Kinetic energy of harmonic oscilator

[stunner5000pt]

[SOLVED] Kinetic energy of harmonic oscilator

Homework Statement

Find the expectation value of the kinetic energy of the nth state of a Harmonic oscillator

Homework Equations

<T> = \frac{<p^2>}{2m}
p_{x} = \frac{1}{i} \sqrt{\frac{m\hbar\omega}{2}} (\hat{a} -\hat{a}^\dagger)
a^\dagger \psi_{n} = \sqrt{n+1} \psi_{n+1}
a\psi_{n} = \sqrt{n-1} \psi_{n-1}

The Attempt at a Solution

So p_{x}^2 = \left(\frac{1}{i} \sqrt{\frac{m\hbar\omega}{2}} (\hat{a} -\hat{a}^\dagger)\right)^2


So to calculate the <T> do i just do this:
&lt;T&gt; = &lt;\Psi_{n}|\frac{p^2}{2m}|\Psi(n)&gt;
&lt;T&gt; = -\frac{\hbar\omega}{4} \int \psi_{n}^* (aa - aa^\dagger - a^\dagger a + a^\dagger a^\dagger) \psi_{n} dx

&lt;T&gt; = -\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

only one term will survive, the nth state ones because the wave functions are orthogonal
&lt;T&gt; = -\frac{n\hbar\omega}{4}

Is this correct??

Thanks for your help!

 

[malawi_glenn]

Negative Kinetic energy is not allowed.

In your final integral, two of four terms will survive.

And also:
a\psi_{n} = \sqrt{n-1} \psi_{n-1}
should be:
a\psi_{n} = \sqrt{n} \psi_{n-1}

 

[stunner5000pt]

thanks i got the required answer!
where do i mark this solved?