# Suppose you wanted to describe an unstable particle, that spontaneously disintegrates with a lifetime . In that

## Question:

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 1.24 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:

where V_{0} is the true potential energy and Γ is a positive real constant?

(a) Show that (in place of Equation 1.27) we now get

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

**Equation 1.24**

**Equation 1.27**

## Step by Step Answer:

**Related Book For**

## Introduction To Quantum Mechanics

**ISBN:** 9781107189638

3rd Edition

**Authors:** David J. Griffiths, Darrell F. Schroeter