Difference between expectation value of ##x## and classical amplitude of oscillation for an harmonic oscillator

[Gabri110]

Homework Statement

The oscillator is in the state $\lvert \psi (t)\rangle = \dfrac{1}{\sqrt{2}} \left( e^{-i (n-\frac{1}{2})\omega t}\lvert n-1 \rangle + e^{-i (n+\frac{1}{2})\omega t}\lvert n \rangle \right)$.

Calculate the amplitude of oscillation of a classical oscillator of this frequency and energy $E = \langle\psi (t)\rvert H \lvert\psi (t)\rangle$ and show that it differs from your result for $\langle\psi (t)\rvert x \lvert\psi (t)\rangle$ by a factor independent of $n$.

Relevant Equations

$\lvert \psi (t)\rangle = \dfrac{1}{\sqrt{2}} \left( e^{-i (n-\frac{1}{2})\omega t}\lvert n-1 \rangle + e^{-i (n+\frac{1}{2})\omega t}\lvert n \rangle \right)$

Using the ladder operators I can easily compute $E = \langle H\rangle = \hbar \omega n$, so I can find the amplitude of the classical oscillator, as $E = \frac{1}{2} m \omega^2 x_{max}^2$, thus, $x_{max} = \sqrt{\dfrac{2 E}{m \omega^2}} = \sqrt{\dfrac{2\hbar n}{m \omega}}$.

The expectation value of $x$ can be also easily computed using the ladder operators. I find $\langle x\rangle = \sqrt{\dfrac{2\hbar n}{m \omega}}\cos{\omega t}$. This is clearly a problem, as I find that $\langle x\rangle$ is time dependent (and the classical solution isn't!). The difference is $x_{max} - \langle x\rangle = \sqrt{\dfrac{2\hbar n}{m \omega}} (1 - \cos{\omega t})$, which isn't independent of $n$, as the exercise statement says.

Can someone help me find where I have made a mistake?

 

[vela]

Reread the question. What does “factor” mean?

 

[Gabri110]

Reread the question. What does “factor” mean?

Oh my... Thank you, I feel so dumb right now...

 

[TSny]

I find $\langle x\rangle = \sqrt{\dfrac{2\hbar n}{m \omega}}\cos{\omega t}$.

I got a slightly different result for the expression inside the square root. Of course, I might be the one making a mistake. But I calculated it two ways: using the ladder operators and using the known wavefunctions for the harmonic oscillator.

Also, since $\langle x\rangle$ oscillates harmonically, I wonder if it would be more appropriate to compare the amplitude, $x_{max}$, of the classical oscillator with the amplitude of the $\langle x\rangle$ oscillation. I'm not sure.

 

[Gabri110]

Yeah, my bad, I had forgotten to divide by 2 the ladder operators...