- #1

nille40

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I'm having some problem calculating the probability for a particle to penetrate a barrier (potential well). This is a math assignment in school, and we haven't learned anything about this area, so I may be fumbling in the dark completely.

Anyway, we have the Schrödinger time-independent equation:

[tex]\frac{h^2}{2m}\frac{d^2}{dx^2}\psi (x)+U(x)\psi (x) = E \psi (x)[/tex]

The equation before the barrier is [tex]\psi (x) = e^{ikx} + Re^{-ikx}[/tex] (incoming and reflected wave).

The equation in the barrier is [tex]\psi (x) = Ae^{imx} + Be^{-imx}[/tex]

The equation after the barrier is [tex]\psi (x) = Te^{ikx}[/tex]

If I've calculated right, the constants (k and m) should be:

[tex]k = +- i \frac{\sqrt{2mE}}{h}, x < a or x > b[/tex]

[tex]m = +- i \frac{\sqrt{2m(E-U_0)}}{h}[/tex]

We know that [tex]U(x) = 0[/tex] when [tex]x < a[/tex] or [tex]x > b[/tex] (outside of the barrier) and [tex]U(x) = U_0, a < x < b [/tex] (inside the barrier).

We have for unknown variables - A,B,R and T. We are supposed to get the probability from [tex]|T|^2[/tex]. We was instructed to derive the functions so that we got 4 function, which should yield an equation system when we attach the functions to each other. So we set:

[tex]e^{ika} + Re^{-ika} = Ae^{ima} + Be^{-ima}[/tex]

[tex]Te^{ikb} = Ae^{imb} + Be^{-imb}[/tex]

[tex]ke^{ika} - Rkae^{-ika} = mAe^{ima} - mBe^{-ima}[/tex]

[tex]kTe^{ikb} = imAe^{imb} - mBe^{-imb}[/tex]

How can I solve this system? Of course, I do not expect you to do this for me, I was just hoping you could help me do it.

Oh, and english isn't my native language, so if the lingo is messed up sometimes - sorry. And I hope the tex stuff works...

Thanks in advance,

Nille