Homework Help: Rewriting the propagator for the free particle as integral over E

1. Nov 18, 2009

quasar_4

1. The problem statement, all variables and given/known data

(This is all with respect to a free particle)
Show that the propagator $$U(t) = \int_{-\infty}^{\infty} |p><p| exp\left(\frac{-i E(p) t}{\hbar}\right) dp$$ can be rewritten as an integral over E and sum over the $$\pm$$ index as:

$$U(t) = \sum_{\alpha = \pm} \int_{0}^{\infty} \left(\frac{m}{\sqrt{2mE}}\right) |E, \alpha><E, \alpha| exp\left(\frac{-i E t}{\hbar}\right) dE$$.

2. Relevant equations

The Hamiltonian for a free particle is just H = P^2/2m (all kinetic energy) and the allowed values of momentum are plus or minus 2mE. So, for a given energy eigenvalue, there is a degenerate 2 dimensional subspace.

3. The attempt at a solution

Well... I can see that when we change to sum over E instead of P (or rather, integrate) that we have to change the limit of integration from 0 to infinity because the energy cannot be negative for a free particle. Also, it makes fine sense to me to sum over alpha because we're dealing with a degenerate subspace. What I can't figure out is the constant in front! Can someone help point me in the right direction?

2. Nov 19, 2009

lanedance

could the constant come from the intgeration variable change?

ie what is dE in terms of dp?