Introductory quantum mechanics problem

[phys-student]

Homework Statement

Consider A(x) is an arbitrary function of x, and p_x is the momentum operator. Show that they satisfy the following condition:
[p_x,A(x)] = (-i/ħ)*d/dx(A(x))

where [p_x,A(x)] = p_xA(x) - A(x)p_x

Homework Equations

ħ = h/2π
p_x = (-iħ)d/dx

The Attempt at a Solution

Starting with the expression given at the bottom of the question and subbing in the expression for the momentum operator and factoring out -iħ, I get:

[p_x,A(x)] = -iħ*(d/dx(A(x)) - A(x)d/dx)

and I'm stuck there for the time being... I don't really understand how to get ħ into the denominator or what to do with the term in brackets. Any help would be appreciated.

 

[TSny]

Homework Statement

Consider A(x) is an arbitrary function of x, and p_x is the momentum operator. Show that they satisfy the following condition:
[p_x,A(x)] = (-i/ħ)*d/dx(A(x))

There must be a misprint in the problem as given to you. The ħ should be in the numerator. You can verify this by dimensional analysis.

3. The Attempt at a Solution

Starting with the expression given at the bottom of the question and subbing in the expression for the momentum operator and factoring out -iħ, I get:

[p_x,A(x)] = -iħ*(d/dx(A(x)) - A(x)d/dx)

When unraveling commutators like this, it is usually a good idea the think of the commutator as acting on some arbitrary function f(x).

Thus, try to simplify [p_x, A(x)]f(x).

 

[DuckAmuck]

h-bar should not be in the denominator. Do a simple unit analysis to see that's not the case.

As for evaluating the commutator, I think it might help if you put the commutator in front of function. Remember p and A are operators, they need to operate on something.

$$[p,A]\psi = (pA - Ap)\psi = pA\psi - Ap\psi$$

Now A and psi are just functions of position. But p is a derivative operator acting on everything to its right. Recall the product rule.

$$pA\psi = p(A)\psi + Ap(\psi)$$

I'll let you do the rest. Hopefully this makes sense.

 

[phys-student]

Thanks to both of you for the help. I've done as you suggested and solved the commutator and found that it was equal to dA(x)/dx, now the answer I have is:

[px,A(x)] = -iħ*(dA(x)/dx)

Another source I've seen online seems to suggest that the first term should be ħ/i (which is equal to my answer -iħ) I will ask my professor whether there is a typo in the assignment tomorrow.