How to Solve Commutators Using the Jacobian?

[Rorshach]

Then it is $\hat{x}$, but I don't know its formula, or if it even has one besides $\hat{x}$

 

[vela]

In the coordinate basis, which is the basis you're working in, the operator $\hat{x}$ when acting on a function $f(x)$ simply multiplies that function by $x$. That is to say, $\hat{x}f(x) = xf(x)$.

 

[Rorshach]

So in this case it does nothing that would influence its right side, except for multiplication, right? Then I only see multiplication of two operators, $\hat{p}x$, which is just $-i\hbar$, and $x\hat{p}\psi$. What can I do with this?

 

[vela]

No. Say A and B are matrices and x is a vector. What you keep doing is equivalent to saying ABx = (Ax)(Bx). That's not how it works.

Just work right to left and remember that $\hat{p}$ acts on everything to its right.

 

[Rorshach]

the only thing I see is $-i\hbar\frac{∂x}{∂x}\frac{∂ψ}{∂x}$. Where am I making mistake?

 

[vela]

$\hat{p}$ acts on everything to its right.

 

[Rorshach]

$-i\hbar\frac{∂x\frac{∂\psi}{∂x}}{∂x}$?

 

[Rorshach]

Gosh, I finally understood my mistake:P I just treated the whole thing as simple standard multiplication, just like I should do in normal mathematical case, however in case of operators I should just multiply every operator by the variable on its right side:D In normal situation it seems illogical, looking like this: $(A+B)CD=(AD+BD)CD$, but in case of operators it should look like this $(-i\hbar\frac{\partial }{\partial x}x+i\hbar\frac{\partial}{\partial x})(-i\hbar\frac{\partial }{\partial x})\psi=(-i\hbar(x\frac{\partial \psi}{\partial x}+\psi\frac{\partial x}{\partial x})+i\hbar\frac{\partial \psi}{\partial x})(-i\hbar\frac{\partial \psi}{\partial x})=(-i\hbar x\frac{\partial \psi}{\partial x}-i\hbar\psi i\hbar x\frac{\partial \psi}{\partial x})(-i\hbar\frac{\partial \psi}{\partial x})=(-i\hbar\psi)(-i\hbar\frac{\partial \psi}{\partial x})=-\hbar^2\psi \frac{\partial \psi}{\partial x}=-\hbar^2\frac{\partial }{\partial x}$

 

[Rorshach]

Unfortunately, I still do not know how to solve it using Jacobi identity... I am not even sure if I interpret it correctly. I tried to use the commutator $[p^2_x,x]=[p_x p_x,x]$, use the given fact $[p_x,x]=-i\hbar$, or $[x,p_x]=i\hbar$ and substitute it to Jacobi identity formula $[p_x,[p_x,x]]+[p_x,[x,p_x]]+[x[p_x,p_x]]=0$, but those commutators zero themselves even before I sum them up. How to do this?

 

[vela]

Gosh, I finally understood my mistake:P I just treated the whole thing as simple standard multiplication, just like I should do in normal mathematical case, however in case of operators I should just multiply every operator by the variable on its right side:D In normal situation it seems illogical, looking like this: $(A+B)CD=(AD+BD)CD$, but in case of operators it should look like this $(-i\hbar\frac{\partial }{\partial x}x+i\hbar\frac{\partial}{\partial x})(-i\hbar\frac{\partial }{\partial x})\psi=(-i\hbar(x\frac{\partial \psi}{\partial x}+\psi\frac{\partial x}{\partial x})+i\hbar\frac{\partial \psi}{\partial x})(-i\hbar\frac{\partial \psi}{\partial x})$

Unfortunately, you haven't understood your mistake. You're still doing the same thing you did before. It doesn't work at all like (A+B)CD = (AD + BD)CD.

After you apply the rightmost $\hat{p}$, you have $\hat{p}\hat{x}\,\left(-i\hbar \frac{\partial\psi}{\partial x}\right)$. Now $-i\hbar\frac{\partial\psi}{\partial x}$ is just another function. Let's call it $f$. So now you have $\hat{p}\hat{x}f$. What is $\hat{x}f$ equal to? Call that result $g$. Then apply $\hat{p}$ to $g$. What do you get?

 

[Rorshach]

$-i\hbar\frac{\partial g}{\partial x}$?

 

[vela]

Ignoring the constants, yes, that's what you get. So what's the derivative of g equal to?

 

[Rorshach]

$-i\hbar\frac{\partial \psi}{\partial x}$?

 

[vela]

No. What is $g$ equal to?

 

[Rorshach]

$-i\hbar x\frac{\partial x}{\partial x}$

 

[Rorshach]

Oh right, I confused just derivative with applying the operator:P

 

[Rorshach]

it came down to $-i\hbar(\frac{\partial g}{\partial x})=-i\hbar (-i\hbar\frac{\partial \psi}{\partial x})$, which is basically a half of the result, so it is correct, right?

 

[vela]

$-i\hbar x\frac{\partial x}{\partial x}$

That's not correct. Where's you get $\frac{\partial x}{\partial x}$ from?

 

[Rorshach]

I just made a mistake while writing the result in code, it should be $-i\hbar x\frac{\partial \psi}{\partial x}$

 

[vela]

OK, so what do you get when you differentiate that?

 

[Rorshach]

$-i\hbar\frac{\partial \psi}{\partial x}$

 

[vela]

How'd you get that?

 

[Rorshach]

I differentiated it by x, but from my experience I again made a mistake somewhere...

 

[vela]

What if you wrote $g=xf$? What do you get when you differentiate g?

