Suppose you wanted to describe an unstable particle, that spontaneously disintegrates with...

Suppose you wanted to describe an unstable particle, that spontaneously disintegrates with a “lifetime” τ. In that case the total probability of finding the particle somewhere should not be constant, but should decrease at (say) an exponential rate:

A crude way of achieving this result is as follows. In Equation 1 we tacitly assumed that V (the potential energy) is real. That is certainly reasonable, but it leads to the “conservation of probability” enshrined in Equation 2. What if we assign toV an imaginary part:

where V0 is the true potential energy and Γ is a positive real constant?

(a) Show that (in place of Equation 2) we now get

(b) Solve for P(t), and find the lifetime of the particle in terms of Γ.

Equation 1

Equation 2