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sunquick

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## Homework Statement

This is problem 17 from Chapter 3 of Quantum Physics by S. Gasiorowicz

"Consider the eigenfunctions for a box with sides at x = +/- a. Without working out the integral, prove that the expectation value of the quantity

[tex] x^2 p^3 + 3 x p^3 x + p^3 x^2 [/tex]

vanishes for all the eigenfunctions."

## Homework Equations

The eigenfunctions for a symmetric box with sides at x = -a and x = a are

[tex] u_n = A_n \sin\left(\frac{n \pi x}{a} \right)[/tex]

or

[tex] u_n = A_n \cos\left(\frac{n \pi x}{a} \right) [/tex]

whether n is even or odd.

[tex] p^3 = (- i \hbar \frac{d}{dx})^3 = i \hbar^3 \frac{d^3}{dx^3} [/tex]

## The Attempt at a Solution

The integral is over a symmetric region so it should vanish if the integrand is an odd function of x.

Starting with the odd eigenfuctions of the infinite well

[tex] x^2 p^3 sin(x) = x^2 i \hbar^3 \frac{d^3}{dx^3} \sin(x) = - i \hbar^3 x^2 \cos(x) [/tex]

which is an even function, so its integral from -a to a does not vanish

[tex] 3 x p^3 x \sin(x) = 3 x i \hbar^3 \frac{d^3}{dx^3}(x \sin(x)) [/tex]

x*sin(x) is a product of two odd functions so it would be an even function. The derivative of an even function is an odd function, so its second derivative should be an even function, and the third derivative will be an odd function again. But then this third derivative function is multiplied to the right by x, and the product of two odd functions is again an even function, So the integral does not vanish.

[tex] p^3 x^2 \sin(x) i \hbar^3 \frac{d^3}{dx^3}(x^2 \sin(x)) [/tex]

x^2*sin(x) is the product of an even function with an odd function, so it's an odd function. The first derivative is an even function, the second derivative is an odd function, and the third derivative will once again be even so the integral does not vanish.

For the even eigenfunctions, that are something like A*cos(b*x), I can make the same argument, only replacing every "even" with "odd" and the expectation value would indeed vanish. But for the odd eigenfunctions that go like A*sin(b*x) this does not happen. How do I prove that the expectation value does vanish for eigenfunctions like sin(n*pi*x/a)?