# Help Verifying Schrodinger's equation in different potentials

• ja07019
In summary, the conversation discusses the use of three regions in a function, with the first region having a potential of zero and the second region having a potential of V. The second region also involves two principles related to wave incidents and tunneling. The conversation also covers the probability current and the process of normalization. The main issue being discussed is how to find the probability current of the transmitted wave and the discrepancy between the book's answer and the calculated one.
ja07019
Homework Statement
I am asked to find the probability current density. The set up is one dimensional along the z-axis and it goes as follows: for z>0, V=0 which has the solution of the time independent SE. For 0<=z<d, V= Constant, which has the solution of the time dependent SE. In addition to V being nonzero, V> E, the energy of the electron. For z>d, then V=0 which is once again the time independent SE. I need verification on the equations I will be using. In addition I amy
Relevant Equations
Schrodinger's Wave equation
There are 3 regions, to which I split the function as follows. I can derive the solutions myself. However I need to verify whether I am using them properly.

There are two principles/ideas that I am not sure if I am misinterpreting.
1) Anytime a wave is incident on a discontinuity(such as when a light ray enters a water, there is a transmitted wave and a reflected wave.
2) Tunneling is the phenomena when an electron can appear on the other side of a potential barrier and still have the same energy as before.In the first region where z>0 and V=0, I have ##exp(-i \frac{E}{ \hbar }t) \left[ Aexp(ik_1z) + Bexp(-ik_1z) \right]##
where ##k_1^2= \frac{2mE}{\hbar ^2}## . E is the energy of the electron. The ##+i_k1z## term represents a wave traveling in the +z direction and the ##-k_1z## term represents the wave reflected at z=0.

In region 2, 0<=z<d and V=V. Here I have ##exp(-i \frac{E}{ \hbar }t) \left[ Cexp(ik_2^{'}z) + Dexp(ik_2^{'}z) \right]## where ##(k_2^{'})^2 = \frac{2mE}{ \hbar ^2}(E-V)## . Let ##k_2=\frac{2m}{ \hbar^2}(V−E)## and thus ##(k_2^{'})^2 = -k_2^2##. Which leads to ##k_2^{'}=ik_2##.
So in effect this is the actual wave equation in Region 2, ##exp(-i \frac{E}{ \hbar }t) \left[ Cexp(-k_2z) + Dexp(k_2z) \right]##.
So here I am saying that there is a wave moving in the +z direction, and because of the discontinuity at z=d, there is a reflected wave.

Finally in region 3 I have ##exp(-i \frac{E}{ \hbar }t)Fexp(ik_3z)##.
Since there are no more discontinuities in front of region 3, there is no reflected wave. In addition I am saying that because the electron has the same energy as before crossing the potential barrier, then k3=k_i and thus I can simply find that. So I end up with ##exp(-i \frac{E}{ \hbar }t)Fexp(ik_1z)##.

I am unsure about how to find the probability current of the transmitted current. My issue here is that I end up with the constant C. I am unsure about how to normalize the wavefunctions here.

Last edited:
Please use LaTeX to write your equations. They are very hard to read. If you don't know how to use LaTeX, click on the LaTeX Guide link lower left, above "Attach files".

kuruman said:
Please use LaTeX to write your equations. They are very hard to read. If you don't know how to use LaTeX, click on the LaTeX Guide link lower left, above "Attach files".

Thank you for the edits.

Do yourself a favor and drop the time dependence, i.e. solve the TISE. Then you will have
$$\psi(z) = \begin{cases} Ae^{ik_1z}+Be^{-ik_1z} & z \leq 0 \\ Ce^{k_2z}+De^{-k_2z} & 0 \leq z \leq d \\ Fe^{ik_1z} & z \geq d \end{cases}$$
Note that the term ##Ae^{ik_1z}## is the incident wave on the barrier. At the first boundary ##z=0##, the fraction of the wave that is reflected is what counts not the value of ##B## by itself. So set ##A=1##, apply the 4 boundary conditions and then use the expressions for the probability current.

kuruman said:
So set ##A=1##, apply the 4 boundary conditions and then use the expressions for the probability current.
Thank you for the response. I have been trying to figure this out but I can't seem to get the answer the book gets. Here is my work for the Probability Current
In region 3,
##J_3 \propto \psi_3^{*} \frac{d \psi}{dz}-\psi_3 \frac{d \psi_3^{*}}{dz}##
## = Fexp(-ik_1z)(ik_1)Fexp(ik_1z)-Fexp(ik_1z)(-ik_1)Fexp(-ik_1z) = ik_1F^2--ik_1F^2 = 2ik_1F^2##

