Problem 1.17 – Griffith’s Intro to QM

Problem 1.17

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

$P(t) \equiv \displaystyle\int_{- \infty}^{\infty}|\Psi (x,t)|^2 dx = e^{-t/\tau}.$

A crude way of achieving this result is as follows. In Equation 1.24 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 1.27. What if we assign to V an imaginary part:

$V=V_0-i\Gamma ,$