The difference btn positive and negative potential quantum well

[ismaili]

Homework Statement

(a) Determine the reflectivity spectrum [tex]R(E)[/tex] of a free particle of mass [tex]m[/tex] reflected from an infinite jump [tex]V(x)[/tex], where
[tex]V(x)=0[/tex] if [tex]x\leq0[/tex], [tex]V(x)=-\infty[/tex] if [tex]x>0[/tex]

The other similar problem is:
(b) Determine the energy levels of a particle of mass [tex]m[/tex] confined to an infinite barrier [tex]U(x)[/tex] of width [tex]L[/tex], where
[tex]U(x) = -\infty[/tex] for [tex]x<0, x>L[/tex], [tex]U(x) = 0[/tex] for [tex]0\leq x \leq L[/tex]

Homework Equations

The Attempt at a Solution

Actually, I can solve these problems. One just solves the Schrodinger equation and matches the boundary conditions at each area. But what I wondered is, how about if we change the problem from a negative potential to a positive potential? (i.e. change the [tex]-\infty[/tex] in the problems to [tex]\infty[/tex])
I think the method to solve the problems are still the same, and the boundary conditions should be the same! So, the solutions would be the same if we change the potential from a bluff to a wall! But intuitively, the bluff and the wall are quite different.
Can anyone explain this physically to me? any ideas would be appreciated, thanks!

 

[HallsofIvy]

Homework Statement

(a) Determine the reflectivity spectrum [tex]R(E)[/tex] of a free particle of mass [tex]m[/tex] reflected from an infinite jump [tex]V(x)[/tex], where
[tex]V(x)=0[/tex] if [tex]x\leq0[/tex], [tex]V(x)=-\infty[/tex] if [tex]x>0[/tex]

The other similar problem is:
(b) Determine the energy levels of a particle of mass [tex]m[/tex] confined to an infinite barrier [tex]U(x)[/tex] of width [tex]L[/tex], where
[tex]U(x) = -\infty[/tex] for [tex]x<0, x>L[/tex], [tex]U(x) = 0[/tex] for [tex]0\leq x \leq L[/tex]

Homework Equations

The Attempt at a Solution

Actually, I can solve these problems. One just solves the Schrodinger equation and matches the boundary conditions at each area. But what I wondered is, how about if we change the problem from a negative potential to a positive potential? (i.e. change the [tex]-\infty[/tex] in the problems to [tex]\infty[/tex])
I think the method to solve the problems are still the same, and the boundary conditions should be the same! So, the solutions would be the same if we change the potential from a bluff to a wall! But intuitively, the bluff and the wall are quite different.
Can anyone explain this physically to me? any ideas would be appreciated, thanks!

Physically, the difference is that the particle is in different areas. First, remember that potential is always "relative" to some value. In the first problem, the potential is 0 for [itex]x\le 0[/itex], [itex]-\infty[/itex] for x> 0. That means that the particle can only be found at x> 0 (the probability of it being found with [itex]x\le 0[/itex] is 0. If you were given that the potential is [itex]+\infty[/itex] for [itex]x\le 0[/itex], 0 for x> 0, that would only change the "reference" and the solution would be exactly the same: the particle can only be found at x> 0.

If you were given either
1) potential 0 for x< 0 or x>L, [itex]-\infty[/itex] for [itex]0\le x\le L[/itex] or
2) potential [itex]\infty[/itex] for x< 0 or x> L, 0 for [itex]0\le x\le L[/itex]
then the particle can only be found between 0 and L for both problems.

In your second problem you have an infinite potential barrier between x= 0 and x= L. In that case, the particle can only be found at x< 0 or x> L. Where there is an infinite barrier, whether it is potential 0 as compared to a potential of [itex]-\infty[/itex] or infinite potential as compared to potential 0, you cannot find the particle in the barrier.

You will probably start shortly on finite potential wells or barriers. In that case, the Schrodinger equation cannot be solved exactly and you will need to find approximate solutions, probably using the "WKB approximation". In that case, you can find the particle in the barrier or sides of the well. In fact, if the barrier is narrow enough the particle may be able to "tunnel" to the other side.