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Momentum Squared Operator

  1. Jun 4, 2012 #1
    1. The problem statement, all variables and given/known data

    This problem is about the momentum squared operator. First, I state how I saw the derivation for the momentum operator. Then I state how I attempt to (and fail to) derive the momentum squared operator using the same methods.

    2. Relevant equations

    <p> = ∫ ψ*(ħ/i ∂/∂x)ψ dx

    The construction for this definition is straightforward. We start with the fact that

    <p> = m d/dt <x>
    = m ∂/∂t ∫ ψ* x ψ dx
    = m ∫ x ∂/∂t ψ*ψ dx
    = m ∫ x ∂/∂x [iħ/2m ψ*(∂ψ/∂x)-(∂ψ*/∂x)ψ] dx (from the probability flux)

    Here, we use the substitution x(∂/∂x)f(x) = ∂/∂x[xf(x)]-f(x) and say that since square integrable functions (that is, normalizable functions) must have derivatives that vanish, the first part is zero. We're left with "-f(x)", which is:

    = m(iħ/2m) ∫ -[ψ*(∂ψ/∂x)-(∂ψ*/∂x)ψ] dx
    = ħi/2 ∫ [(∂ψ*/∂x)ψ-ψ*(∂ψ/∂x)] dx
    = ħi/2 ∫ (∂/∂x)ψ*ψ -2ψ*(∂ψ/∂x) dx

    Again, the derivative vanishes.

    = ħ/i ∫ ψ*(∂ψ/∂x) dx

    Hence, we define the momentum operator as:

    p = ħ/i (∂/∂x)

    3. The attempt at a solution

    To obtain the momentum squared, I go through the following steps (the first few steps are about the same):

    <p> = m2 d/dt <x2>
    = m2 ∂/∂t ∫ ψ* x2 ψ dx
    = m2 ∫ x2 ∂/∂t ψ*ψ dx
    = m2 ∫ x2 ∂/∂x [iħ/2m ψ*(∂ψ/∂x)-(∂ψ*/∂x)ψ] dx (from the probability flux)

    From here, I use:

    x2(∂/∂x)f(x) = ∂/∂x[x2f(x)]-2xf(x) (1)

    The first part vanishes. We're left with 2xf(x)

    = m2(iħ/2m) ∫ 2x [ψ*(∂ψ/∂x)-(∂ψ*/∂x)ψ] dx
    = mħi ∫ x [ψ*(∂ψ/∂x)-(∂ψ*/∂x)ψ] dx
    = mħi ∫ x [(∂/∂x)[ψ*ψ]-2(∂ψ*/∂x)ψ] dx

    using another x(∂/∂x)f(x) = ∂/∂x[xf(x)]-f(x) type substitution:

    = mħi ∫ -[ψ*ψ]-2x(∂ψ*/∂x)ψ] dx
    = -mħi (1+∫ 2x(∂ψ*/∂x)ψ] dx)

    It's obviously wrong since it's imaginary. Going back to step (1), which I bolded, and making the substitution

    x(∂/∂x)g(x) = ∂/∂x[xg(x)]-g(x), where g(x) = (∂/∂x)g(x)=f(x), I'm left with 2g(x)=2∫f(x)dx

    = 2mħi ∫∫ [(∂/∂x)[ψ*ψ]-2(∂ψ*/∂x)ψ] dx2

    I don't understand why this happens. I get an imaginary answer every time. :(
     
    Last edited: Jun 4, 2012
  2. jcsd
  3. Jun 5, 2012 #2
    your equation <p2> = m2 d/dt <x2> is wrong
     
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