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Lecture 07. Tunneling Microscopy and Vibrations


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Title Lecture 07. Tunneling Microscopy and Vibrations
Alternative Title Lecture 07. Quantum Principles: Tunneling Microscopy and Vibrations
Title of Series Chemistry 131A: Quantum Principles
Part Number 7
Number of Parts 28
Author Shaka, Athan J.
License CC Attribution - ShareAlike 4.0 International:
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 and the work or content is shared also in adapted form only under the conditions of this license.
DOI 10.5446/18885
Publisher University of California Irvine (UCI)
Release Date 2014
Language English

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Subject Area Chemistry
Abstract UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. Index of Topics: 0:03:27 STM 0:19:19 Vibrations 0:46:33 Zero-Point Energy 0:49:19 Isotope Effects
welcome back to camp 131 A where we last left our hero we had decided that it was possible for a light particle to tunnel through a fair region much like a high jumper going over a bar but not going over it just appearing on the other side collecting the trophy but without having enough energy to actually go over a phenomenon that we call tunneling to indicate that we went through but we did not go all over today what we're going to talk about is we're going to talk about tunneling microscope and that's an application that turns out to be very very interesting for a lotta reasons and they were going to introduce a little more complex problem on vibrations the reason why vibrations are more complex problem is that the potential energy for a vibrational problem this is not the square Weller something that's mathematically so easy it's trickier because we get X squared in there we're going to see how we have to handle there but it certainly seems at 1st blush that this phenomenon of quantum mechanical tunneling is just a small nation field for experts and people in ivory towers to study but just like a lot of basic research oftentimes leads to killer applications and this is very very true in this case that we saw for example that the phenomenon that when you measure something new cause it to change allowed us to do quantum cryptography so that we could have this team that we can tell somebody was spying on us and we could establish an unbreakable code between us likewise this phenomenon of pummeling because when the barrier is big it depends exponentially on the distance and that means is very sensitive to it in some sense and that means that something that's closer tends to dominate everything and that lets us have a trick to make very shocked looking .period and 1 of these applications than of tunneling is called scanning tunneling the scanning tunneling microscope were the Sts sometimes called scanning probe it was good Beni and 100 grower and when they were working at IBM were granted a patent on the scandal and scanning tunneling microscope in 1982 while they were at IBM and I've given reference here on Google they have a list basically of all the patents that have ever been granted the public knowledge you can search them you can find out and boy are there a lot of them and this pattern was 4 million 343 thousand 993 and that was back in 19 82 I see take sharpened metal tip and I mean really shocked as sharp as you can make it but this will see it it may not matter how sharp you make it because when you look closely it's going to be extremely sharp no matter what if you if you fairly lucky and you bring it up to to a clean surface like a gold surface and you put 0 voltage on the ticket then they there's nothing between the on the surface except the vacuum and the way we interpret that is that the electron can come off from the gold too far because its energy is its potential is getting too too high as it goes away from the out the atom has a big positive nuclear charge that's going the Adam down that's holding it down and therefore crossing through this region all of space is like jumping through that region the In our totalling problem it is classically it should not happen we need to have a conductor to have occurred With electron but if the electrons a wave that if we get near the surface of the gold or something else then the fact that the wave function can sneak out a little bit means if it can sneak into the tip then there's some chance that the electron materializes in the tent and that gives us a crack in this current is going to be some noisy thing because it occurs because of tunneling but it's going to be a current and it's going to depend if we moved to tip it's going to get much much bigger because as the barrier gets thinner the gets exponentially more likely and what that means is that the position of this tip floating over the surface it is extremely sensitive
