Introduction to Quantum Mechanics · Problem 1.17

The Wave Function — David J. Griffiths

Physics — Quantum Mechanics · unstable_particle_complex_potential

Problem

Suppose you wanted to describe an unstable particle, that spontaneously disintegrates with a "lifetime" τ\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)≡∫−∞+∞∣Ψ(x,t)∣2 dx=e−t/τ.P(t) \equiv \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 VV (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 VV an imaginary part: V=V0−iΓ,V = V_0 - i\Gamma, where V0V_0 is the true potential energy and Γ\Gamma is a positive real constant? (a) Show that (in place of Equation 1.27) we now get dPdt=−2ΓℏP.\frac{dP}{dt} = -\frac{2\Gamma}{\hbar}P. (b) Solve for P(t)P(t), and find the lifetime of the particle in terms of Γ\Gamma.

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