WEBVTT chemistry/physical-chemistry/hovasapian
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Hello, welcome back to www.educator.com and welcome back to Physical Chemistry,
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Today, we are going to round out our discussion of the quantum mechanical harmonic oscillator.
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Let us jump right on in.
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We said that our energy levels for the quantum mechanical harmonic oscillator, our energy levels are as follows.
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We found that E sub R = H ̅ ω × R + ½, where R is that harmonic oscillator μ quantum numbers.
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R= 012 and so on, all the way through.
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Now let us talk about the wave functions.
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We saw the Schroeder equation, we are finding energies and we are finding wave functions.
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The wave function represents what the particle is doing.
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We do things to the wave function in order to extract information about the actual physical system that we are dealing with.
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The wave functions ψ sub R corresponding to the energies.
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For each R value you have the energy and you have the wave function ψ corresponding to the wave functions.
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No, the wave functions corresponding to the energies.
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They look like this.
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Ψ sub R= N sub R H sub R Α ^½ × X E ^-Α X²/ 2.
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This is reasonably complex.
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Let us see what each of these are.
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Α here is going to equal to K × μ divided by H ̅ ^½.
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We introduce some symbols just to make it look a little more clean.
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N sub R, let us go ahead and do red.
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N sub R is the normalization constant.
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We want to normalize so the integral was equal to 0.
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The integral =1, the normalization constant.
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N sub R = 1/ 2 ⁺R × R factorial ^½ × Α/ π¹/4, there we go.
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The H sub R they are called the Hermite polynomials.
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H sub R Α ^½ X, they are called Hermite polynomials.
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They are part and parcel of the solution of the particular Schrödinger differential equation.
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H sub R is called the first degree hermite polynomial.
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It is not going to be the end of the world if you say hermite, it is not a big deal.
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Let me go ahead and lists the first few hermite polynomials,
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then I will go ahead and list the first few of the actual complete wave functions for the harmonic oscillator.
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Let me do this in blue.
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H sub R, this is in parenthesis Α ^½ X.
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It is whatever is in the parenthesis, goes into the variables.
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The first few hermite polynomials
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We have H₀ of Z = 1.
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H₁ of Z = 2 Z.
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H₂ of Z = 4 Z² – 2.
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H₃ of Z = 8 Z³ - 12 Z.
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Let us go ahead and do H₄ Z that is equal to 16 Z⁴ -48 Z² + 12.
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We said H sub R of Α¹/2 X.
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If we did H of 2, this one right here.
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4 Z² - 2 it becomes 4 × Α¹/2 X² -2.
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Whatever is in parentheses goes into the variable.
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It is just basic functional notation.
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Let us go ahead and do the first few, these are just the hermite polynomials.
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Let us do the first few wave functions, the complete wave functions.
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When we put everything together, normalization constant, hermite polynomial, and the rest of it.
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The first few wave functions we have, Z₀ of X = Α/ π¹/4 E ^-Α X²/ 2.
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Z₁ of X = 4 Α³/ π¹/4 × X × E ^-Α X²/ 2.
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As you can see, these tend to be very complicated very quickly.
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But again, most of it is just that the constants that tend to be unwieldy.
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The functions themselves are not that difficult to deal with.
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Ψ₂ that is what we are, we are at number ψ₂.
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It is going to end up being Α/4 π¹/4 × 2 Α X² -1 × E ^-Α X²/ 2.
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Let us go ahead and finish up with ψ₃ so we can move on to some other things.
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It is going to equal Α³/ 9 π¹/4 × 2 Α X³ -3 X × E ^- Α X²/ 2.
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This is just a representation of the first 4 wave functions.
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Let us go ahead and actually plot these 4 wave functions to see what they look like.
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Let us plot the probability density which is ψ².
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Let us go ahead and do that. Let us do this on one page here.
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Let me go ahead and just do this.
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This is going to be the wave function, this is going to be ψ of R of X.
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And over here we are going to do the probability density that is you remember, the modulus.
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That is the ψ sub R² which is nothing more than ψ complex × ψ itself.
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Since this is real, this is going to be to ψ² but this is the symbol, just in case.
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This is going to be the 0, this is going to be the 0.
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We have something like a high point there.
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This is going to be to R =0 and here we are going to have the probability density of that.
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This is the wave function, this is the probability density, this is where you are most likely to actually find the particle.
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This says that when you are at the R= 0, when your energy level is E₀ = ½ H ̅ ω.
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When you are at the first energy level, the chances are that
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you are more likely to find the particle near the equilibrium position than you are near the extremes, near the amplitudes.
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We will go ahead and go to R = 1.
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We are going to end up with something which is this way, this way.
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Then we will go ahead and do the probability density for this.
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We are going to end up with node there and there.
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It is going to be like this, something like that.
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This is the first energy level.
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We will go ahead and do R2, R=0 R =1 R =2.
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We have something like this.
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That is the wave function and now we will go ahead and do the probability density.
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This is going to end up with 0 here, and 0 here, a high point and high point.
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This is going to be something like that.
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This is the second energy level.
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We will go ahead do that and that.
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Let me see what we got.
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We got 123, 123, we have a low point, a high point, low point, and a high point.
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Of course we are going to have 1, 2, 3.
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We are going to have a high point 01234, something like that.
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The wave function as the wave function goes from as R goes from 0 to 1 to 2 to 3, the energy is rising.
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This is just the wave function.
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It is a probability density that gives us the most information.
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At the lower energy levels, the 0, 1, 2, we tend to find it mostly in the center.
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As the quantum number rises 12345, you notice that it is more evenly distributed.
