite make said this is something classic or write him sigh may wish should put arrows data would be worth it last that Bush it seems to be what it is right that White Rock or prints are a B C a B C C a B C ADB with a minus sign and then we have a couple of extra turns terms that might PARIS ACB plus a PCB they yup throughout the goods will be young now nothing of what I've done tho so far tells you that this should be identified exactly what it might be identified with this times of numerical multiple reports on racket might be 7 times a country commutator a quarter of the country and they all still Mike Saenz de facto ruler by Mitchell difference between these of false alarm are derivatives AT with respect our X rivers and be with respect the key may have Hewlett's which are different than just a beat the difference of units is actually stopped up plants can't constant if we work with plants constantly equal 1 import song brackets adjust just miners are 8 times coming was bought for dimensional reasons as each part of it but try our way with pH bomb left him aside certainly classical Our we think of commutator very small but we think of them as essentially 0 being lines BAT so in the classical limit the commutator is something which is 0 has happened well the bar he age very very small units which each bars were very small which just ordinary units units meters 2nd kilograms it follows an incredibly small numbers and the miners were 30 forget off a very small number show 0 in the classical approximation commutator zones negligible negligible small what I want keeping track of real quantum a character the commutator is nonzero sewer that's not start rats means emotional coming here is the a nice property now is that this relationship here it is simply the quantum analog of an equation that we started last quarter a D L it is equal to the past on bracket of L with from where chimeric truce quite well with others that is the essential reason at this stage for identifying each with a Hamiltonian classical mechanics were packaged In this fishy called on song formulation fishy porcelain formulation and capture fixed organizers Everything relations in bars across on rackets in particular of says that feed Elka be anything failure was not the angular momentum it's anything PLO by IDT is equal to a posh on bracket of old age and quantum mechanics In the sense of average value we found that deal by DT is mine the size the commutator so let's see essence of the connection between the Hamiltonian of classical car mechanics of glasses require mechanics yacht did act From Wash Velez any the any of them I use balefully operator but of course it should be mission when operators are my views each but then I would've gotten court and a firmer each oasis Somerset and so was our values they'll for a generic Baja observable so assumed that l is her mission originally IUI use it for linear but now I'm using your community leader and her mission L. could be a which L could be a ditch the so I would study the equation of motion of how each changes with time or better yet however had an average of each changes with time with those here we are D. H B H by D. Vt as you go from minus or ii times a commutator of each with a of the coming federally anything with 0 so insolently as they passed all right if anything with
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itself because our from rackets or on change signed when you changed objects that means the porcelain bracket of a thing with its offer 0 so from classical poor interview we could have derived the conservation of energy from the principle of the car and the root of the energy is the past on racket of energy itself here we find Kwan mechanics Avery simple things system or time derivative I'll be again the Hamiltonian is plus Albright is coming of a Hamiltonian itself and therefore 0 energy is conserved corner carousel at least B average of the energy concern mounted that it is a very definite sense in which the energy is exactly conserved With Jaafar to that they he you commentary were what To what it the 1st young absolutely absolutely whenever we build a system out of 2 systems they in general observables for 1 of the systems and the observable for the other systems commute with each other that I you could measure the position of me and son Javier simultaneously because we're too because propositions commute either tournaments are awash in there I think it's a question that is a question your asking not not bright young boy had cognac I knew was that's what was go later Brown get it on the question of composing systems taking systems and making bigger systems however composing that would bring us issue of entanglement I have a feeling we probably won't quite get there but maybe would lose potluck the next thing we wanna this solved Ingres equation I don't mean 1 the Eigen argues in the eigenvectors that we solved by standard methods of if we Have we happen to to know what the Hamiltonian is particularly we happen to know in some matrix form them we just calculating the eigenvectors Amargon values of some matrix of supposedly done we have the eigenvectors we have the Eigen values but now find the general solution of from Ingres equation moorage how things evolve with price who want more what will but no I realize this abstract rector of the only system that we have to apply sort through the world's poorest was again and I am very much so at this stage you probably saying to yourself whether we going to be thinking about real they objects the systems we will we will borrow meters but had I think having more of this apparatus abstract form new additional corner mechanics will make it very