Show by symmetry arguments that the expectation value of the momentum for an even- n state of th

Show by symmetry arguments that the expectation value of the...

Key Concepts

Parity Symmetry

This concept refers to the invariance of a system under spatial inversion. In quantum mechanics, if the potential is symmetric under reflection (x ? -x), the eigenstates can be chosen to have a definite parity, meaning they are either even (symmetric) or odd (antisymmetric). This property plays a crucial role in determining the behavior of observables.

Even-Parity States

Even-Parity States are those quantum states whose wavefunctions are symmetric about the origin, meaning ?(x) = ?(?x). In the context of the one-dimensional harmonic oscillator, even-n states exhibit this symmetry, leading to specific consequences for the evaluation of expectation values, particularly when combined with operators that are odd functions.

Momentum Operator

The momentum operator in quantum mechanics, typically given by -i?(d/dx), transforms as an odd operator under parity transformation because its action changes sign when x is replaced by -x. When evaluated in an even state, the integrand for the expectation value of momentum becomes an odd function, resulting in an integral that is identically zero.

Expectation Value

The expectation value is the weighted average of an observable over all possible outcomes in a given state. Mathematically, for an observable represented by an operator, it is computed as the integral of the wavefunction times the operator acting on the wavefunction. When symmetry properties ensure that the integrand is an odd function, the expectation value will vanish.

Transcript

00:01 Hello everyone.

00:02 In this problem, we're asked to use symmetry arguments to show that the expectation value of the momentum operator on even numbered wave functions of the one -dimensional harmonic oscillator are always zero.

00:19 So the approach to this problem should be that you kind of have a general sense of what sort of combination the expectation value of the momentum operator is going to be on these vape functions, right? so the general nth wave function of the simple harmonic oscillator looks like this.

00:40 So it's psyn of x is equal to 1 over n factorial times 2 n times 2 to the n times pi to the minus 1 over 4 times the hermite polynomial of x over sigma times e to the minus x squared over 2 sigma squared.

00:57 So this is the genetic rate function.

00:59 And the hermit polynomial is the following.

01:02 So it's minus 1 to the n, e to the minus x, e to the x squared over sigma squared, and then the nth derivative with respect to x of e to minus x squared over sigma squared times sigma to the n.

01:14 Now, if you then combine the hermite polynomial into the wave function, so you write the whole thing as just one function and replacing the hermite polynomial with its expression, then you get that this is minus sigma to the n, that's pi to the minus 1 over 4, over the square root of a factorial times 2 to the n times sigma times times e to the x squared over 2 sigma squared, and then the nth derivative of e to them minus x squared over sigma squared.

01:43 Now what the really interesting thing to think about here is kind of this function.

01:50 This is where the ambiguity lies, because we know that this part here, this function is symmetric, right? so if that function is symmetric, then if it's going to be multiplied by an anti -symmetric function, then the overall function is going to be anti -symmetric.

02:08 So that's something that we want to figure out.

02:10 So we ask how could the nth derivative of the function e to the minus x squared over sigma squared look? so in order to do that, we kind of going to use integration or like the chain rule to figure that out.

02:27 So we're going to first make a substitution for y is equal to minus x squared over sigma squared.

02:33 Then the derivative, the first, derivative becomes the y by the x is equal to minus 2x over sigma squared.

02:41 This also tells us, as later we will need, that the second derivative of y in respect to x is minus 2 sigma squared, minus 2 over sigma squared, and then the third derivative actually vanishes.

02:55 So there's no more than 3 derivatives that you can apply, or no more than 2 derivatives that you can apply to y.

03:01 So, you know, just to kind of have a sense for what the symmetry of this function is going to be.

03:09 I made the nth, so n equals 3, we calculated the n equals 3 derivative of this function.

03:19 So i made the substitution that y is equal to minus x squared over sigma squared, and i rewrote the function.

03:26 So i said that i'm looking for the derivative of d3 by d x, x cubed of e to the y.

03:34 So then i apply the chain rule here in the innermost bracket once.

03:40 Then i find that that is equal to d .y by the x, d by d .y of e to the y.

03:46 And of course, the derivative of the exponential function is just itself.

03:50 So i end up with this combination over here.

03:52 And now i'm going to take the derivative in respect to x of that combination.

03:57 So i apply the d by dx operator.

04:00 So i apply the operator to this function first.

04:05 And leave everything else the same.

04:07 So this will give me actually d by d so that it was like i missed an e to y over here.

04:17 So it's going to become d by dx of d2 by d x squared of y times e to y.

04:27 And then there's going to be an other factor here where we apply to dy by dx.

04:36 All right, so let me, since i made a mistake, i'm going to have to go over the whole thing one more time.

04:43 So let's do that.

04:47 Let's do that.

04:48 So here we're going to have d by d x squared of e to the y plus d .y by dx times d by dx of etha to y.

05:02 Right.

05:02 That's just coming from applying d by dx to the first term...

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