Finding the Wavefunction for Tunneling,with tunnel lenght L

[Arman777]

Homework Statement

Let us suppose we have a particle with energy $E$ and $E<U$ and the potential defined as

$U(x)=0$ for $x<0$ (I)

$U(x)=U$ for $0<x<L$ (II)

$U(x)=U_0$ for $x>L$ (III)

In this case $E>U_0$ and $U>U_0$

Homework Equations

$$HΨ=EΨ$$

The Attempt at a Solution

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I find the general form of solution, which it is

$Ψ(x)=Ae^{βx}+Be^{-βx}$ for $β^2=2m(U(x)-E)/\hbar^2$

For region (I) I find that

$Ψ_1(x)=c_1sin(\frac {\sqrt{2mE}} {\hbar}x)+c_2sin(\frac {\sqrt{2mE}} {\hbar}x)$

For region (II)

$Ψ(x)=De^{\beta x}+Ee^{-\beta x}$ for $\beta=\frac {\sqrt{2m(U-E)}}{\hbar}$

Is this true ? Because in the site of the hyperphysics it says it should be,

$Ψ(x)=Ee^{-\beta x}$ for $\beta=\frac {\sqrt{2m(U-E)}}{\hbar}$

I am not sure how we can derive this mathematically ? Why the

$De^{\beta x}$ term vanishes ?

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html

For the region (III)

I find that

$Ψ(x)=Ge^{iαx}$ for $α=\sqrt{2m(E-U_0)}/ {\hbar}$

So $D=0$ or not ? If so why its 0 ?

If $U(x)=U$, $x>L$ then it was logical thing to say that $D=0$, but $U$ is just for some distance.

 

[drvrm]

If U(x)=UU(x)=UU(x)=U, x>Lx>Lx>L then it was logical thing to say that D=0D=0D=0, but UUU is just for some distance.

If one is dealing with tunneling - in your situation if U(0) is less than U; effectively a barrier of height ( U - U(0) ) = V is effective and only with the condition of E < V one has to explore the tunneling process.

for calculating the tunneling probability one needs the amplitudes A and E,
the coefficients/amplitudes are A, B, C, D., E, F In The three regions

if the wave/particle is incident from left B=0 and C, D, will survive as reflection from the barrier wall and E will survive,

one can see the following reference-
<https://pdfs.semanticscholar.org/61f0/15a4466eda8a25bce28e766ffd60edd3d934.pdf>

 

[Arman777]

If one is dealing with tunneling - in your situation if U(0) is less than U; effectively a barrier of height ( U - U(0) ) = V is effective and only with the condition of E < V one has to explore the tunneling process.

This helpmed me a lot actually. I was so confused about the process of it and how to deal with it. Thanks a lot.

if the wave/particle is incident from left B=0 and C, D, will survive as reflection from the barrier wall and E will survive,

I did not understand this part. If you are talking about the article then why B should be zero its the incident wave and it should have reflective parts ?

In the article it says nothing about the D and E ( In terms of my description)

I still didnt understand..

 

[drvrm]

I did not understand this part. If you are talking about the article then why B should be zero it is the incident wave and it should have reflective parts?

Actually, B will not be zero, you are right I might have made the omission as I was focussing on A and E.thanks lot

 

[Arman777]

I am confused about something. You said that I can take $V=U-U_0$ in this case the potential for the second and third region is $V$ or only for the third region its $V$.

And in the first region $U=0$ definitely right?

I guess its $U(x) = 0 (I), U(II),V(III)$ But I can't be sure.

 

[drvrm]

I am confused about something. You said that I can take V=U−U0V=U−U0V=U-U_0 in this case the potential for the second and third region is VVV or only for the third region its VVV.

 

[drvrm]

I am looking at the tunneling process

you are right that first region has potential zero-
If one takes a potential V which is a barrier of height (U - U(0)) then for particle energy E which is less than V has a quantum probability of tunneling. the base potential is U(0) in the third region and E is greater than U(0) but less than U.

the nature of the solution is well known inside the barrier of length L or outside the barrier- as a standard solutions of Schrödinger equation.

the solution does depend on the value of E relative to the height of the barrier and only two types of wave functions either exponential decay or oscillatory is available with continuity at the boundary.
the continuity equations further relate the amplitudes and thus determines the transmission or reflection at the boundary.