- #1
fluidistic
Gold Member
- 3,953
- 265
Homework Statement
A particle of mass m is find to be inside a uni-dimensional potential well of the form: [itex]V(x)=0[/itex] for [itex]x \leq -a[/itex] and [itex]a\leq x[/itex] and [itex]V(x)=-V_0[/itex] for [itex]-a <x<a[/itex].
1)Write down the corresponding Schrödinger's equation.
2)Consider the case [itex]-V_0<E<0[/itex]. Determine the contour conditions and conditions of continuity that an eigenfunction must satisfy inside this potential well.
3)Show that the eigenfunctions have a definite parity and that they are real for the "linked states". I'm not really sure about the word "linked state", this is my own translation of the original problem.
4)Show that the energy of the linked states only take discrete values. (Maybe linked states means eigenstates.)
5)Consider now that E>0. Find and graph the wave function of the particle. Calculate the transmission coefficient when the particle moves along the potential well.
Homework Equations
[itex]-\frac{\hbar ^2 }{2m} \frac{d ^2 \Psi (x)}{d x^2}+[V(x)-E]\Psi (x)=0[/itex].
The Attempt at a Solution
Part 1) is solved if you take the equation above and replace V(0) by its values, for each region of the potential well.
I won't put all my work for 2), since it's extremely long.
I reached that in the first region [itex](x \leq -a)[/itex], [itex]\Psi _I (x)=De^{\alpha x}[e^{x(\alpha+\beta )}-e^{x(\alpha - \beta)}][/itex].
For the region of the well, region II, [itex]\Psi _{II} (x)=D(e^{-\beta x}-e^{\beta x})[/itex].
For the third region, when [itex]x \geq a[/itex], [itex]\Psi _{III}(x)=De^{-\alpha x} [e^{x(\alpha - \beta)}-e^{x(\alpha + \beta)}][/itex].
Where [itex]\alpha = \frac{\sqrt {-2mE}}{\hbar}[/itex] and [itex]\beta = \frac{\sqrt{2m(-V_0-E)}}{\hbar}[/itex].
I never met any complex exponential since despite its negative look, all the arguments I met under the square roots were positive.
For the contour condition, I assumed that Psi should not diverge when x tends to negative and positive infinity. From this, I could "drop" 2 constants, or more precisely, equal them to 0.
For the continuity, I assumed that the function Psi and its first derivative must be continuous over the whole 3 regions. This is how I could reduce the number of unknown constant to 1 (I started with A, B, C and D).
My question is now... how do I obtain D? Should I normalize something?
I don't think I'm done for part 2).
For the first part of part 3), it's easy. In previous exercise I showed that if V(x)=V(-x) then Psi is either odd or even. And this is the case in this exercise so by the previous exercise, the eigenfunctions have a definite parity. I'll have to show that they are real and take discrete values.
But first, I want to deal with D... but have no idea how to.
Any help will be appreciated! Thanks!