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## Homework Statement

Find the momentum-space wave function, [tex]\Phi (p,t)[/tex], for a particle in the ground state of the harmonic oscillator. What is the probability (to 2 significant digits) that a measurement of [tex]p[/tex] on a particle in this state would yield a value outside the classical range (for the same energy)?

*Hint:*Look in a math table under "Normal Distribution" or "Error Function" for the numerical part -- or use Mathematica.

(Problem 3.11 - Intro to QM, 2nd Edition, by Griffiths)

## Homework Equations

[tex]\Phi (p,t) = \frac{1}{\sqrt{2\pi \hbar}} \int^{\infty}_{-\infty} exp[{\frac{-ipx}{\hbar}}] \Psi (x,t) dx[/tex]

## The Attempt at a Solution

The above equation is the only one that I know should be right. That being said, here is what I've tried so far:

[tex]\psi_{o} (x,t) = (\frac{m\omega}{\pi \hbar})^{\frac{1}{4}} exp[{\frac{-m\omega x^{2}}{2\hbar}}] exp[{\frac{-iE_{o}t}{\hbar}}][/tex]

With [tex]E_{o} = \frac {\hbar \omega}{2}[/tex]

[tex]\psi_{o} (x,t) = (\frac{m\omega}{\pi \hbar})^{\frac{1}{4}} exp[{\frac{-m\omega x^{2}-i\hbar \omega t}{2\hbar}}][/tex]

[tex]\Phi (p,t) = \frac{1}{\sqrt{2\pi \hbar}} \int^{\infty}_{-\infty}exp[{\frac{-2ipx -m\omega x^{2} -i\hbar \omega t}{2\hbar}][/tex].

And if that's right, then I'm not sure how to integrate that. Then after I get what that equals (lets call it ANS), then to find the probability outside the classical value, I need to square [tex]\Phi (p,t)[/tex] right? I'm guessing that I would have something like:

[tex]2\int^{\infty}_{ClassicalValue}ANS dp[/tex]

I'm taking 2 times the integral because I'm guessing since it's an even function that I could take the negative classical value to -infinity as well; but it would equal this. But I'm not sure what to use for the classical value because of how the problem is worded. Am I on the right track?