easy and quick to start applying it a lot of things we were here a while our case autonomy and right now armed that depends on what we do with has been in free space far from anything else there and its energy your Tamil basically 0 insolently any luck constant for Hamiltonian doesn't do anything and the reason is because of constant commutes with everything so apart from a possible a additive constant energy which is now around anything else just as a Hamiltonian 0 and that means that doesn't change with time if you put it a magnetic field happens to be real spin electron you put in the magnetic field magnetic field induces a Hamiltonian which I hope will get tonight and it causes a sprinter things most of the time dependence and then what the Hamiltonian would be would have to do with which direction the magnetic field was Hamiltonian commute change them you could change Hamiltonian by changing the direction of a magnetic field Nauru will calm where I hope we get to this example for minor birth let's just 1st right down the general solution of referred questions to don't act we start by saying whether it be a basis of eigenvectors I guess we call it was the same as Esau if I don't want to write a piece of wire I would just right eye eigenvectors in means the eigenvectors with energy he surmised we can take any state I worries about Our basis of states we can take any state let's call it a butcher Siletz you record be a courtside revered forward side sigh but now analysts think about sigh at times 0 well now mostly with Prime arbitrary is somewhat of of the Eigen states so the Eigen values of some sort of coefficients alphasub I I use the term now from the past and the note the coefficients of states of bases vectors a state
56:45
whatever could be written as a superposition of eigenvectors of the energy since the eigenvectors Our religion operator or a complete based a society let's say a prior people 0 now is the time not a party houses state change
57:05
with time what's changing here well the state fire just a fixed set of states is they Athens which changed the coefficients here which change with time I took coefficients he expansion of arbitrary state typically kind D pending coefficient that's what gives rise to a time timedependent in expectation values and other things now let's write the Schroedinger equation we have quite so I've got sorry have so I got which we can now arrived at 6 I've got CibaGeigy of aside decide that's equal From here some on ah be alphasub buyout tea with respect the tee times I differentiate the state all we do is differentiate individual coefficients assuring yours equation tells us that it will minus 4 different are you always smile OK jp Jerry OK set that equaled the Martinez columns each act enormous OH could be brought he a summation here and this could just be written as additional are recall coefficient numbers are offered off star James J. Harford Javed T H R J was a judge From this is 0 but could now Moon Rueda is now a
59:15
load of that ii in the use of IOC actors got Top said of her public origins 8 on I there was energy money I eigenvectors OK solve this just becomes east of J. here we have our lefthand side a call storm of basis vectors with coefficients Alfred J. sorry where are you make a mistake in argued the shutter Mister Ron Baron Pierre was wrong the America yeah this is some RJD Debye Didier about for J. D. how and then them up From lost that but on check that brawl this is correct Gupta here this is correct to here a pay for now write us go back he now let's calculate the hour I was calculated time derivative of various where we don't we put in my ears I kind of H on J here is where I used H R J is equal to 0 he J & J OK I think therefore I had a moment ago anyway anyways exactly what I had my side of my side I this thing in this thing on the same so this arm must be be exactly the same as if you have to look at some of bases vectors the coefficients must be the same and you have to set embraces vector sorry sorry fear of tool linear superposition of the same 7 bases vectors and their equal to each other than the individual coefficients have be equal to each other story follows that the alpha by did as you work on my I try and see how far J. Brady J offered J all are done is yet they offer DT equal to the coefficient of this equation and that's all outshot it tells you how the office change with time for each J. this is a differential equation just think of this as just a differential equation for half of what the solution of the differential equations we've done this equation over and over again the time derivative of something is proportional to the same thing bring itself exponential social right so Resource this equation it How often J. equals offer J. at prices 0 times minors a T if we differentiate Alfred J. with respect to it will bring down this fact miners by subjected found the same thing so it we can now write a general solution of assuring requires we solved it for each individual 0 eigenvectors This is the way individual individual coefficients half a change with time but now we can go back go back careerism or quite equation but the state vector is a sum over the eigenvectors and West and what we now know about our from analysis off our offer and here we can write the general state vector at time t if you not always have time 0 but suppose we know it at times 0 at times all it was just for Jr turns euro plans J. all I happened each how often J. picks up a kind dependents and time dependence there's just governed by the energy of the state madam all yes I was quick Syria press do defined a general solution is you take this state you break it up in so