Why can't a particle at rest decay when its energy is real-valued?

[Tiberius47]

Homework Statement

Show that for real valued E a particle at rest can not decay.

Homework Equations

Width of resonance.
\Gamma=\hbar/\tau
Relativistic Breit-Wigner distribution
P_rel_=\Gamma/((E^2-M^2)^2+M^2\Gamma^2)
Wave function of particle at rest.
ψ(t)=ψ(0)e^(-iEt/\hbar)

The Attempt at a Solution

I guess Lambda would be 0 if tau (the lifetime of the particle) were infinite but I have no idea what that has to do with real valued E.

 

[maajdl]

Put a imaginary part on E and see the effect on |ψ|² and the total ∫|ψ|²dx .
That's just a simple model to represent the decay of a particle.
The total probability of presence would decay.

 

[Tiberius47]

Oh wow, duh. It's been over a year since I've taken quantum and I assumed the distribution would be relevant somehow.