to the distance How can you use that to do something well this is not a microscope in the conventional sense of a light microscope where you might look at a hair look sells this is only for looking at the top ,comma fever surface but it can be fantastic because you can put the pieces of DNA on the surfaces of something like that and you can use variants of the atomic force microscope for example to look at these things in and see all kinds of things the amount current that you get is a measure of of the local density of states in other words it has to do with how many electrons can be there and how the way functions are on the surface and the distance of of the tip to the actual surface itself and this works with a conducting surface if we Rasta the tip over the surface by moving it back and forth and we keep track of where we are and with the little system of Pierce that's just like a GPS for your car it knows exactly where the thing it then what we can do it is if the current gets too big we assume were too close to the surface and we pull the tip of things and we try to keep the current at a certain value and what we keep track of as a function of of the value the current trying to keep it locked onto a certain value that's convenient that means wherein were close enough to get something we know we are too far away we are getting anything but were far enough away that we can move reasonably the problem is therefore to close and we come along with the 2 there's of Mesa or something on the surface it we come along then it'll increase it'll increase it'll increase but then will crash the tail end of the surface and then with put a notch in the surface sorta like scratching an LP in the old days if you're careless and we may change the tip because the tip as Adams on it and they may get knocked off and then get knocked around so we have to strike a compromise we wanna get a current that we can tell that were in contact without actually touching but we're in traveling contact with the surface but we don't want them to be so high that if we moved too quickly that were likely to crash the tip in the early days of of doing these experiments people crashed the tip into the surface all the time now there are commercial kinds of machines that even people have made St hands on their own and avid hobbyists you can you can actually make this device is not so difficult to make because of the exponential dependence that let's just imagine the tip of the tail whatever it is has been sharpened it's very sharp it's as sharp as you can make a sharp thing and why do want to be so sharp because you want to be able to see a little egg carton things of Adams and things on the surface and so your tail should be very sharp if you're campus this big wide and then you kind of blurred everything together you can't see anything but it turns out because of the exponential dependence if there's a cluster of grapes hanging off the tip all Adams the piled up there and it could be anywhere doesn't have to be in the middle of the tip but whatever is closest all the current is going to go through there and so that's good perfect because it means that you just get lucky with the tip and then all the current goes through there and the tip sort of makes itself sharper by the way In the current depends on on the proximity and so unless you're very unlucky unlucky would be to tip ss coming down 2 bunches of grapes both the same size that would give you a very confusing image that doesn't happen I'm very likely the other ones contribute exponentially less current and that means that they don't bother you much you can experiment with different tips and people did a lot of that and find 1 a good 1 and then you conduct experiments with this good tip you try to get your Ph.D. with it if you can until you're unlucky and you're too aggressive or something and you crash the tip of the service and of course most of the time when you're looking at things with the STM you're looking at things that you kind of assume are pretty smooth you're trying to look at the details of a flat surface of Adams you're trying to see for example if you have a material whether some of the Adams might if you have 2 different kinds of Adam some of the Adams might like to be on the surface of the material in a different amount and the
ball and you can see that kind of behavior With and you can also use the STM 2 new chemical reactions if you have I'm a surface of some molecules and you bring down here now you can influence you've got current going through what you can do it you can do a pulse so like having a media come in here and you can make a small chemical reaction at that point and you can ride a little docked there for example and then you can move the tip and impulse that again and you can modify the surface by doing this over and over and there was interest in that at 1 point here's us figure from the patterns you can see back in 1982 you do things by hand but a new due to a lot of things by hand and factories something called a Leroy set back then to get the numbers to look nice and so forth and Hinduism in India ink this is from the original scan of the original document here they're showing this tip their showing a flat surface they're showing seeing which is the up and down there showing X and Y which they're controlling with these piezoelectric controllers there showing a plot of what they get and they're showing a screen which is going to show the top pornography what the current debt as a function of z and here's a figure 3 from the same pattern where you can see here is that they're showing the current I as a function of the distance from the tip from the surface and the drawing an exponential which is exactly what we derive from tunneling and they're showing that the current should go like to the minor something and then you can see that there is a distance and in the exponential that's the exponential distance dependence that we derived for very unlikely event and you want this to be unlikely you'd want to be too likely and here is that is a ski Maddock again figure 4 from their patent there's a surface there's a tip the tip is very