eigenvectors of a Hamiltonian and then whatever was a Times 0 the only thing that happens the cities of each entry year change with car with each of the miners on 8 times the energy of the state where were age Bob was formulated by number of other I going ahead on the maybe energy or the car and was our it goes into a printer nominated here this is a connection between energy and oscillations I think this is a oscillations of the wave function oscillations a real things about oscillations of the arms of a Accor efficiency of the state while erratic let's 1 more thing let's use our general solution of the Schroedinger equation a calculator How the average of L changes with time we had equation for it but now we actually know how site changes with time and this is just a little bit of a gym not gymnastics just a little bit of our fooling around with equations some of you may recognize the equation were about to get what recognize or not it's just that explaining how the symbols I used your broker let's calculated give is that the state changes like 0 West ask how the expectation value on or arbitrary arms oral changes where you have them nor how the average of the expectation value of changes Vela some operator and I'll do what operator of course you need to know operator order tour to say how it changes of course you do but still we can now we can go away is let's plug in this formula society somewhat J officer some J. we from minors R. J. T France J now we have dealt with over here relative same thing that has brought Rector over here but why have this will we do abroad withstood the brother brought back there sigh of His summation we better use a different summation variable K E and then we will have star of K 0 from brought 2 cats you have to complex conjugate will also have to change this come up plots I he times the Bravo the case I'm Iife change summation index years of commuters confuse it with a 1 have
1:08:28
complex conjugate complex conjugated here and slipped from caps the Brough's Hasakah put this over here a summation or Katie Katie fourstar for starch I 0 is is a prime 0 times each other plus ii EKT sorted Belsano of double some or Katie and JET that's not terribly illuminating but let's just the just after a poor far worse worker our North Sea in a product of K with Jerry we see here in the although not so it's part of care with Jerry or we see here is L sandwich between J and K L sandwiched between J we operated acts on state J and the new projector carries let's give them a much call L K J that's exactly the kind of matrix elements we computed when we talk about the matrix representation of observables E In a product on Alpha sorry are filled fog J. with Vasily called LKJ just the matrix representation of the observer and Emily Harris How also star K 0 0 far J of 0 or so it will talk about moment that we have a phase E R he sub Katie minus ease of J. the he summed of James K. now the main reason I did this was just illustrate the machinery I did not do this because this is the formula of us of any importance right now what some of you may nevertheless represent recognized that it is Heisenberg matrix Our formulation of quantum mechanics Heisenberg formulation of the way average change with time if you go back a Heisenberg 1st paper on quantum mechanics he gets formula basically Disney gets formula every observable has phone frequencies associated aware that dollars weird it was common for people to understand that Our observable Mike b an oscillating observable might be something arousal rates and would have a frequency associated with a Heisenberg said there are still frequencies associated with every matrix our from observable this came from not as it came out hasn't birds head but this is the way Dirac explained what this is a way Dirac understood what Heisenberg was saying through his machinery 1st just the pieces the pieces this represents the observable where observe wears matrix representation these things have to do with the initial state the stated plants equal 0 and these things here I have to do with the way things change with time everything changes with time with 2 separate truck frequencies and then use somewhat altogether and that show you Carol L. changes how the average of El change with so putting together a bunch of machinery W. burning questions Yap while a a up people the authors affects the opposite of the currently equal 0 period yacht yet you know I I think your back here young yes that's correct magnitudes of the office effects as the relative phases between them talk all the time evolution has to do with these relative phases of a partial is white always that actually doesn't matter this just means the holder of prospective partners usually written as a partial derivative but just means it could be written as a focal derivatives saying things I only depends on which his body in OK backtracked we will eventually come the think of sigh as a function of time and position and that's why the best way for the moment it doesn't matter whether we use posh over a pot over it just means a change in science small time Arthur and here of matter were now that has the edge he said the war is a sale I think that's a well it is a rhetorical fire call Bob the law the fact I think it's much more than that on the evidence of much more than there was also an empirical fact insecurity Grover Pirkle fat people try to change course this seems bit hard to change Kwan mechanics as it is the change arithmetic for logic Braun it is not fooling around with try to change the rules you very quickly find something really bad happens probability is not conserved arm probability is not not positive terrible novel