Particle in a potential well

  • #1

Main Question or Discussion Point

Let's suppose I have a potential well: $$
V(x)=
\begin{cases}
\infty,\quad x<0\\
-V_0,\quad 0<x<R\\
\frac{\hbar^2g^2}{2mx^2},\quad x\geq R
\end{cases}
$$

If ##E=\frac{\hbar^2k^2}{2m}## and ##g>>1##, how can I calculate how much time a particle of mass ##m## and energy ##E## will stay inside the well?

I'm thinking of using the expression of the transmission coefficient: $$T=e^{-2\int_{x_1}^{x_2}dx\sqrt{\frac{2m}{\hbar^2}[V(x)-E]}}$$
and ##\lambda=vT##, where ##\lambda## the probability per unit of time for the particle to pass through and ##v## the number of collisions per second. Any help would be appreciated.
 

Answers and Replies

  • #2
The expression you wrote down for $$ \lambda = \nu T $$ should give you the quantum tunneling rate. To find the lifetime of the particle in the well, find the reciprocal of the rate.
 

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