Particle in a potential well

In summary, the conversation discusses calculating the time a particle of mass ##m## and energy ##E## will stay in a potential well with given conditions. The suggestion is to use the transmission coefficient formula and the quantum tunneling rate equation, where the lifetime of the particle can be found by taking the reciprocal of the rate.
  • #1
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.
 
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  • #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.
 

1. What is a "Particle in a potential well"?

A "Particle in a potential well" refers to a theoretical model used in quantum mechanics to describe the behavior and energy levels of a particle confined within a potential energy well, which can be thought of as a "trap" created by a potential energy barrier.

2. How does a particle behave in a potential well?

In a potential well, the particle's behavior is determined by the shape and depth of the potential energy barrier. If the barrier is high enough, the particle will be confined within the well and can only exist at specific energy levels, known as quantized energy states.

3. What is the significance of a particle in a potential well?

The concept of a particle in a potential well is important in understanding the behavior of particles at the quantum level. It helps explain phenomena such as particle tunneling and the stability of atoms and molecules.

4. How does the shape of the potential well affect the particle's energy levels?

The shape of the potential well determines the number and spacing of the particle's energy levels. A deeper and narrower potential well will have more closely spaced energy levels, while a shallower and wider well will have more widely spaced energy levels.

5. Can a particle escape from a potential well?

Yes, a particle in a potential well can escape if it has enough energy to overcome the potential energy barrier. This phenomenon is known as quantum tunneling and is a key concept in understanding the behavior of particles in confined systems.

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