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This seems like a really straightforward problem that nevertheless stumps me.

We have $\frac{i}{\hbar}[p,px^n] = \frac{i}{\hbar}(p[p,x^n] + [p,p]x^n)$

Now I have previously found that $[p,x^n] = - i \hbar n x^{n-1}$ and we know that $[p,p] = 0$ by axiom.

So we should have $\frac{i}{\hbar}(p[p,x^n] + [p,p]x^n) = \frac{i}{\hbar}p(-i \hbar n x^{n-1}) = pnx^{n-1}.$ But I know this cannot be the right answer because it should not be the partial derivative of $px^n$ with respect to x. I'm obviously missing something, but what?

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closed as off-topic by Kyle Kanos, sammy gerbil, Qmechanic Sep 16 '17 at 7:02

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It looks right to me. A function $f$ satisfies $\frac{i}{\hbar}[p,\,f]=[\partial_x,\,f]=\partial_x f$ since $[\partial_x,\,f]g=\partial_x (fg)-f\partial_x g = (\partial_x f)g$.

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  • $\begingroup$ The answer to the question should supposedly show that in general, $\frac{i}{\hbar}[p,G(p,x)] \neq \frac{\partial}{\partial x} G(p,x)$. As opposed to if G is a function of x only. Also, the practice system I feed the answers into doesn't accept it... $\endgroup$ – LC Nielsen Sep 13 '17 at 13:02
  • $\begingroup$ Got it - all I needed to do was to expand to $- i \hbar n (n-1) x ^{n-2} + n x ^{n-1} p$... I feels silly now. $\endgroup$ – LC Nielsen Sep 13 '17 at 14:13
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In general, for any function of $f(x)$, the action of $\hat x$ and $\hat p$ is given as $$ \hat xf(x)=xf(x)\, ,\qquad \hat pf(x)=-i\hbar \partial_xf(x)=-i\hbar f'(x) $$ so, for $\hat p\hat x^n$, you simply apply in proper sequence.

Thus \begin{align} [p,px^n]f(x)&=-i\hbar \partial_x (-i\hbar \partial_x) x^n f(x) - (-i\hbar \partial_x)x^n(-i\hbar \partial_x)f(x)\, ,\\ &=-\hbar^2 \partial_x^2(x^n f(x))+\hbar^2(\partial_x x^n f'(x))\, . \end{align} Since you can use the replacements $$ f'(x)\to \frac{i}{\hbar}\hat pf(x)\qquad f''(x)\to -\frac{1}{\hbar^2}\hat p^2f(x) $$ and $f(x)$ is arbitary, you should be able to eliminate $f(x)$ at the very end to get a "clean" answer in terms of operators only (which will not be a simple derivative.)

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  • $\begingroup$ Aha, so let's see... $\begin{align} -\hbar^2 \partial_x^2(x^n f(x))+\hbar^2(\partial_x x^n f'(x))\, . \end{align}$ Well, we have $f'(x) = \partial_x f(x)$ so the answer should be... $\hbar^2(-\partial_x^2 x^n + \partial_x x^n \partial_x) = \hbar^2(-n (n-1) x^{n-2} + nx^{n-1} \partial_x)$ Is that right? $\endgroup$ – LC Nielsen Sep 13 '17 at 13:17
  • $\begingroup$ You can further simplify but you get the idea. $\endgroup$ – ZeroTheHero Sep 13 '17 at 13:20
  • $\begingroup$ It actually will be the simple derivative. $n p x^{n-1}$ I mean $\endgroup$ – OON Sep 13 '17 at 13:20
  • $\begingroup$ All I needed to do was expand it to $- i \hbar n (n-1) x ^{n-2} + n x ^{n-1} p$... I feel really silly now. $\endgroup$ – LC Nielsen Sep 13 '17 at 14:18
  • $\begingroup$ when you don't know it's more formidable than it looks. Best of luck. $\endgroup$ – ZeroTheHero Sep 13 '17 at 14:22

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