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I'm having trouble with quantum harmonic oscillators and I'm not sure how to approach these questions:

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I'd really like to get my head around these concepts but I'm struggling to understand fully. Could somebody please briefly explain the method I should be using to tackle these problems? Any help would be greatly appreciated.

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  • $\begingroup$ For the speculation in (c), you might start by considering this anharmonic oscillator in classical mechanics. Draw a graph of its potential energy function, and consider how it would move if it were started with initial velocity $0$ and initial position rather negative. $\endgroup$ – Andreas Blass May 5 '15 at 14:37
  • $\begingroup$ Do you know how x and p operators are expressed from creation and annihilation operators? $\endgroup$ – Incnis Mrsi May 5 '15 at 19:54
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$\newcommand{\ket}[1]{\left| #1 \right>}$ $\newcommand{\ad}[0]{\hat{a}^\dagger}$ $\newcommand{\ao}[0]{\hat{a}}$

Firs of all I'd like to tell you that the position operator $x$ is given in terms of the ladder operator by the following relation:

$$\hat x = d (\ao+\ad )$$, where $d=\sqrt{\frac{\hbar}{2m \omega}}$.

Computing $\hat{x}^3$ gives then, as you can easily verify by expanding out $(\ao+\ad)\cdot(\ao+\ad )\cdot (\ao+\ad )$ the following:

$$\hat x^3 = d^3(\ao \ao \ao +\ao \ao \ad + \ao \ad \ao + \ao \ad \ad + \ad \ao \ao + \ad \ao \ad + \ad \ad \ao + \ad \ad \ad)$$

Since I found the notation $:H:$ to be confusing instead I would like to write in calligraphic notation ie I define $\mathcal{H} \equiv \; :H:$ .

$\Delta= \mathcal{H}_{an-harm}-H_{an-harm}$ is then fairly easy to calculate. Notice that $\mathcal{X}^3$ is given by binomial theorem since you don't really care in which order you write the operators as long as $\ad$ is on the far left side. Notice also that $\mathcal{H}_{harm}=H_{harm}$ then it follows that

$$\Delta = \eta d^3(-\ao \ao \ad - \ao \ad \ao - \ao \ad \ad + 2 \ad \ao \ao - \ad \ao \ad + 2 \ad \ad \ao ) $$

You may want to tidy up $\Delta$ by using the commutator identities if you want to have really good answer but I don't find it too necessary to do it for this particular problem.

For the second part you can calculate $\Delta \ket{0}$ and $\mathcal{H}_{an-harm} \ket {0}$ if you remember that $\ao \ket{0} = 0$ and $\ket{n}= \frac{1}{\sqrt{n!} } (\ad)^n \ket{0}$. I'll leave this part to you.

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  • $\begingroup$ Wow. Thank you so much. I appreciate the effort put into this post and this certainly clears a lot up. Thanks again, this is much more clear now! $\endgroup$ – Ben Gerry May 8 '15 at 23:39

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