near the surface the 2 pass some shape and all the current it is coming From the part of the team that's closest to the surface so that part of the tip is a super sharp part that's giving all the current and the other part is giving a little current which is kind of a noisy background but it's not a big deal and then you're going to move the tip along and you're going to keep the current constant and by keeping track of how you have to adjust the the the height of the tip but to keep the current constant you get a picture of the lay of the land the tips can also be used that actually pick up Adams and move them around and here's a spectacular example of that which Don Eigler published in Nature In 1990 and again working at IBM here you have a very very flat atomic precision nickel surface and on it are scattered some xenon atoms of xenon atoms have a lot of electron density and they show up there as these round things just like you might imagine as xenon Adams should look and you can see in frame a they're all randomly positions and what they were able to do is 1st use the microscope to see where everything is be quite careful and then go 2 inexact position where you know there is a and of course this has to be extremely cold this is basically a liquid helium temperature because if this is not at room temperature or even liquid nitrogen temperature it's going to be like drops of water off right on that they'll be moving all over the place and it won't matter that you scan through and see where they Arkansas next time you come through and be somewhere else and half the time just
pop off the surface entirely there won't be enough sticking up for them to stay on so this is extremely extremely cold which is also a challenge because you have to cool your microscope you have to cool the surface yet cool everything done have to be extremely careful and then you come down the tip on top you know it's still there cause it's so cold and you actually push on it and then you drag it somewhere and with your ex-wife magical GPS system there you pocket and then you go get another Adam and you drag it and another 1 and day again and then in between times you then image the surface
very gently so that you are not dragging any Adams and then you can see your program and what they show here is they can ride out IBM In xenon atoms on a nickel surface using the asking and this was really just a spectacular example of how you can manipulate this the very smallest things with such exquisite detail using this device but unfortunately I think nobody has figured out how to use the device to make extremely small things like computer chips or for other things that might be ultra ultra ultra miniaturized and because it's too slow it takes too long and has limited ability to make any kind of 3 shapes here's another image this is from a group of current Carnegie Mellon in the physics department and what this is is this is a picture of the surface of silicon and the 1 1 1 just means that it's a certain crystal claims In the end overtake silicon and has a certain crystal and structure I can cut at certain angles to cutting and diamond and I cut at certain angles and I would expect to have certain kinds of atomic but what tends to happen once I come is that the atoms are very unhappy if they're sticking out too far they're very unhappy because they don't have enough bonding neighbors so-called dangling bonds bonds that are going out into space doing nothing and what they may decide to do if they're unhappy enough like a lonely person going to a bar they may pull and try to make extra bonds with other atoms which has nothing to do with the original structure that you would expect and that's called reconstruction and here you can see is the so-called 5 by 5 reconstruction of silicon and you can even see there's 1 defect in the middle of the picture where there's an that's kind dislocated that's out of out of position but most of them seem very perfect and very interesting of course the color is faults the color here is just to guide the alright now let's go on to vibrations vibrations are important because when we do a chemical reaction we take a chemical bond
and we usually break we break it and we make a new bond and the whole business of chemistry is to take stuff where things are organized in the same manner but there worthless just junk manure and then we make a break bonds will witchcraft and outcomes some very very important to by article which is worth a lot more money and in order to understand how that works in detail we need to have a very good idea of how strong the bonds are we need to be able to predict if we're going to make something if it's gonna have strong bonds are weak bonds and we need understand also if it's going to absorb light so that we can do an asset to see if we've made what we think we've made like I our spectroscopy here what I've shown is a potential energy curve for the hydrogen molecule H to the simplest molecule on the proton Proton distances along the x-axis and the commuters and the energy the electronic energy so we calculate this curve by moving the protons together at different distances now we freeze and even tho we know they can be frozen by the uncertainty principle and we calculate the electronic energy and if the protons sir to close they tend to repel each other plus the the electrons at the orbital sought to close together it's not optimum if they're the right distance than the electrons can be in between each proton sees both electrons as part of the principle of bonding as will see is that they share each proton each hydrogen thinks it's a helium atoms and that's a very stable configuration and then as we tend to pull the protons apart the electron clouds can overlap this Proton cannot see anything to do with this electron and so the strength of interaction decreases and finally when they're far apart they're just 2 hydrogen atoms and macho men in this so-called potential energy curve which is just the electronic energy plotted as a function of the frozen distance of the 2 protons when they're too close you see that the electronic energy is above 0 that means that when they're that close but there unless they're more unstable than just 2 hydrogen atoms apart but there is a well there is a
