Show that the expectation value of momentum is zero

[member 553083]

Homework Statement

Demonstrate that the expectation value of momentum (p) for the wave function: ψ(x)∝e^(-γx) when x>0, ψ(x)=0 when x<0. Hint: Pay special attention to the discontinuity at x=0.[/B]

Homework Equations

<p>=<ψ|p|ψ>=∫dxψ*(x)[-iħ∂/∂x]ψ(x) from -∞ to ∞. [/B]

The Attempt at a Solution

I have normalized the wave function such that ∫ψ*(x)ψ(x)dx from -∞ to ∞ =1. I get a constant of √(2γ) so that ψ(x)=√(2γ)e^(-γx).

Then, I attempt to set up the <p> integral as:
<p>=∫√(2γ)e^(-γx)[iħ∂/∂x]√(2γ)e^(-γx)dx from -∞ to ∞.
Simplifying, I get that the integral is: ∫2iħγ^2e^(-2γx)dx from 0 to ∞.
I am fairly confident in my evaluation of this integral (which i get to be γiħ), but I do not know how to approach the discontinuity at x=0. I attempted to use a delta function, but I cannot seem to get the overall expectation value to equal 0.

Thanks for any and all suggestions![/B]

 

[mfb]

I think you could replace it by some smooth function that connects both parts, and then take the limit of zero width for this smooth function.