localities happen we you could communicate the infinite distances are so Kwan characters is not a thing which is easily changed in fact there was no known changes the commune maybe I are took cost there is 1 on changed you could just replaced by classical mechanics that's a consistent theory the reason to consistent theories because of good limit of quantum mechanics but was always the seems it's an elite the form will bear my friend David Finckel Stein who was very wise is wise man and a great scientist wants explained to me that there are 2 kinds of revolutions in physics he called 1 plastic flow 0 and the other a brittle cracking what he meant by that he considered for example general relativity could be a plastic floor you could change a little bit you could perform a little bit Cuban change Einstein's equations by Yarborough changing variables to its it's not something that you will you cannot change at all change in little ways in the form of special relativity is very hard to change wants to change special relativity are at the a very brittle turned down the whole structure of it goes down the drain but you probably but nevertheless you could change a little bit it's harder the change in general also Torbert quantum mechanics is absolutely brittle and by that he meant not that the theory is in danger of being broken by anything but that if you try change you will get broken there are so contrary to their marked a change and I think the evidence from rappers just all of years or which people try to change yacht police say they go real right is rooted in the area the city we show the right yes yes we derive from unit we derive from U.S. charity unit Harry Is this condition that you dagger you cyclical 1 all
1:17:39
right or unitary from that we derive the Hamiltonian from mission from that we can define we can derive Eigen values a real . but now alive of what facts age or you're lot these alive for a missile with a laugh not well OK it's this solving as an eigenvalue eigenvectors both eigenvectors with I mean value he survived so you and your labeling the eigenvectors by their Eigen value here's a sister died in value but and this is a case where repeated indices do not get some Velva is a case of be non summation convention let's say 3rd applies some of this apparatus for the problem of single spin spin in a magnetic field Elian calculated in particular Holly average of the components of the spins change only know there's depending on your memory for classical physics you may start before in some of familiar marked remind who would hope Hope what we want full who Oh boot there's your life no no it's not now robbers now have but that wasn't there bad assume macaque nite OK we have a will spend less the spindle magnetic field the report the magnetic field along let's take the zaxis force be simple mistake caxis spin on Berkeley magnetic field along Rosiak exorcist as In particular the spin of a charged particle is a little electromagnet charged particle spins around and of spins around their recurrence of floor and it creates a little magnet put a magnificent magnetic field and you get an energy and basically the energy is proportional to to a missile alignment of the magnetic field would be spent with the axis of the spin and the simple formula are is but energy is proportional to product of the spin with a magnetic field that showed stock where start with energy which is proportional to magnetic field and the magnetic field along with actors so what is it about product of the spins with a magnetic field that's going to be proportional Paluzzi component of spend about product spin with the magnetic field will be the component of the all the euro along the axis of a magnetic field that's a guest trained as a guest which we could then study and then ask if a experiments on spin everywhere OK so we have a spin we are magnetic field along the axis of course be busy family have a spin and spin as components Sigma exit wanted to see people ask me incidentally wiII in the book that I'm writing about our mechanics lot more figures there are lots of figures in the classical mechanics brook no reason this is a simple real reason you can't draw quantum mechanical systems quantum mechanical systems over very nature not things that you could drop to try withdrawal them you will always be trying to draw things which cannot simultaneously make sense in particular RightWriter draw other components of the spin on trying to draw components of the spin which don't Camille with each other end that doesn't make sense to draw them nevertheless their narratives magnetic field the spin and were just going right now take this as a guest but the postulate the Hamiltonian is just going to be proportional component of spin along the zaxis find something proportional to the magnetic field are not going to right out of it contains electric charge a contains a magnetic moment contains a magnetic field but let's just called Lee coefficient here let's call it they got over that's a definition of Omega if you like whatever the right combination of magnetic fields charge magnetic moment particular charge on magnetic moment magnetic field will want them up and quotes all sorry coral may go over tool and the reason for the Omega will become clearer shortly after the reason for the tool both the army and the 2 will be completed shortly OK now I wanna Capulin while I calculate 1 I calculate the kind Riverdale on averages Ave Oh look components of the spin nor have we v component the Hamiltonian but I want to find out how all the components in Texas in Hawaii and Cigna how they vary with time so in particular I want to calculate the time derivative boss time