position where the 2 hydrogen atoms working together are much more stable than 2 hydrogen atoms apart and that's a stable H 2 molecules and then the potential curve goes back to 0 as they go back to work just to isolated hydrogen now near the bottom of the well and at the equilibrium distance which was 74 Pico meters in the previous figure the potential has a minimal and where the potential has minimum calculus tells us that the slope must be 0 and that means that if we expand the potential the of ah that curve whatever the form of that is in Taylor series which we do biting function Diamond the derivatives the song around the equilibrium position we can write the it equal to the of Part B which is the equilibrium the lowest point plus AA-minus sorry so get a check of rearranging something by making it seem more complicated and we can write that as the Givati plus delta are where Delta r is how how much the bond is stretched or compressed from the equilibrium what we get is In the Taylor series we get the of and the evaluated the body which will just call the of already plus the derivative Of the alarm evaluated already times Delta plus 1 hour to factorial times a 2nd derivatives view of our evaluated the Gari times dulled our squared plus blocking keeps on going said if we look at the bottom of the well the directive 0 and therefore it simplifies quite a bit near the bottom of the well the derivative is 0 right there so we throw that term away because we evaluate that term right at Oracle's already if we're near the bottom of the well that are as close to ah eh what we will assume then is that AA-minus already squared is something but are minors are accused and all the higher ones are too small because our is very close to party and so therefore much smaller and if we do that we end up with is following very simple form for the potential which is what we're going to use it and when we do the shoring equation because we want we don't wanna use the feel potential will never get out alive it'll be far too difficult for us to do we get the Amari plus one-half carry guns AA-minus are equal quantities square where K is the 2nd derivative of the with respect are evaluated Oracle's and Katie here is the is the force constant not the so I apologize for using K again but carries a conventionally the force constant the spring and before was the way vector UDI kayaks this is a different there's another cable 2 months constant which might put case of be to tried to keep that 1 separate but will use K all we talk about vibrations to me the force constants of a spring and the 1st small displacements them we have that the motion should be harmonic unless K 0 if Kerry happens to be 0 than that term goes away and then the whole motion is described by something very funny which ever terms of leftover but usually carries not 0 because the thing comes down and goes back up and so it has some
part of it that's quadratic and around the bottom and that's going to be the main part of the actual potential therefore we can model a chemical bond as a one-dimensional harmonic oscillator we totally ignore any kinds of other displacements in other directions or anything funny and we just say Look these 2 things are lying here there's a desk distance between them we know the energy what we want to figure out as was the wave function as a function of this distance between them given the form of a potential we can always adjust the energy 0 the we can call the bottom of the well 0 you know it's not even over hydrogen is minus 4 . 4 1 half the the down we can call it 0 and then we can just add it later if we want to get the real energy so we don't have to worry about that in the math and so Will call 0 but when when the displacement is 0 and just to keep in in keeping with what we've been done for consistency rather than using AA-minus sorry Altus introduce a variable X and so I will be a function of accidents so effects is is 0 then there at the equilibrium and affects some males plus or minus than of its away from equilibrium we get the same tired old time independent Schrödinger equation to solve some mine minus 8 bars square over to m the square outside the EC squared now plus one-half kayaks were signed is equal to the side and given K and M Our task is to figure out what the allowed values of eh all and what the fuck functional form of science for the particle box the liabilities of the word like and squared and so I was assigned wave now we've got a different
potential completely it's get the SEC squared toes keeps continuously changing so guess it's going to be quite a bit harder to do it and that would be a very good guesses will say but if we have the 2 nuclei connected together and let me just remarked that the and here is this the Mass the reduced mass of the