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It means that the particles or the particle is spending more time sort of evenly distributed.
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This getting a little bit closer to the edges.
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Notice, it is spending more time everywhere instead of spending most of its time towards the equilibrium position.
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That is all that is happening here.
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This is the wave function, this is the probability density, this gives us the probability of where you actually going to find the particle.
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The places where it is 0, you are not going to find the particle here, that is what this is saying.
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You are not going to find the particle there.
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Those are nodes of the places where this is X.
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Here is the 0, the equilibrium position.
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As you go farther and farther away, that this is the oscillation point.
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The amplitude this way - amplitude this way, this is the equilibrium position.
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As you get much higher and higher and higher in energy, you are going to find
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the particle distributed more evenly between the amplitude and the - amplitude between here and here.
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But there are places where you absolutely not find a particle, that is what this means.
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Places where the probability density is 0, you will never find the particle there.
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Because we are talking about something that is quantize.
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For large values R, as R gets really big and R goes on to infinity.
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Where large values of R, the probability density looks like this.
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We will end up with something looking like this.
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The correspondence principle, you remember we talk about it once.
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The correspondence principles says as quantum numbers increase,
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the quantum mechanical system starts to display classical mechanical behavior.
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In this case, as the energy and displacement rise,
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the particle is more likely to be found at the extremes points, the turning points.
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Because it is moving more slowly at the extreme points.
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It is basically telling me that as the quantum number rises, as energy rises,
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as displacement start to increase, you are more likely to find the particle near the edges.
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The particle is going like this, back and forth.
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At the extremes, it is actually moving quite slowly because it slowing down at 0.
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Its kinetic energy is virtually 0 so it is moving more slowly.
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Because it is moving more slowly, you are more likely to find it at the extremes of the equilibrium position, the center.
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You are more likely to find it because it is passing through the equilibrium so fast, the kinetic energy is maximized.
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It is going this way and zipping through the center and coming here.
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It is zipping through the center, it is slowing down going there.
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This is what the classical harmonic oscillator would do.
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It would spend more time at the extremes, less time in the center.
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Again, as the quantum number increases, the correspondence principle says that
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the quantum mechanical behavior is going to start to look more like a classical mechanical harmonic oscillator.
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It is going to spend more time at the extremes and less time at the equilibrium position.
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In other words, it is going to be vibrating so much.
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You are virtually not going to find the particle at its equilibrium position.
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You are going to be finding it more often as its extreme points.
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Let us go ahead and finish off with a discussion of the hermite polynomials.
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Let us recall, an odd function is if you were to put -X in for X for the function, you end up getting negative of the original function back.
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You put in -X into the function and what you get back is actually negative of the original.
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That is an odd function.
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An even function says that if you were to put -X into your function, you just get the original function back.
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And odd function is symmetric about the origin.
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An even function is symmetrical about the Y axis.
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Something like this.
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The cubic curve as an odd function, it is symmetric about the origin.
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X² is symmetric about the Y axis.
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The hermite polynomials, we are doing this discussion in order to help us with our math.
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The hermite polynomials are even when R is even and they are odd when R is odd.
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Recall also that if F of X is odd, then the integral from - infinity to infinity of F of X DX = 0
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because we are integrating below the X axis, above the X axis the integral cancel.
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Ψ sub R= N sub R H sub R of Α¹/2 X E ^-Α X²/ 2.
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When R is even, ψ sub R is even.
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When R is odd, the ψ sub R is odd because this is an even function.
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Therefore, when the hermite polynomial is even, when R is even, the hermite polynomial is even which makes this whole thing even.
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When R is odd, the hermite polynomial is odd which makes and odd function × an even function, it makes it odd.
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In either case, ψ sub R² is even.
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If ψ sub R is even, the square of it is even. If the ψ sub R is odd, the square of it is even.
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The expectation value of X, the average value of X is equal to the integral – infinity to infinity.
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Ψ sub R complex ψ sub R × X × ψ sub R = the integral from –infinity to infinity of X ψ sub R².
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That is odd because F of X = X is odd.
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Add an odd function × an even function, you have a integral of odd function.
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This integral is equal to 0.
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The average value of X =0.
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If I take a 1,000, a 1,000,000 measurements, on average I’m going to find that the particle spends time in the middle,
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simply because it is spending an equal amount of time on this side and an equal amount on this side.
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It averages out to 0.
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The average momentum is going to be the integral –infinity to infinity.
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Remember, ψ sub R complex × the momentum operator which is - I H ̅.
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The average value of any particular thing that we are trying to measure
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is going to equal the wave function × the operator of the wave function, that is the definition.
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- infinity to infinity ψ sub R complex - I H ̅ DDX of ψ sub R.
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If ψ sub R is even, it implies that the derivative is odd.
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And if ψ sub R is odd, that implies that ψ sub prime R is even.
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In either case, ψ sub R × the momentum operator acting on ψ sub R is odd.
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If this is even, this part is odd.
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If this is odd, this part is even.
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In either case, I have an odd × an even function which is an odd function.
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In either case, the integrand is going to be odd.
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Therefore, the average momentum is also 0.
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All that means is that it is spending as much time going this way, as it is going this way.
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It is oscillating back and fourth.
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That is another preference for going this way or that way.
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On average, I might find it going this way or this way.
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At any given moment, I might find it going this way or this way.
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On average, it is going in both directions or going 0.
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The harmonic mechanical oscillator, the average value of X is 0 and the average value of P is 0.
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Thank you so much for joining us here at www.educator.com.
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We will see you next time, bye.