- #1
Bobbo Snap
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Homework Statement
A particle of mass m is in the potential
[tex] V(x) = \left\{
\begin{array}{rl}
\infty & \text{if } x < 0\\
-32 \hbar / ma^2 & \text{if } 0 \leq x \leq a \\
0 & \text{if } x > a.
\end{array} \right.
[/tex]
(a) How many bound states are there?
(b) In the highest energy bound state, what is the probability that the particle would be found outside the well?
Homework Equations
Schrodinger's Time Independent Eqn. [itex] -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} - V(x)\psi = E \psi [/itex]
The Attempt at a Solution
Starting with [itex]0<x<a[/itex], I rewrote the S.T.I. eqn to read [itex] \frac{d^2 \psi}{dx^2} = -k^2\psi[/itex]
where [itex]k=\sqrt{2m(E-V)/\hbar^2}[/itex].
The solutions being [itex]\psi=A\cos{kx} + B\sin{kx}[/itex], I dropped the cosine term because [itex]\psi(0) = 0[/itex] leaving [tex]\psi(x) = B\sin{kx} \text{ (eqn 1)}[/tex]
When [itex] x>a, V=0 [/itex] and the S.T.I. eqn. gives [itex]\frac{d^2\psi}{dx^2} = l^2\psi[/itex] where [itex]l = \sqrt{-2mE/\hbar^2}[/itex] The solutions are exponential and I dropped the positive one because it blows up as x gets big. So I have
[tex] \psi = Ce^{-lx} \text{ (eqn 2)} [/tex]
I set eqns. 1 and 2 equal by continuity at a. I set their derivatives equal by continuity of the derivative at a. So I have
[tex] B\sin{ka}=Ce^{-la} \text{ (eqn 3)} \qquad \text{ and } \qquad kB\cos{ka} = -lCe^{-la} \text{ (eqn 4) }[/tex]
Now I divide (4) by (3) and get
[tex] k\cot{ka} = -l[/tex]
From here I'm not sure where to go. The book (I'm using Griffith's) has a similar example for the fully finite square well. In it, he defines [itex]z=ka, z_0 = (ka\text{ with }E=0)[/itex] and then does some wizardry where he plots solutions and finds their intersection points. I have tried to do this with my slightly different problem and failed. But do I even need to plot intersection points to answer part (a)? I'm kind of lost. And then how would I proceed with part (b), just calculate the probability density?
*I can't edit my typo in the thread title? Embarrassing.
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