oscillator or M 1 and M 2 over M 1 plus too but we won't worry for what we're doing which we just want to get a qualitative field so just keep an eye on em as his son mass associated with the ocelot now before we solve the differential equations but it's it's a good idea to take a 2nd and try to figure out what what it is we would predict that we should see that way if we get some ridiculous because we make a mistake will know at all so the question is what properties should be the way functions have and that's pretty easy to suss out 1st the energy will be quantized why because the potentials going like that and something that's going like that is tending to confine the particle particle cannot just go anywhere because the potential gets bigger and bigger bigger bigger escaped the trap
it's trap it's got to be quantized discover go way out there in the energy it has to fit into the space and therefore can be quantized we don't know what shape its can secondly the lowest energy Oregon State can be 0 energy it wasn't 0 energy for the particle on a boxy the problem with that is if we picked 0 there it went away and that we have a similar problem here if we want to have a real way function is going to have to be in there it's going to have to satisfy the uncertainty principle and therefore it's going to have to have non-zero Peace Square and it has non-zero EC squared because it's not not an infinitely narrow box and therefore it's going to have non-zero energy and thirdly the wave function has to die away somehow insects gets far from 0 the reason why there is that the potential keeps getting larger and also addressed the guy 0 because it has to be and finally the ground state I should have no notes and the reason for that is that we by analogy with the particle on a box warning to the particle box it was 0 0 at the edge because it had to be but it wasn't 0 anywhere in between it was just a lot and if we change this to this we expect this month To change but we don't expect that to change into
2 lumps therefore we expect something like a turtle in there somehow sitting there not not very exciting but just sitting there and if we have an excited state if they're excited states in this potential we would expect them to have notes just like the higher excited states Of the particle in a box but what how should we solve the differential equation remember I said the most powerful method to solve differential equations is often guessing and I'm going to try to get I have something attends a derivative of the derivative of the use of the wave function twice plus some other stuff it is equal to the way function and a number but I've gotta get the other stuff to go where therefore I need to have a function that generates itself again that's so I can get this part and I needed to generate a little of the garbage times itself so that I can get rid of them one-half game squared which I want to go away because there's no 1 have TAX word on the right side and I could use just the 2 the X to do that because he the accident give itself times number but I could use speed the something else and I would expect that I'm going to have to use an exponential function because there are always the solutions of these differential equations furthermore I couldn't guess then look this thing has cemetery if I draw potential at the bottom Texas 0 and then it's going up and a symmetrical and therefore the wave function has to be symmetrical too and that means that we can have anything like the 2 the minus Alpha extra something like that because that's not symmetrical around 0 we could have that policy to the class of expert we can see right away that neither of those would be any good since those don't look good I tried the next power up and if you try the next power up which is a Gaucin function you get very lucky In fact it seems to work so let's guess let's gasps acts is equal to 8 times the exponential function of minus a time were little eh they based the normalization constant we won't worry about what that is at this time and little something that we're going to have to pick I want to make it work and we'll see what the condition you wouldn't necessarily know that this would work but you can easily work it out so let's go ahead and and work it out if we take we want to take the 2nd derivatives and multiplied by each part squared over to well the 1st derivative of that function is that function again times the derivative remember that the jury derivative of either the EU is speedily the unity you DX we have minus a X squared as you the derivative of that is minus 2 AXA so the 1st derivative is 8 times exponential minus X squared times minds to it at all the 2nd
