Momentum Squared Operator

Harrisonized

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. :(

sgd37

your equation <p²> = m² d/dt <x²> is wrong