# Math Help - Qm #2

1. ## Qm #2

1.) By using the commutator $[x,p] = i\bar{h}$ calculate the commutator: $[xp^2, px^2]$

Note that the commutator xp - px is denoted by the commutator bracket [x,p], etc.

2.) Verify the following properties of commutators:

a.) $[\hat{A},\hat{B}] = -[\hat{B},\hat{A}]$

b.) $[\hat{A},\hat{B_1} + \hat{B_2}] = [\hat{A},\hat{B_1}] + [\hat{A}, \hat{B_2}]$

c.) $[\hat{A}\hat{B},\hat{C}] = [\hat{A},\hat{C}]\hat{B} + \hat{A}[\hat{B},\hat{C}]$

d.) $[\hat{A},\hat{B}\hat{C}] = [\hat{A},\hat{B}]\hat{C} + \hat{B}[\hat{A},\hat{C}]$

2. Originally Posted by DiscreteW
1.) By using the commutator $[x,p] = i\bar{h}$ calculate the commutator: $[xp^2, px^2]$

Note that the commutator xp - px is denoted by the commutator bracket [x,p], etc.

2.) Verify the following properties of commutators:

a.) $[\hat{A},\hat{B}] = -[\hat{B},\hat{A}]$

b.) $[\hat{A},\hat{B_1} + \hat{B_2}] = [\hat{A},\hat{B_1}] + [\hat{A}, \hat{B_2}]$

c.) $[\hat{A}\hat{B},\hat{C}] = [\hat{A},\hat{C}]\hat{B} + \hat{A}[\hat{B},\hat{C}]$

d.) $[\hat{A},\hat{B}\hat{C}] = [\hat{A},\hat{B}]\hat{C} + \hat{B}[\hat{A},\hat{C}]$

3. For #1 you might find these theorems helpful:
$\left [ x , G(p) \right ] = i \hbar ~ \frac{dG}{dp}$

and
$\left [ F(x) , p \right ] = i \hbar ~ \frac{dF}{dx}$

I couldn't find a link to provide an explanation, but you should be able to find a derivation in any entry level QM text.

-Dan