Proving the Commutator Relationship with Power Series Expansion | Homework Help

[cahill8]

Homework Statement

Show \left[x,f(p)\right)] = i\hbar\frac{d}{dp}(f(p))\right.

Homework Equations

I can use \left[x,p^{n}\right)] = i\hbar\\n\right.p^{n}\right.
f(p) = \Sigma f_{n}p^{n} (power series expansion)

The Attempt at a Solution

I started by expanding f(p) to the power series which makes

\left[x,\Sigma\\f_{n}\\p^{n}\right)]

and I know I must use the commutator identity [A, BC] = [A,B]C + B[A,C]
but the power series cannot be split up into two products(BC) ? So I'm not sure how to go on

 

[Hurkyl]

and I know I must use the commutator identity [A, BC] = [A,B]C + B[A,C]

How do you know that?

 

[cahill8]

In a textbook it says it can be shown using that equation

Trying a different method:

[x, f(p)] = [x,\sum_{n}\\f_{n}p^{n}] = [x,f_npⁿ + \sum_{n-1}\\f_{n}p^{n}]

using [A, B+C] = [A,B] + [A,C]

= [x, f_npⁿ] + [x, \sum_{n-1}\\f_{n}p^{n}]

using [A, BC] = C[A,B] + B[A,C]

= f_n[x, pⁿ] + pⁿ[x, f_n] + [x, \sum_{n-1}\\f_{n}p^{n}]

using [x, pⁿ] = i\hbarnp^n-1

= f_ni\hbarnp^n-1 + pⁿ[x, f_n] + [x, \sum_{n-1}\\f_{n}p^{n}]

[x, f_n] = 0 as f_n is a const.

= f_ni\hbarnp^n-1 + [x, \sum_{n-1}\\f_{n}p^{n}]

am I on the right track?

 

[Hurkyl]

I'm curious why you used

[A, rC] = [A,r]C + r[A,C]​

to pull out a scalar, rather than just using

[A, rC] = r [A,C]​

I'm also curious why you stopped using

[A, B + C] = [A,B] + [A,C]​

after a single addition.

But that aside, everything you wrote looks correct. We won't know if you're on the right track until we see where this path leads, though!

 

[cahill8]

I see what you mean. [x, f_npⁿ] = f_n[x, pⁿ] is fine.

I kept going with the addition and noticed a pattern and managed to solve it. Thanks for the hints :)