Introduction: Equations You Can't Solve Exactly
Matching boundary conditions for the finite well leads to transcendental equations—equations mixing algebraic and trigonometric functions that must be solved graphically or numerically.
The Matching Conditions
Continuity of \(\psi\) and \(\psi'\) at \(x = a\) gives:
Even states: \(k\tan(ka) = \kappa\)
Odd states: \(-k\cot(ka) = \kappa\)
where \(k^2 + \kappa^2 = 2mV_0/\hbar^2\)
Graphical Solution
Define \(z = ka\) and \(z_0 = a\sqrt{2mV_0}/\hbar\). Then:
Plot \(\sqrt{z_0^2 - z^2}\) (circle) and \(z\tan z\) (for even) or \(-z\cot z\) (for odd)
Intersections give allowed energies.
Worked Example
For \(z_0 = \pi\) (moderately deep well):
- One intersection with even curve near \(z \approx 1.3\)
- One intersection with odd curve near \(z \approx 2.8\)
- Two bound states total
The Quantum Connection
Transcendental equations appear throughout quantum mechanics. They can't be solved in "closed form" but numerical methods handle them easily. The number of bound states increases with \(z_0\)—deeper or wider wells hold more states. A shallow well (\(z_0 < \pi/2\)) has exactly one bound state.