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stunner5000pt
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[SOLVED] Kinetic energy of harmonic oscilator
Find the expectation value of the kinetic energy of the nth state of a Harmonic oscillator
[tex] <T> = \frac{<p^2>}{2m} [/tex]
[tex] p_{x} = \frac{1}{i} \sqrt{\frac{m\hbar\omega}{2}} (\hat{a} -\hat{a}^\dagger)[/tex]
[tex] a^\dagger \psi_{n} = \sqrt{n+1} \psi_{n+1} [/tex]
[tex] a\psi_{n} = \sqrt{n-1} \psi_{n-1} [/tex]
So [tex] p_{x}^2 = \left(\frac{1}{i} \sqrt{\frac{m\hbar\omega}{2}} (\hat{a} -\hat{a}^\dagger)\right)^2[/tex]
So to calculate the <T> do i just do this:
[tex] <T> = <\Psi_{n}|\frac{p^2}{2m}|\Psi(n)> [/tex]
[tex] <T> = -\frac{\hbar\omega}{4} \int \psi_{n}^* (aa - aa^\dagger - a^\dagger a + a^\dagger a^\dagger) \psi_{n} dx [/tex]
[tex] <T> = -\frac{\hbar\omega}{4} \int\psi_{n}^* \sqrt{n(n-1)} \psi_{n-2} + \sqrt{n(n+1)}\psi_{n} + \sqrt{n^2} \psi_{n} + \sqrt{(n+1)(n+2)} \psi_{n+2} dx [/tex]
only one term will survive, the nth state ones because the wave functions are orthogonal
[tex] <T> = -\frac{n\hbar\omega}{4} [/tex]
Is this correct??
Thanks for your help!
Homework Statement
Find the expectation value of the kinetic energy of the nth state of a Harmonic oscillator
Homework Equations
[tex] <T> = \frac{<p^2>}{2m} [/tex]
[tex] p_{x} = \frac{1}{i} \sqrt{\frac{m\hbar\omega}{2}} (\hat{a} -\hat{a}^\dagger)[/tex]
[tex] a^\dagger \psi_{n} = \sqrt{n+1} \psi_{n+1} [/tex]
[tex] a\psi_{n} = \sqrt{n-1} \psi_{n-1} [/tex]
The Attempt at a Solution
So [tex] p_{x}^2 = \left(\frac{1}{i} \sqrt{\frac{m\hbar\omega}{2}} (\hat{a} -\hat{a}^\dagger)\right)^2[/tex]
So to calculate the <T> do i just do this:
[tex] <T> = <\Psi_{n}|\frac{p^2}{2m}|\Psi(n)> [/tex]
[tex] <T> = -\frac{\hbar\omega}{4} \int \psi_{n}^* (aa - aa^\dagger - a^\dagger a + a^\dagger a^\dagger) \psi_{n} dx [/tex]
[tex] <T> = -\frac{\hbar\omega}{4} \int\psi_{n}^* \sqrt{n(n-1)} \psi_{n-2} + \sqrt{n(n+1)}\psi_{n} + \sqrt{n^2} \psi_{n} + \sqrt{(n+1)(n+2)} \psi_{n+2} dx [/tex]
only one term will survive, the nth state ones because the wave functions are orthogonal
[tex] <T> = -\frac{n\hbar\omega}{4} [/tex]
Is this correct??
Thanks for your help!