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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.

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.