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**Quantum Theory: Commutators of Functions of Observables**

## Homework Statement

First is a question from Sakurai

*Modern Quantum Mechanics, 2nd ed.*, 1.29a.

Show that

[tex][x_i,G(\mathbf{p})] = i\hbar\frac{\partial G}{\partial p_i}[/tex]

and

[tex][p_i,F(\mathbf{x})] = - i\hbar\frac{\partial F}{\partial x_i}[/tex]

for any functions [itex]F(\mathbf{x})[/itex] and [itex]G(\mathbf{p})[/itex] which can be expanded in power series of their arguments.

## Homework Equations

Definition of a Taylor series of a function [itex]f[/itex] of variable [itex]x[/itex] expanded around point [itex]a[/itex]:

[tex] f(x) = \sum^{\infty}_{n=0} \frac{f^{(n)}(a)}{n!}(x-a) [/tex]

Commutator of position and momentum operators: [itex][x_i,p_j]=i\hbar\delta_{ij}[/itex].

## The Attempt at a Solution

I tried a general solution, i.e. looking at the commutator [itex][F(\mathbf{x}), G(\mathbf{p})][/itex]. The first problem I'm having is with the concept of taking derivatives with respect to operators. Can I simply treat [itex]x[/itex] as a scalar while computing [itex]\frac{\partial F}{\partial x}[/itex], for example? Is

[tex]\frac{\partial}{\partial x}(x^2) = 2x[/tex]

valid?

Assuming I have that right, I am really stuck on the Taylor expansion itself. I don't see why powers of higher than first order would disappear. I assume my solution will look something like

[tex][F(\mathbf{x}),G(\mathbf{p})] = [\mathbf{x},\mathbf{p}]\frac{\partial F}{\partial x_i}\frac{\partial G}{\partial p_i}[/tex]

which would satisfy the problem, but I don't know how to get there.

Thank you!

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