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Math Help - Qm #2

  1. #1
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    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}]
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  2. #2
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    Quote Originally Posted by DiscreteW View Post
    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}]
    Try reading this.
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  3. #3
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    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
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