In region 1,
##J_1 \propto \psi_1^{*} \frac{d \psi_1}{dz}-\psi_1 \frac{d \psi_1^{*}}{dz}##
## = Aexp(-ik_1z)(ik_1)Aexp(ik_1z)-Aexp(ik_1z)(-ik_1)Aexp(-ik_1z) = ik_1A^2--ik_1A^2 = 2ik_1A^2##
Note here I use only the transmitted wave and not the Transmitted+Reflected Wave Function. Doing so would lead to,
##J_1^{'} = [Aexp(-ik_1z)+Bexp(ik_1z)][ik_1Aexp(ik_1z)+-ik_1Bexp(-ik_1z)] - [Aexp(ik_1z)+Bexp(-ik_1z)][-ik_1Aexp(-ik_1z)+ik_1Bexp(ik_1z)] = ik_1A^2exp(ik_1z-ik_1z)-ik_1ABexp(-2ik_1z)+ik_1ABexp(2ik_1z)-ik_1B^2exp(ik_1z-ik_1z)-[-ik_1A^2exp(ik_1z-ik_1z)+ik_1ABexp(2ik_1z)-ik_1ABexp(-2ik_1z)+ik_1B^2exp(ik_1z-ik_1z)] = ik_1A^2--ik_1A^2-ik_1ABexp(-2ik_1)+ik_1ABexp(-2ik_1)+ik_1ABexp(2ik_1z)-ik_1ABexp(2ik_1z)-ik_1B^2+-ik_1B^2 = 2ik_1(A^2-B^2)##

For continuity I have,
##\psi_1(0,t) = \psi_2(0,t)##
##Aexp(ik_1*0)+Bexp(-ik_1(0)) = Cexp(k_z*0)+Dexp(-k_2*0) = A+B = C+D##
##\frac{d \psi_1(0,t)}{dz} = \frac{d \psi_2(0,t)}{dz}##
##ik_1Aexp(ik_1*0)+-ik_1Bexp(-ik_1(0)) = k_2Cexp(k_z*0)+-k_2Dexp(-k_2*0) = ik_1A-ik_1B = k_2C+-k_2B##
##\psi_3(d,t) = \psi_2(d,t)##
##Fexp(ik_1d) = Cexp(k_2d)+Dexp(-k_2d)##
##\frac{d \psi_3(d,t)}{dz} = \frac{d \psi_2(d,t)}{dz}##
##ik_1Fexp(ik_1d) = k_2Cexp(k_2d)+-k_2Dexp(-k_2d) ##

I just can't see how the book arrives at their answer of
##\frac{J_3}{J_1} = [cosh^2(k_2d)+\frac{1}{4}(\frac{k_2}{k_1} - \frac{k_1}{k_2})^2sinh^2(k_2d)]^-1##
Looking at the ##\frac{J_3}{J_1}##, it seems like both regions should have continuity satisfied via exponential's but only Region 3 has these continuities. Setting ##A=1##, I still have to deal with B.

## 1. What is Schrodinger's equation?

Schrodinger's equation is a fundamental equation in quantum mechanics that describes the behavior and evolution of quantum systems over time. It was first developed by Austrian physicist Erwin Schrodinger in 1926.

## 2. How is Schrodinger's equation verified in different potentials?

Schrodinger's equation can be verified in different potentials by solving the equation for a given potential and comparing the resulting wave function with experimental observations. This can be done through analytical solutions or numerical methods.

## 3. What are potentials in the context of Schrodinger's equation?

Potentials in Schrodinger's equation refer to the forces or fields that act on a quantum system, such as an electric field or a potential energy well. These potentials affect the behavior and evolution of the system as described by the equation.

## 4. Why is it important to verify Schrodinger's equation in different potentials?

Verifying Schrodinger's equation in different potentials allows us to understand and predict the behavior of quantum systems in a variety of environments. This is crucial for applications in fields such as quantum computing, materials science, and chemical reactions.

## 5. What are some common methods used to verify Schrodinger's equation in different potentials?

Some common methods used to verify Schrodinger's equation in different potentials include the finite difference method, the variational method, and the matrix diagonalization method. Each method has its own advantages and limitations depending on the system being studied.

Replies
2
Views
429
Replies
9
Views
935
Replies
2
Views
816
Replies
2
Views
1K
Replies
1
Views
874
Replies
1
Views
612
Replies
22
Views
2K
Replies
12
Views
2K
Replies
12
Views
1K
Replies
8
Views
5K