# Thread: Qm #2

1. ## Qm #2

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

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

2.) Verify the following properties of commutators:

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

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

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

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

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

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

2.) Verify the following properties of commutators:

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

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

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

d.) $\displaystyle [\hat{A},\hat{B}\hat{C}] = [\hat{A},\hat{B}]\hat{C} + \hat{B}[\hat{A},\hat{C}]$
$\displaystyle \left [ x , G(p) \right ] = i \hbar ~ \frac{dG}{dp}$
$\displaystyle \left [ F(x) , p \right ] = i \hbar ~ \frac{dF}{dx}$