1-d Time Independent Schrodinger Equation Problem

In summary, the homework statement is that a particle of mass m and energy E, where E >V1 >V2 travels to the right in a potential defined as V(x) = V1 for -b < x < 0, V(x) = 0 for 0 < x < a, V(x) = V2 for a < x < b. The boundary conditions are 1)A$_{1}$e$^{-ik$_{1}$b}$ + B$_{1}$e$^{ik$_{1}$b}$ = 0, 2)A$_{1}$ + B$_{1}$ = A$_{2}$ + B$
  • #1
zZhang
11
0

Homework Statement


A particle of mass m and energy E, where E >V1 >V2 travels to the right in a potential defined as

V(x) = V1 for - b < x < 0

V(x) = 0 for 0 < x < a

V(x) = V2 for a < x < b

(a) Write down the time-independent Schrodinger eq. and its general solution in each region. Use complex exponential notation.
(b) Write down the boundary conditions which fix the undetermined constants in the solution of part (a).
(c) Eliminate from those eqs. The normalization for the wave in region 3 and calculate the ratio of intensities of waves traveling to the left and to the right in region number 2.



Homework Equations





The Attempt at a Solution


I really have no idea what I am doing. After writing the solutions as
ψ(x) = A*exp(ikx) + B*exp(-ikx) with appropriate k for each of the 3 situations,
I just cannot solve for the undetermined coefficients (and I've been doing this for the last 3 hours -_-). So yea, I'm completely lost.

EDIT: Boundary Conditions are
1)A$_{1}$e$^{-ik$_{1}$b}$ + B$_{1}$e$^{ik$_{1}$b}$ = 0
2)A$_{1}$ + B$_{1}$ = A$_{2}$ + B$_{2}$
3)k$_{1}$(A$_{1}$ - B$_{1}$) = k$_{2}$(A$_{2}$ - B$_{2}$)
4)A$_{2}$e$^{ik$_{2}$b}$ + B$_{2}$e$^{-ik$_{2}$b}$ = A$_{3}$e$^{ik$_{3}$b}$ + A$_{3}$e$^{-ik$_{3}$b}$
5)k$_{2}$(A$_{2}$e$^{ik$_{2}$b}$ - B$_{2}$e$^{-ik$_{2}$b}$) = k$_{3}$(A$_{3}$e$^{ik$_{3}$b}$ - B$_{3}$e$^{-ik$_{3}$b}$)
6)A$_{3}$e$^{ik$_{3}$b}$ + B$_{3}$e$^{-ik$_{3}$b}$ = 0

Look right to u guys?
 
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  • #2
the potential is not specified in all space... for the first line describing the potential you wrote down, is the "-b" supposed to be a negative infinity? otherwise, what is the for of the potential for x<-b
?
 
  • #3
It's not specified for all space, just a certain region, so b and -b is just any random number basically.
 
  • #4
okay... so then continue with your solution attempt. what boundary conditions are you going to use.
 
  • #5
Oh wow hold on that post just made me realize I left out 2 boundary conditions :-p (silly me)
 
  • #6
I can't read the correction you made to your original post but the problem only makes sense to me if there is an incoming particle (wave actually) from the left, a step in potential at x=0 and another one at x=a. Then there is a reflection at the first boundary, a middle zone with waves going both ways, and a transmitted beam at the second boundary.

I think you already have expressions for the waves, which is to say their k numbers, based on the three potentials. So there are 5 undetermined coefficients, in general complex, for the wave amplitudes.

However, as I see it, it gets a lot simpler when you consider what they are asking for: just the ratio between the left- and right- propagating waves in the middle zone. Let's take the outgoing wave in zone 3 to have unit amplitude; in fact, make the wave number something simple like 1 so it is just:

exp(ikx - wt)

(of course we won't worry to much about the wt).

Then you just need to match up the waves in zone 2, let's give them a wave number like 3 or something:

Aexp(i3x) + Bexp(-i3x)

And I believe the boundary condition is that both the amplitudes and their derivatives have to match up at the transition. With the arbitrary numbers I put in above, I can solve pretty easily: I get A + B = 1 for the amplitudes, and 3A - 3B = 1 for the derivatives.

So it seems I can solve the problem as stated without even worrying about what happened at the first boundary, with the incoming particle. Does this look right?

Marty
 
  • #7
how did u have only 5 undetermined coefficients, not 6?
 
  • #8
Because if the wave starts from the left, then in region 3 there is no left-moving (incoming) wave, only the outgoing one which I arbitrarily assigned a unit amplitude.
 
  • #9
I guess that would work, I just wasnt sure that there was not in fact some other potential past alpha, granted we are not told anything about it, so I guess you can take away the incoming wave.
 

What is the 1-d Time Independent Schrodinger Equation Problem?

The 1-d Time Independent Schrodinger Equation Problem is a mathematical equation that describes the behavior of a quantum mechanical system in one dimension, such as a particle confined to a one-dimensional space. It is an important tool for understanding the behavior of quantum systems and making predictions about their properties.

What is the significance of the Schrodinger Equation?

The Schrodinger Equation is a fundamental equation in quantum mechanics, and it allows us to calculate the wave function of a quantum system which contains all the information about its properties. This equation has been extensively tested and has been found to accurately predict the behavior of quantum systems.

What are the key components of the 1-d Time Independent Schrodinger Equation?

The 1-d Time Independent Schrodinger Equation is made up of three key components: the Hamiltonian operator, the wave function, and the energy of the system. The Hamiltonian operator represents the total energy of the system, the wave function describes the probability of finding the particle at a certain position, and the energy of the system is the sum of the kinetic and potential energies of the particle.

What are the solutions to the 1-d Time Independent Schrodinger Equation?

The solutions to the 1-d Time Independent Schrodinger Equation are the allowed wave functions for a given system. These solutions can be used to calculate the probability of finding the particle at a specific position, the energy levels of the system, and other properties of the system.

How is the 1-d Time Independent Schrodinger Equation solved?

The 1-d Time Independent Schrodinger Equation is solved using various mathematical techniques, such as separation of variables, perturbation theory, and numerical methods. These methods allow us to find the allowed wave functions and energies for a given system and make predictions about its behavior.

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