Solving for Probability Current Density in Region 2

[UpQuark]

Ok, I am need of some serious help here. ,I don't want an answer just some guidance.
In the attachment you'll see a diagram of a step potential. The particle is traveling from the left. There is nothing incident from the right. We need to find the probability current density in region 2 in terms of V1, V2, a, E, m, and the incident probability current density.

What I have (hopefully correctly so far) is that we need seven equations to find the seven variables.

Simplified I've come up with,
K1=sqrt(2mE)/(hbar)
K2=sqrt(2m(E+V1))/(hbar)
K3=sqrt(2m(E-V2))/(hbar)

psi(x)=Aexp(i(K1)x) + Bexp(-i(K1)x) from negative infinity to zero
psi(x)=Cexp(i(K2)x) + Dexp(-i(K2)x) from zero to a
psi(x)=Eexp(i(K3)x) + Fexp(-i(K3)x) from a to infinity
also note that the scattering matrix is
S(Energy)=Aexp(i(K1)x) - Dexp(-i(K2)x) - Fexp(-i(K3)x) from a to infinity

Setting the derivatives from either side equal and setting up to solve the constants we find,

K1(A-B)=K2(C-D)
C(K2)exp(i(K2)a)=E(K3)exp(i(K3)a) - F(K3)exp(-1(K3)a)
A-D-F=AS(Energy)
E(K3)exp(i(K3)a) - F(K3)exp(-i(K3)a)=AS(Energy)(K1)exp(i(K1)a)
S(Energy)=Aexp(i(K1)a) - Dexp(-i(K2)a) - Fexp(-i(K3)a)

those I'm pretty sure of, except maybe in third and fifth equations adding in the B term... not sure though... and lastly, the ones that I keep changing my mind over,
A-B=C+D
C-D=E+F

should that read A+B=C and C+D=E? perhaps, but it seems that the transmission coefficient minus the reflection coefficient of one region should equal the transmission coefficient for the next.

If this much is right, I'll go on and use a computer to solve the system for each variable and then, go ahead solve for the probability current density in region 2 using
J2= abs(S(Energy)^2))J1
where J1 is the initial probability current density and J2 is the final probability current density.

If anyone can throw me a bone that'd be great.

 

[xepma]

Well, my first impression is that you have too much constants A,B,C etc.

I'm guessing that for the region where V = 0, you should get psi[x] = A exp[kx] + B exp[-kx]. So in that case A must be 0, because that term blows up when x -> minus infinity. This also applies for the region V2. So then you will get 4 constants, which you can solve by stating that psi[x] and it's derative must be continious (which you already did ofcourse).

 

[wais]

Thank you for sharing your progress and thought process so far. It seems like you have a good understanding of the equations involved and are on the right track. Here are a few suggestions that may help guide you towards finding the probability current density in region 2:

1. Double check your equations: It's always a good idea to double check your equations and make sure they are correct before moving on to the next step. For example, in your fourth equation, you have K1 instead of K2 in the denominator of the right side.

2. Consider using boundary conditions: Boundary conditions can be very helpful in solving for the constants in your equations. For example, in this problem, the wavefunction and its derivative must be continuous at the boundaries between regions. This can help you set up additional equations and solve for the constants.

3. Use conservation of probability current: In a system like this, probability current must be conserved, meaning that the total probability current in region 1 must equal the total probability current in region 2. This can also help you set up equations and solve for the constants.

4. Use a computer to solve the equations: As you mentioned, using a computer to solve the system of equations is a good idea. This can help you find the values of the constants and then use them to calculate the probability current density in region 2.

I hope these suggestions help guide you towards finding the solution. Remember to double check your equations and use boundary conditions and conservation of probability current to help you solve for the constants and find the probability current density in region 2. Good luck!