- #1
xWaffle
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Homework Statement
[itex]\Psi (x) = C e^{i k_{0} x} e^{\frac{-x^{2}}{2 a^{2}}}[/itex]
Find [itex]\left\langle x \right\rangle, \left\langle x^{2} \right\rangle, \left\langle p \right\rangle, \left\langle p^{2} \right\rangle.[/itex]
Homework Equations
Operators make a "psi-sandwich":
[itex]\left\langle x \right\rangle = \int_{- \infty}^{\infty} \Psi^{*} \left( x)\right) \Psi[/itex]
[itex]\left\langle x^{2} \right\rangle = \int_{- \infty}^{\infty} \Psi^{*} \left( x^{2}\right)\Psi[/itex]
[itex]\left\langle p \right\rangle = \int_{- \infty}^{\infty} \Psi^{*} \left( \frac{\hbar}{i}\frac{\partial}{\partial x}\right)\Psi[/itex]
[itex]\left\langle p^{2} \right\rangle = \int_{- \infty}^{\infty} \Psi^{*} \left( \frac{\hbar}{i} \frac{\partial}{\partial x}\right)^{2}\Psi[/itex]
The Attempt at a Solution
I found <x> to be zero, because of an odd-integrand. (Multiplying psi by psi* in the integrand to simplify). I did the same thing for <x2>, except the integrand was not odd this time because of the x^2 in there. Using a special Gaussian integral formula listed in my book, I was also easily able to find <x2> = x2 / 2.
Since <p> is just m*(d<x> / dt), <p> is also zero in this case.
For <p^2>, I take the second derivative of psi, and multiply this result by psi*.
But when finalizing all the derivatives and simplification in the integrand for <p2>, there is always an i left over (I don't think there should be an i in the momentum..), and the integral is not easily doable (where the professor has said all integrals needed will be listed in the back cover of our book).
So I must conclude that my approach is fundamentally wrong, even though I'm using the equations I have been given.
I have found in a separate part of the question the constant [itex]C = \sqrt{\frac{1}{\sqrt{\pi} a}}[/itex]
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