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$$\hbar$$np^n-1

= f_ni$$\hbar$$np^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$$\hbar$$np^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 :)