Schrodinger equation for step potential

[Abdul.119]

Homework Statement

Consider the one-dimensional Schrödinger equation for the step potential, that is for U(x) = 0 for x<0, and

for

,

. Consider a particle with mass m and energy E < U. Assume the particle is initially at x<0.

a) Calculate the penetration depth Δx at which the probability density of the transmitted wave "decayed" to half the value that it has at x=0.

b) Apply your result of part a) to an electron inside a metal block for which the work function is 4 eV.

c) Apply your resilt of part a) to a "macroscopic" particle with mass m=10^-15 kg and velocity v=10^-2 m/sec. Assume a barrier height that is 1.5 as high as the kinetic energy of the particle.

Homework Equations

The Attempt at a Solution

I have no idea where start with this one, I have a clue on how to find the transmission and reflection probabilities but I am not sure this is relevant for finding Δx. In this question are we suppose to use the equation Δp Δx >= ħ/2π ?

 

[PietKuip]

The question does not ask you to calculate transmission probabilities. Just write down the form of the wave function for x > 0.

 

[Abdul.119]

The question does not ask you to calculate transmission probabilities. Just write down the form of the wave function for x > 0.

For x>0 I believe the wave function would be u(x) = Ae^(ikx)

 

[PietKuip]

So imaginary k will give you the decay. Adjust with a factor ln(2).

 

[Abdul.119]

So imaginary k will give you the decay. Adjust with a factor ln(2).

I don't quite understand what you mean by adjust by a factor of ln(2). Also still not seeing how k will help find penetration depth Δx