# Particle in a potential well

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## 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.

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.