Schrodinger's Equation, Potential Energy Barrier U>E

In summary, the conversation discusses particles incident on a potential energy step and the resulting wave functions in different regions. The boundary conditions for the wave functions are applied and it is shown that the full wave intensity is reflected at the step. The solution involves manipulating equations and using algebraic skills to find the unknown variables.
  • #1
Oijl
113
0

Homework Statement


Consider particles incident on a potential energy step with E<U.
(That is, a particle with total energy E travels along one dimension where U=0, then crosses, at point x=0 into a region where U>E.) (The particle is incident on the potential energy step from the negative x direction.)

Starting with the wave functions,

x < 0: Ψ0 = A’e^(ikx) + B’e^(-ikx), k = 2mE/h_bar2
x >= 0: Ψ1 = Ce^(k1x) + De^(-k1x), k1 = 2m(U-E)/h_bar2

Apply the boundary conditions for Ψ and dΨ/dx and show that the full wave intensity is reflected at the step [i.e., |A'|^2 = |B'|^2].


Homework Equations


Ψ0(x=0) = Ψ1(x=0)
dΨ0(x=0)/dx = Ψ1(x=0)/dx


The Attempt at a Solution


I set C=0, or else the wave function Ψ1 may become infinity.

The boundary conditions are stated above. They become
A' + B' = D
and
ikA' - ikB' = -k1D

How, from this, do I find that
|A'|^2 = |B'|^2
?

Thanks.
 
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  • #2
you have two equations and three unknowns..

my hint for your:

write A and B in terms of D .. and i noted something in your solution (you may not need it, but i will say it anyway) when you wrote k = 2mE/h_bar2, and k1 = 2m(U-E)/h_bar2, they should actually be k^2 and k1^2 .. finally after you find A and B in terms of D find AA* (which is |A|^2) and BB* (which is |B|^2) you need to write them in that form since you will have to find a complex conjugate of both of them ..

good luck with this .. and tell us what you get .. :)
 
  • #3
Yay, I got it! I didn't have enough faith in algebra, was my problem. I often don't.

Thanks!
 
  • #4
:) u r welcome .. next time don't give up so fast..
 

What is Schrodinger's Equation?

Schrodinger's Equation is a mathematical equation that describes the evolution of a quantum mechanical system over time. It is used to calculate the probability of finding a particle in a certain location at a specific time.

What is a potential energy barrier?

A potential energy barrier is a region in space where the potential energy of a particle is higher than the energy of the particle. This creates a barrier that the particle must overcome in order to move through the region.

How does Schrodinger's Equation relate to potential energy barriers?

Schrodinger's Equation is used to calculate the probability of a particle overcoming a potential energy barrier. It takes into account the energy of the particle and the shape of the barrier to determine the likelihood of the particle passing through the barrier.

What happens when the potential energy barrier is higher than the particle's energy?

If the potential energy barrier is higher than the particle's energy, there is a high probability that the particle will be reflected back instead of passing through the barrier. This phenomenon is known as quantum tunneling.

Why is it significant when U>E in Schrodinger's Equation?

When U>E in Schrodinger's Equation, it means that the potential energy barrier is higher than the energy of the particle. This can lead to interesting quantum mechanical effects such as quantum tunneling, which allows particles to pass through barriers that they would not be able to overcome based on classical physics principles.

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