Estimating Electron Lifetime in 1D Quantum Well with WKB Approximation

[FiatLux]

Homework Statement

Using the WKB approximation, estimate the lifetime of an electron in the ground state of a 1D quantum well with 10 nm width, surrounded from both sides by 0.3 eV high and 8nm wide barriers.

Homework Equations

Hint: Estimate the tunneling probability and find the "tunneling attempt frequency" assuming that the electron moves back and forth in the quantum well with the velocity, v = sqr(2E/m)

The Attempt at a Solution

This is from a collection of practice questions our professor provided to help us prepare for our final exam. I'm not great at this WKB (semi-classical approximation) method. We are really just looking for the start of this problem since we are having trouble visualizing it. Any help would be much appreciated.

 

[Jhero]

Dear fellow student,

Thank you for posting this question on the forum. The problem you are trying to solve is a classic example of using the WKB approximation to estimate the lifetime of a particle in a potential well. I will provide some guidance on how to approach this problem and hopefully it will help you understand the concept better.

First, let's start by defining the WKB approximation. It is a semi-classical method that is used to solve the time-independent Schrodinger equation for a particle in a potential well. It is based on the assumption that the wavefunction can be expressed as a product of an amplitude and a phase term, and the phase term varies slowly compared to the amplitude. This allows us to approximate the wavefunction as a series of classical turning points, where the phase changes rapidly.

Now, let's apply this to your problem. You are given a 1D quantum well with a width of 10 nm and surrounded by barriers that are 0.3 eV high and 8 nm wide. The electron is in the ground state, which means it has a specific energy, E, and a specific momentum, p. Using the WKB approximation, we can estimate the tunneling probability for the electron to escape from the well and the "tunneling attempt frequency" which is the frequency at which the electron will try to escape from the well.

To find the tunneling probability, we need to consider the potential energy barrier that the electron needs to overcome in order to escape. We can use the fact that the electron is in the ground state, which means its energy, E, is equal to the potential energy of the well, V. We can then use the WKB approximation to estimate the tunneling probability, which is given by:

P = exp(-2γ), where γ = ∫sqrt(2m(V-E))dx/h

Here, m is the mass of the electron, h is Planck's constant, and dx is the width of the barrier. You can use these values to calculate γ and then find the tunneling probability.

Next, we need to find the "tunneling attempt frequency" which is given by f = (v/2L)P, where v is the velocity of the electron, L is the width of the well, and P is the tunneling probability we calculated earlier. You can use the given values to calculate f and then use it to estimate the lifetime of the electron in the well