derivative of Sigma X and more answered Missouri was the rule on afraid rules been erased but I think you probably remember it and by definition now this means the average average of stigma acts and not going to bother writing of the Brock Kent angular brackets we now talking about B averages of Mexican wines to and the kind derivatives of the average's plots of remember this is able to minus are these kinds of commutator of sigma X with the Hamiltonian right so the Hamiltonian simazine we yet another factor of all may go over 2 5 yacht that's correct likewise here With more white certain Missouri fans Omega overtones likewise here now where they said Missouri but no 1 of these is very easy which is easy 1 the bottom 1 everything commutes itself right every thing committed offer services 0 was the Gaza tells us that the the component of the spin along the magnetic field the average of it does not change with time that also tells us what clad if we would do in classical physics which were not if we're doing classical physics a with tell angle of that a year with magnetic field stays the same but when I would just take this forward says it says the Zeke component doesn't change with time the average for good measure that's the 1st thing we discover but the goal Beyond that we have to know what the commutation relations are Of the signatures so let's work them out we have a rough time let's work out the commutation relations of the stigma is all Perez around and see what we get the poor have who no war who who a
1:26:53
start with sigma x Cigna z but instead on whether they would be opposite way when calculator string with zany next as a saying think except for a minus sign new interchange 2 of them sought to remember on calculating movies X will have to remember changes so I'm OK so was this this is the product of everybody remember Watson was 8 years as a matrix Wallumbilla now matrices straight away calculate remember that movie and my ex on matrices orally Sara matrix representation and we just have to multiply the makers was act that was 1 0 I 1 starter warned greatest more Rosser next 0 1 1 0 as 1st the commutator there was time certain X and then the other terms just put them in the opposite order 0 1 1 0 times warned 0 0 minus 1 no With respect to the Taconic headers from Rio on matrices as where calms down to we can't do it without the matrices just by applying the operators of the vectors collectors often bound and we know what we know enough to actually do that without actually using matrices for the matrices are very convenient tool for manipulating operators operator product is saying there was a major part of stock reliable job but so this 1 here the upper left corner is the product of this role with this column onetime 0 0 1 0 the upper right corner is a product of this role With this column 1 times 1 0 0 1 the lower left corner is a product of this role with this column minus 1 event last warned this role this column this Raul 0 times 1 member of started now 0 airstrike from still 0 OK now let's do the other 1 over here minus 0 0 times warned us for just 4 0 this foreign firms as 1 might 1 this 1 just 1 1 0 are I think our Mr. signs from wars truck truck that Yeh murmur fright right Mrs. minors OK so that's Reagan they just gave the same thing bought a minus sign start this miners this gave 2 terms which head so the comedy data it is just equal to the commutator itself has just equal 0 1 you twice minus 1 0 now recognize this thing over here if I put I know it here I only hear you might recognize it as being related to signal wife me right now serve more single y a busy quarter minus party party zeros why have sensify multiply signified by Odile might try minor size as 1 and cars are I was minus 1 so what this is it this is just twice by a said y RIA more attending the closing up on itself commentators of Cygnus was us and that's good because it's going to allow me to write equations for the Pine derivatives of signals which it's discovered he said was again selects votes you what we have with comedy testing Missouri was a X is equal to twice I Hawaii and now all right down the rest of them incidentally as I said If you want to get the opposite order you just change assigned these X was more wiII is equal to 2 are a signal Z N C X Y or y x y z z was acquitted Tewari Soon next the truck these over 3 connotations these it is line of anything they run up for song rackets of regular memory from course all my efforts of angular momentum work els L X equals a Y L Y X L XOY User Group G and L Y musical elects so passed from a factor of 2 wires and there is an ornery in the connection between our brackets the and commentators essentially a partial factor to which we can absorb Sicily angular momentum about brackets that doesn't portend close there from meeting of angular momentum are but don't want from moment this these are the commutation relations of the components of the spin and they just get back from components of the spinning in with this funny ordering GXY X Y Z Y zee X just from new Sequoyah great OK so let's go over a year now this now becomes minus I may go over to you know what's commutator of Cigna X with Missouri it's Martinez To Tewari more why so we get our rather like this 2 party Soon more white the chose can't that was the reason I put 2 in the nominating here in the 1st place was canceled so let's cancel the we have I like IRA which is my sworn in so what we get is minus Omega stigma why that I went away the I went away with I was worried about it about a half an hour ago that all my goodness time derivatives of real things really get imaginary but they didn't they didn't because the commentators themselves imaginary coefficients and that general as general OK so where where are we Harris stigma X . 2 years ago from might also Major lorry more wine . 2 here we have really relationship was more worried and it involves twice my acts the correct relationship before work of our forces corporate plus New plus or make a stigma OK so so busy 0 0 but what's happening to Exs signal white Stewart sampling the same access to worried we recognize this is aquariums these are the accord germs profane moving in circular motion right what's happening is simple me X in the sense of expectation about expectation values or real numbers and guard the sense of expectation values the