derivative that we've got a product of 2 things we've got the derivative of you times V and that is the derivative of you times the plus the derivative of the times you and the first one week done the derivative before so therefore the derivative the 2nd derivatives Kapolei Eden minus a X squared times minus 2 a it plus 8 times even the minor 6 squared times minus 2 AX times minus 2 area together the 1st 2 A X comes from the UDA the 2nd minus two-way act comes from the fact that that 2nd 1 is there and we can now put this together and we see but we got what we wanted we got 8 times even minus X where so that gave the same thing reproduced and then we have this term and it has 2 parts it has a for a squared acts square which if we take a right is going to cancel out the 1 have kicked square and then got the other part to which is going to have something to do With the let's have a look around if we put everything and then into the Schrödinger equation will come the following conclusions minus squared over 2 hours capital a the minus X squared times for a square decked squared minus 2 plus one-half case square again capital a minus its worth is equally the times the same thing and so now we can divide both sides by Capital a Eden minus said square and get a relationship between a little late and care but if we want those terms to cancel because there's no X squared term on the right-hand side it's just eh the terms expert cancel that means that minus 8 bars square over 2 hours times for a square decked square plus one-half kayaks where is equal to and for that to be equal to 0 for all values of acts a little lady has to equal to the square root of them upon to each bar and a therefore the exponential argument has to do with the masses In the spring constant and planks and boy is that sweet because this is Quan mechanics and that's exactly the kind of behavior that we would have expected to see and we can make a connection with the classical loss later if you've done a classical oscillator if you haven't then you should but the angular frequency of the classical oscillator was the angular frequency well if if I see this thing going back and forth like that I can interpret it as something as a projection of something going around as if something's going around it's going like this and that angular frequency is the square root of K overran and since Omega's the angular frequency is the square of K overran the square root of MK is equal to m times Omega and using that we can now figure out the energy the energy is sage bars squared over 2 m times minus 2 air that's the only part left over conceal Texas gone by our choice of it and that's 8 per square squared of 2 and carried over to wage and that's H bar over 2 m times animal go the ants cancel and we get Omega Over 2 that is the ground state energy of the oscillator which is not 0 the oscillator is always a little bit excited can be 0 because of the uncertainty principle and we can make a connection between what we found with light by saying Look Omega the angular frequency is to pioneer newest to regular frequencies so a spiral may go over to it is also age Nuova over too and H knew was the quantized edition of a photon so this is similar to the quantities edition of light except that now there is a factor of 2 and the denominator for the energy but other than that it's very closely related knowing the value of a now a little later then we can calculate and normalize the entire way functions and determine the value of big game because now we know how wide discounts in function is we know what area it's going to have when we integrated so we know how big and this it has to be to make the probability of finding that the displacement somewhere equal to and our normalization condition now is that we should take the square integral of this thing from minus infinity to infinity as usual and be integral Evita the minus to LAX where DX that's a standard integral which you can look up that's the square root of by over 2 divided by the square root of saying and capital a square times that should be equal to 1 and therefore protect the square root of both sides we find that a squared should be To a divided by time To the one-fourth power and that's capital where we've picked a as usual to have real face because we always like a having real phase were just biased toward that just as we did with the particle on a box with functions we have that too wide area in ways that will get rid of the eye because of this this therefore gives a star Our very final form for the ground state Of the harmonic oscillator and that is way over prior to the 1 one-fourth even the minuses square and then I can put on all the things with an animal may go for and I did this very nice formula now the problem with this approach of guessing is that whatever you don't guess doesn't turn up and we guess 1 thing and we found 1 thing and it made sense why because it's a is like a big turtle has no knows it has a low energy which satisfies the uncertainty principle but it doesn't have any more energy than that but now what we would have to do is we would have to try to guess so higher things that it's not like the particle in a box where we had an empire we don't have any answers yet here in this problem we just had this 1 wave function and this 1 solution we suspect and were right but because it's like a particle box just slightly different that there should be a ladder of states and in fact the latter is actually equally spaced it and plus one-half time at times age borrow maker which makes this potential really unique because it's the only form where you get an equally spaced ladder of states all the way up to infinity and there are some very very nice mathematical ways of attacking that problem very beautiful but they they take us a little bit too far afield for our costs and they don't have that much to do with chemistry that has more to do with operator algebra and quantum physics and chemistry and so were at work not going to explore those but will just quote that that the general solutions there is some polynomial enacts that you have to pick carefully times each the minus 6 square for that part of it is always the same now what is the interpretation of office it's completely different than a classical because