# Physical interpretation of a complex potential for a particle in quantum mechanics

In Griffiths' Quantum Mechanics, it is mentioned in a problem that

For an unstable particle that spontaneously disintegrates with a lifetime $\tau$, the total probability of finding the particle somewhere should not be constant$(=1)$ but should decrease at an exponential rate given by : $$P(t)=\int_{-\infty}^\infty |\psi(x,t)|^2 dx =e^{-\frac{t}{\tau}}$$

Now the book says that this mathematical result can be proven if we assume that the potential V is a complex quantity, say, $V=V_0-i\gamma$ where $V_0$ and $\gamma$ are real and $\gamma \not =0$.

And I have verified it also, that it can be proved hence.

My question is: What is the physical interpretation of this complex potential and, apart from mathematical reasons, how can I explain the above assumption from a physical aspect?

• a potential with an imaginary part is equivalent to adding the imaginary part to the eigenenergies, $H\psi=(E+i\gamma)\psi$, where $H$ is hermitian. In other words, a complex potential (with constant imaginary part) is the same thing as a complex energy. See physics.stackexchange.com/a/298984/84967, physics.stackexchange.com/q/60185 and physics.stackexchange.com/q/178233 Jan 25, 2017 at 14:03
• What is a complex energy? We normally deal with real energy. So what do you actually mean by an energy represented by a complex number? Jan 25, 2017 at 16:19
• well, that is addressed in the posts I linked before ;-) Jan 25, 2017 at 16:22