1:36:05
spin vector is procession around zaxis processing around the zaxis stigma X not as minus make signal YCY got from X misses the same kind of formula that we discovered in classical mechanics we talk about a rotor remember the rotor also put rotor and a magnetic field the angular momentum executed exactly the same and it's also the gyroscope procession basically gyroscopic procession spin quantum mechanical spin in the center of expectation values it doesn't make any sense to say that the quantum mechanical operators themselves our process that doesn't mean it acquire chemical operators just these matrices when but because the state vectors change with time the expectation values the average values of the spin execute motion which is basically the same as a classical rotor processing in a magnetic field and sorrow we have now solved the problem of a spin and magnetic field we solve the hardest possible problems associated with singles are there is no Hamiltonian which is more complicated than this sales more general you will cost you could put this paper but the magnetic field the erection either as a lawyer yes to the problems off you could but they have the spins a long summer of peculiar axis what you would find FTU work golfing Arizona we're just process around the peculiar axis of the magnetic field and rotated so reprimand medic fuels pointing in different directions it is a set a yes yes they really aren't very many operators OK so that's how it is a general you Corpus Act itself was not part of the world the most general permission operators in tow dimension which is a linear combination of EU owner operator cigna X Y and most general mission operated a linear combination of Connie unit operator indoor Hamiltonian that's like constant for the energy you're operators commute with everything 0 putting EU operators is not interesting owes nothing serious next possibility is a general combination with me a combination in the mark of a coefficient stigma X another coefficient signal while coefficient Missouri that just correspond the putting the magnetic field its some nerves an angle it's just I'll be simple case where we only took 1 of these it is a magnetic field along zaxis we took a general linear combination but say In the sub x x and y wine Boston subsidiary said Wednesday he this would just correspond to the component of the spins along with in access and you would do exactly the same calculation except you might work instead in the basis of eigenvectors of Sigma about and sorrow really is the most general Hamiltonian the Duke America to correct and that's just because a twodimensional space is so simple that just on very many independent operators are we did is nothing else How about payment of places further set right FOR gaffe but remember for every direction that you could think of is a state for which back component of the spinners definite but not secondary without after a further with that play littered very much a role here we just follow the rules and her there are 2 things going on here this is a good place to work focus on those 2 things there 40 vectors sectors in real space over the components of averages of stigma and there's the tool dimensional space of quite different things or they are related now 8 if this is a had about what you have that problem from Meg is basically the product of a magnetic field firms the magnetic moment of moved in but Libya tour after remember knowing love gather the residual got with tool is a product of a magnetic moment of the era with magnetic field so they creating semis it was the mayor had pulled back now what level of last year you're right right of how it is there's a couple of 2 or 3 it's not right for a couple not forget the procession couple spent alumina feel there will be full energy states for example if a magnetic field is along the zaxis then the energy of the upstate and downstate will be different no before amount Omega Opel have added your McGoldrick tool Darrell Hair managing minus and there will be an energy difference of Omega 1 of them has higher energy Warren has Lauren G. what happens to a real spin if you wait warned last yeah fight you put a magnetic field the answer is eventually the higher energy state will Miller photon and more will be a freak Wallaby the energy of a photon Omega just a difference between the Aboriginal energy and the process of admitting the photon will flesh so these things a Raul certainly we connected by but we solve a real genuine quandary tax problem now the what he called what will they do not feel if you want to think of the Magnum magnet as part of a quantum mechanical system whenever you when only of facing a problem a real quantum which is beaten in interaction with a heavy system with a system like a heavy magnetic some other kind of detected an almost classical source from our you don't bother trying to think of the heavy system quantum mechanical you just say it have some our best to be justified this is not obvious to justify our white