the classical oscillator yeah has the highest probability of we will take a film of it it goes out it stretches it stops turns around come back through that's where X is 0 that's words perfect compressors start back through again through again and so on and the chance of finding it if it's oscillating right at the equilibrium position it is not likely if you just grab it at some point and measure its distance apart it's much more likely to be the fully stretched it has to turn around there as was called the turning point or fully compress but will we look at the ground state of the harmonic oscillator what we find is something completely different the highest chance is that the thing is in the middle in the perfect position where it shouldn't be that seems as if is trying to stay in the middle but it's actually spreading out little bit not because it wants to but because it has because of the uncertainty principle and therefore this Krause data the harmonic oscillator is looks completely different than a classical Ottley late and that caused a lot of consternation I think in the year very early days because it looked so different and how do you interpret this thing you want to make sure you have made a mistake or something's not right in the equations but it turns out disagrees exactly with with what we observed and there are many experiments where we take a molecule and we we excited electron and it really looks like it comes mostly from the equilibrium position we very rarely find something coming from the extended position so this interpretation of this it is correct like I said the operate the OSS later can't sit still it's like a small kid has to squirm around the satisfy the uncertainty principle and the 0 . energy depends on the square root spring constant K and on the inverse square root masses and this gives rise to something called the isotope effect let's have a look at the isotope effect suppose we have 2 isotopes for example hydrogen has a single proton deuterium is a single proton plus a new trial it's heavier but the charges the same and deuterium behaves much the same way as hydrogen does there's D to all 0 you can use it to make an Aymara samples for the neutrons don't have any charge no electric charge so the electrons don't really care too much about the neutrons and therefore what we expect to a 1st approximation is that the force constant which has to do with the electronic orbitals and the repulsion and the repulsion of the 2 protons doesn't depend on whether the 2 proton due to do the answer it depends on the charge and the separation and the electrons pretty much don't care there are small effects because the electron sneakier can see the neutron there's some small effects but basically the former the potential energy surface is the same but what's different than is the man and therefore this hope the whole field of isotope effects is a great field to get into if you're interested in theoretical effects of 0 . energy and tunneling and all the subtle things because you have a perfect thing where everything's the same except the mask and so your calculation even doesn't have to be quite so good because even the things the new calculation the slightly wrong are the same except for the mass so much a very unlucky get a pretty good results anyway if we have them on a CD bond a carbon deuterium bond 1st as a carbon hydrogen bond the carbon deuterium bond is stronger because it did it in the sense that they might have manage it takes to go from the ground vibrational state which is as close to the bottom of the well as you can get 2 where the bond is broken which is the same place on the curve is higher for this CD and here's then will close with this here's how it might look on the last we have a molecule that could be the end of a hydrogen or deuterium unless its carbon and then in the chemical reaction this bond breaks and as it's breaking the force costing gets less so when the transition state it's almost off and therefore the potentials very wide because the force constant for something that is is a weak bond is not very stiff spring but when it is actually the reactive it is stiff and therefore there's a big difference 1 of the reactant and there's a small difference was the transition state and that then translates into a different rate of reaction that means that if we have molecules that could be CD or H and we react them somehow we expect the CH ones to react more quickly than the CD ones and this is called the kinetic isotope effect and I've adapted this from Little figure here from Wikipedia to just show how this works In fact this again seems like a very esoteric things but you would be amazed if you go to the entomology literature and you look at how do I know if this enzyme is breaking this ponder them ponder what is the rate-limiting step for the synthesis of cholesterol or the removal of something from the body you put on a deuterium as a spy and look for the dude rated products versus the protonated product and boy do you find a ton of information so this is not translated all the way from this esoterica effective the harmonic oscillator the uncertainty principle 1 0 .period energy all the way to modern medicine where it's used to figure out what's going on in the body the next time will continue on with somebody's one-dimensional model problems and then we will begin to actually do some multidimensional "quotation mark mechanical problems I'll close at there


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