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- 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.

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.

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