 

[Rorshach]

oh shoot... it is $x\frac{\partial f}{\partial x}+f$... but it only complicates the matter, now I have $-i\hbar(x\frac{\partial f}{\partial x}-i\hbar\frac{\partial \psi}{\partial x})$

 

[vela]

That's only the first term (with some typos, I hope). You still have to subtract off the result from the xp term.

 

[Rorshach]

Now I'm completely lost. I repeated that calculation and got the same result, I cannot see the mistake. I thought that xp term was included in that $-i\hbar\frac{\partial g}{\partial x}$ combination?

 

[Rorshach]

my mistake again, forgot about the second term in the bracket. But I cannot use g substitution, since x is on the left side of the operator, I can only use f. What can I do with this?

 

[vela]

Try starting over from the beginning now. You began with
$$[p^2,x]\psi = (p[p,x]+[p,x]p)\psi = p[p,x]\psi + [p,x]p\psi.$$ Now redo your calculations for $p[p,x]\psi$ and $[p,x]p\psi$.

 

[Rorshach]

the result for the first term was correct, it came out as $-\hbar^2\psi\frac{\partial \psi}{\partial x}$. I just have problem with the second term, because I can't think of the solution with changed order of the operator...how to bite this?

 

[Rorshach]

I think I finally did it. Here it goes: $-i\hbar\frac{\partial }{\partial x}(-i\hbar\frac{\partial x\psi}{\partial x}+i\hbar x\frac{\partial \psi}{\partial x})+(-i\hbar\frac{\partial xf}{\partial x}+i\hbar x\frac{\partial f}{\partial x})$, where we let $f=-i\hbar\frac{\partial \psi}{\partial x}$. Then it goes on:
$-i\hbar\frac{\partial }{\partial x}(-i\hbar(x\frac{\partial \psi}{\partial x}+\psi\frac{\partial x}{\partial x})+i\hbar x\frac{\partial \psi}{\partial x})+(-i\hbar(x\frac{\partial f}{\partial x}+f\frac{\partial x}{\partial x})+i\hbar x\frac{\partial f}{\partial x})$, proceeding
$-i\hbar\frac{\partial }{\partial x}(-i\hbar x\frac{\partial \psi}{\partial x}-i\hbar\psi+i\hbar x\frac{\partial \psi}{\partial x})+(i\hbar x\frac{\partial f}{\partial x}-i\hbar f+i\hbar x\frac{\partial f}{\partial x})$, and then
$-\hbar^2\frac{\partial \psi}{\partial x}+(-i\hbar f)=-\hbar^2\frac{\partial \psi}{\partial x}-(-i\hbar\frac{\partial \psi}{\partial x})=-\hbar^2\frac{\partial \psi}{\partial x}-\hbar^2\frac{\partial \psi}{\partial x}=-2\hbar\frac{\partial \psi}{\partial x}$.
Please tell me that this time I am right.

 

[vela]

Just a few minor errors, probably typos, and you're not quite done.

I think I finally did it. Here it goes:
$$-i\hbar\frac{\partial }{\partial x}(-i\hbar\frac{\partial x\psi}{\partial x}+i\hbar x\frac{\partial \psi}{\partial x})+(-i\hbar\frac{\partial xf}{\partial x}+i\hbar x\frac{\partial f}{\partial x}),$$ where we let $f=-i\hbar\frac{\partial \psi}{\partial x}$. Then it goes on:
$$-i\hbar\frac{\partial }{\partial x}(-i\hbar(x\frac{\partial \psi}{\partial x}+\psi\frac{\partial x}{\partial x})+i\hbar x\frac{\partial \psi}{\partial x})+(-i\hbar(x\frac{\partial f}{\partial x}+f\frac{\partial x}{\partial x})+i\hbar x\frac{\partial f}{\partial x})$$
$$-i\hbar\frac{\partial }{\partial x}(-i\hbar x\frac{\partial \psi}{\partial x}-i\hbar\psi+i\hbar x\frac{\partial \psi}{\partial x})+({\color{red}-}i\hbar x\frac{\partial f}{\partial x}-i\hbar f+i\hbar x\frac{\partial f}{\partial x})$$
$$-\hbar^2\frac{\partial \psi}{\partial x}+(-i\hbar f)
= \hbar^2\frac{\partial \psi}{\partial x}-{\color{red} {(i\hbar)}}(-i\hbar\frac{\partial \psi}{\partial x})
= -\hbar^2\frac{\partial \psi}{\partial x}-\hbar^2\frac{\partial \psi}{\partial x}
= -2\hbar^{\color{red}2} \frac{\partial \psi}{\partial x}$$.
Please tell me that this time I am right.

Now use the fact that $-\hbar^2 = (-ih)^2$ and rewrite the final result in terms of $\hat{p}_x$ to show that $[\hat{p}_x^2,\hat{x}] = (-i\hbar)2\hat{p}_x$.

 

[Rorshach]

I don't quite understand what do You mean by presenting it in terms of $p_x$, I thought that my calculations (without those stupid typos) are enough to show the correct result?

 

[vela]

Well, ask yourself what you're trying to show. The problem asks you to calculate $[\hat{p}_x^2,\hat{x}]$ which is equal to $-2i\hbar \hat{p}_x$. You haven't shown that. You've shown that when you work in the position basis, the action of $[\hat{p}_x^2,\hat{x}]$ on some arbitrary function $\psi(x)$ is to differentiate it and multiply it by $-2\hbar^2$. You need to show that that's equivalent to applying $\hat{p}$ and multiplying by $-2i\hbar$.

 

[Rorshach]

So basically I should take function $\psi(x)$, act on it with $\hat{p}$ and multiply it by $-2i\hbar$?