Schrodinger time independent equation for steps and barriers

• WrongMan
In summary, the book the person is following does not help much, they are trying to solve the homework equations using a step and a barrier, but are confused because they are not understanding the coefficients used. They are also trying to solve the barrier equation using e>V, but are stuck because they do not have enough equations.
WrongMan
Ok for starters, I am sorry for the length of the question and jamming multiple questions in here i feel like my doubts are all related and it would be best just to put it all in one place instead of creating 2-4 topics.
I'm beginning to feel confused with the amount of ways i see people solving this.
The book I am following isn't helping much

Homework Statement

so I am dividing this in two; a step and a barrier, both finite.
I want to find the reflection and transmission coefficients of a step/barrier

Homework Equations

Ψ''(x)=-2m/ħ2(E-V)Ψ(x) for V=0 k2=E2m/ħ2 for V=/=0 (E-V)2m/ħ2=q2.

The Attempt at a Solution

a)[/B]so for the step we have two cases e<V and e>V, the step begins at x=0
When e<V:
for area x<0
Ψ(x)= reikx+e-ikx ; Ψ'(x)= rikeikx-ike-ikx
for x>0
Ψ(x)= te-qx ; Ψ'(x)= -qte-qx
This choice of coefficients is confusing to me (also see underlined part bellow), most of the material i see has A, B and C, but i remember using "r" and "t" in class, and since the negative exponential represent the wave moving to one direction and the positive ones represent the opposite direction it'd make sense to use this set up, but i feel the results i get are wrong.
For example: r+1=t this doesn't really make sense does it?
anyway, for "r" and "t" coeficients i get:
t=(2ik)/(-q+ik) and T=(q/k)|t|2 so T= 4kq/(q2+k2)
for A,B,C i get:
C/A=(2ik)/(-q+ik) and T=(q/k)|C/A|2
So this appears to be correct
b)for e>V i get two complex exponential with coeficients A,B,C,D or 1,r,tD
and i have no idea how to solve them

c)
now for the barrier of length L and e<V
x<0
Ψ(x)= Aeikx+Be-ikx ; Ψ'(x)= Aikeikx-Bike-ikx
x>L
Ψ(x)= Ceikx+De-ikx ; Ψ'(x)= Cikeikx-Dike-ikx
and C must be 0 so it doesn't blow up to infinity.
0<x<L
e-V is negative so
Ψ''(x)=-2m/ħ2(E-V)Ψ(x) becomes Ψ''(x)=2m/ħ2(V-e)Ψ(x)
so
Ψ(x)= Eeqx+Fe-qx ; Ψ'(x)= qEeqx-qEe-qx
And now i set the negative exponentials to be the wave moving to the right (which i read it should be the positive exponentials moving to the right due to Euler's formula but there is no positive exponential so ?)
T=|D/B|2
and R = |A/B|2
And so this is solvable... how?
so i set the boundary cond. and got:
Ψ(0)= A+Be=E+F (1.)
Ψ'(0)= Aik-Bike=qE-qE (2.)
De-ikL=EeqL+Fe-qL (3.)
-Dike-ikL=qEeqL-qEe-qL (4.)
I did (2)+ik(1), then (4)/(3) and got stuck here:
[-(EeqL-Ee-qL)/(EeqL+Fe-qL)]*(A-B)=(E-F)
im stuck.
I have that feeling i have to understand something I am missing, or there's a specific path I am not taking and I am just making it harder on me

d) for e>V i feel it would be simmilar since the only difference is that all the exponentials are complex.

Last edited:
Always make sure sure you have the correct number of equations for solving the system. In these problems, if you have n coefficients you would expect n-1 equations such that all the other coefficients can be expressed in terms of the incoming wave coefficient. These equations come from matching boundary conditions (Hint: Match them at every step/barrier, the ks change! I believe that's where you're missing equations). If you have enough eqtns, the rest is algebra.

What is the Schrodinger time independent equation?

The Schrodinger time independent equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system in terms of its wave function. It is used to calculate the energy levels and wave functions of a system, and can be applied to both simple and complex systems.

How does the Schrodinger time independent equation apply to steps and barriers?

In the context of quantum mechanics, steps and barriers refer to changes in the potential energy of a system. The Schrodinger time independent equation can be used to solve for the wave function and energy levels of a particle as it encounters these changes in potential energy.

What are the assumptions behind the Schrodinger time independent equation?

The Schrodinger time independent equation makes several key assumptions, including that the system is in a stationary state, the potential energy is independent of time, and the system is in a one-dimensional space. These assumptions allow for a simplified mathematical description of the system.

How is the Schrodinger time independent equation solved?

The Schrodinger time independent equation is typically solved using mathematical techniques such as separation of variables and boundary conditions. These solutions yield the wave function and energy levels of the system, which can then be used to make predictions about the behavior of the system.

What is the significance of the Schrodinger time independent equation in quantum mechanics?

The Schrodinger time independent equation is a fundamental equation in quantum mechanics and has played a crucial role in our understanding of the behavior of particles at the atomic and subatomic level. It has helped to explain many phenomena, including the behavior of electrons in atoms, and has allowed for the development of new technologies, such as transistors and lasers.

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