don't we have to think of a magnet as part of a quantum mechanical system that's another way of asking the same question and the answer is that so big and so heavy that it doesn't fluctuate much by event in eventually will want to address that problem for the moment we just think of magnet as part of a heavy apparatus which is a fixed Baran nothing about electron is go cause that the magnetic field to change Mr. stock warehouse we get to think of it the terminus equate Wendell eager to think about things Stickley we get the think about things turn mystically when all heavy and so solve many degrees of freedom that we calculated the core mechanics of that we find the fluctuations are really small but that has not been fixed established yet so far we've divided the world into a quantum and the apparatus heavy apparatuses which we don't have to deal with corn chemically that of course has to be a temporary thing real apparatuses are made of the same thing as assistant to study source on par with the compact what was the equal the justification the saying the apparatus was allowed to be considered just classically you just point that some
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direction and that's the way it is being dealt with corner category will have to come back and that left there my usual our a statement at this point is not my head of how to do that asked about so the buyback yes yes yes proverb world War want good yes they also is not unpredictable manner yet it yes yes who will occasionally but that will build corner McCarrick hazard O'Connor character but has civility use of statistics and probability why probability work we make a basic postulates and that's that you do an experiment over enough time the average value that you get in the laboratory will agree to within small precision with the mathematical averages that you calculate the probability distribution I am not going to try to justify that the reason I don't know why more why does probability work White Nevada you flip recording a thousand times come up heads a 500 times the whole no such predictions Avatar Mike among 500 times heads but was no prediction that worker up 500 times also minus square root thousands of minus 30 of margin of error every time they also has no such predictions what can you say you could say it will probably come up from the mother of 1 of from cycle about regular million times would certainly come up 500 thousand plus a the square root of event not Mama Hunley times you do it there is no hard predictions From probability that particular thing will happen wiser work is is a big her where dots war they not that the only 1 who puzzled by their serving a strike joke purposes probably the reason the Bering Stein my quantum mechanics but it is better off the Reagan Khaled site why not that bad war left us Ehrlich 3rd but same calculations by instead of putting here growing yours Segway about entered but send back and is matrix not in that we've been using up from now with the stigma makers is over the user wrote that will be the matrix are Hey and as unit vector and Serbs 18 miners sense of being an ex Miners in white and an X plus ii in white it's just taking the components of and multiplying them with corresponding components stigma and then you could follow this is a this is an hour from a and they just the Hamiltonian recalculate what happens to the different components as well the component of spin all an axis do that we would have to calculate Connie affairs of the various components of the spending was said Nevada and Nevada what about the component along the annexes what happens to it commentators or without and commute so we will find out that a component of the spin on Wall angry and access would be the thing that of change about the components as big perpendicular about acts we work that out and we would discover that the spring processes about the an axis that's a good exercise to try to do so the classes which now up the new chip and you want the same again asked what I about that is why I am water said I not make this is the number it houses dot a what but Oh by Orient pays off some minus 1 William Barak said well not not only doesn't have the door the erection Omega just the magnitude of the magnetic field and the tour of the magnetic field finds the magnitude of the magnetic moment don't bother with just numbers the number of back product is between signal him With maybe it is just a spectator it's as the number of a number all you can do like you the is all they get quota would begin the few want it is not what I thought all but depends on errant Britain's bath confuse them afterward he says less than here the magnetic field wasn't as indirect reviews where I have only his sends was a yes soon Missouri where you the special let us fit for may get back and means by Price that both is that where are you but there's a lot about dash of he got back her whereabouts or was being which whose last book a but they were bad and stands for a stigma X an ex post signal wine and wine Missouri and exactly as Jeff signal was a classical back but of course it's not just a collection of 3 makers use so that fans from matrix stands a single make bricks which happens to be the sum of 3 particular matrices that's the definition of certain Nevada you these other components of and and they determine the the direction of the magnetic field the multiply back which is just a number of just a number of nothing nothing more complicated the number and Watson without any if the magnetic field is all along the zaxis than the only component of in which is nonzero would be and subsidy prevent if the magnetic field was along the zaxis then we would say only and some Zee if you like with your wagonwright found the relationship between man and the magnetic field the relationship between then and the magnetic field that the magnetic field vector is equal to the magnitude of the magnetic field crimes unit vector Her magnetic field as a vector is the the
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magnitude of the magnetic field climbs the rectory and stance from magnetic field of pointing z an axis with magnitude B I want to ignore that then we could basically identified the the magnetic field with an hour apart from a numerical Kohl's Oracle coefficient was buried in all major want to know that we right says the spin Nevada and the B so I'm not sure what the questions are some something which is a barber and I'm not sure what I always say that it depends on the magnitude of the magnetic field nothing more than a little thought from the frequency of a photon now frequency of the photon is although it is just a difference of the energy of the opera energy level and law energy level and always just Omega a dozen repair murder cool right if you take said that in the U.S. Court Eigen values are the Eigen values don't depend on him the cost of minor sworn in In a Barak Chem no this is the year of the Eigen values what's right outside and by his an exercise for you it should be done send Nevada and is in the sub Z kind Mirza Missouri that's zany miners and Missouri plus In the X stigma excess about minus Laura and NYC wife that's and that's certain question what are the ideal values Of this matrix somebody's work and now I know somebody in rooms work amount where a course the Eigen values of this matrix he affect any ideas as well as the unit sectors if any of the components of the unit vector another words an X squared MY square square musical 1 you'll find the eigenvalues of this maker always plus or minus warned there this this is simply the statement that would remember what senior point you detect your apparatus with the apparatus Act gives you also might warned whatever you take here the Eigen values Of the Matrix ah plus or minus 1 Jordan you know little bit about matrices for those who know a little bit about matrices what progress who the trace of a matrix is the sum of the site in value the traced means the sum of bag Norman the 1st step Eigen values of Cuomo and the ran and the a trace of a matrix 0 was from process 1 and that tells us that when no 1 was planned to 0 0 and no 1 is equal to my land until Eigen values in their equal and opposite or about the determinant of if the product Arriaga values so we now know common equal opposite Perez what's products what determined determined follows the matrix is you quote from this and disease square this finds this might this times this and that's minus an X squared minus and why square leg next plus a I am white elects miners and square post squared here it is what is annex square person why square poses a square 1 of the determinant which is a product of the Eigen values his eagle minus 1 there's only 1 solution when the 1 plus slammed the is equal to 0 or as land or physical Simitis and the product is equal to minus 1 is only 1 such PARIS plus 1 minus 1 Oh all such matrices of this form have Eigen values which opossum I was 1 and therefore the energy levels are always plus or minus 0 may go over to 1 of Clouseau may go Italy other minor summer go over to your new flat from 1 to another you get Omega Worster energy the matter which workings of 40 yup Iowa to stem the Golden Gate Park 3 areas Yao walk 1 seat yup yup yup that's in a study researchers studying the procession here with in the north possibility from radio for now the probability to radio photon per unit time I can be pretty small a very very small in particular of the energy differences are very small as they would be in ordinary magnetic field it will process many many times before it emits photons for of us so if it does not stop there will be old trees they had ended the bottom of the barrel joined him back the told eigenvectors Auburn down or left and right or in an hour correspond opposite orientations of the or spend even those the state vectors are orthogonal the directions of space 180 degrees apart so Asia Satellite suddenly some 30 what I I thought there was a probability from your charm the and they're photons it is and that probability per unit time is typically small so they could be unlucky and after a quarter of a revolution might photon but that's extremely unlikely you can estimate how many riders at my injury a question on the it's a rest that I it send him year he these questions should not do thought less wrong from these Africa Noonan know it's in the position of being up and narrow as 1 of the oil facility average which is Kelly want misdirection it's a superposition of being often down even then only and that was a images state it won't really up radio so what electron some other direction there will be some probability for it to radiate and a probability for not to radiate the probability for it to radiate the probability that it was up Dr. very confusing very confused Frank Brogan RICO barring the magnetic field as that way and this is a that way you might have thought that with jumps you only may half the energy of a fulltime before half the energy not what's correct is that the probability for a amid a photons 1 half white because as have a probability that this is actually down here for probably sup with you to measure the this so average energy that America's half but either that's all they go for opposing the of for more please visit us
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at Stanford guard EDU