AcademyQuantum Wave Functions
Academy
Potential Wells
Level 1 - Physics topic page in Quantum Wave Functions.
Principle
A finite potential well confines bound states by oscillation inside the well and exponential decay outside it.
Notation
\(V_0\)
well depth or outside potential height
\(\mathrm{J}\)
\(a\)
half-width for a symmetric well
\(\mathrm{m}\)
\(E\)
bound-state energy
\(\mathrm{J}\)
\(k\)
oscillation wave number inside the well
\(\mathrm{m^{-1}}\)
\(\kappa\)
decay constant outside the well
\(\mathrm{m^{-1}}\)
\(\ell\)
penetration depth
\(\mathrm{m}\)
Method
Derivation 1: Two spatial behaviors
The time-independent equation changes character depending on the sign of kinetic energy.
Classically allowed region
\[E>V\Rightarrow \frac{d^2\psi}{dx^2}=-k^2\psi\]
Classically forbidden region
\[E<V\Rightarrow \frac{d^2\psi}{dx^2}=\kappa^2\psi\]
Derivation 2: Wave numbers
The allowed region oscillates; the forbidden region decays for a bound state.
Inside a zero-potential well
\[k=\frac{\sqrt{2mE}}{\hbar}\]
Outside a finite well
\[\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}\]
Penetration depth
\[\ell=\frac{1}{\kappa}\]
Derivation 3: Matching conditions
Finite jumps in potential require smooth joining of the state and its slope.
Continuity of wave function
\[\psi_{\mathrm{left}}=\psi_{\mathrm{right}}\]
Continuity of derivative
\[\frac{d\psi_{\mathrm{left}}}{dx}=\frac{d\psi_{\mathrm{right}}}{dx}\]
Rules
For a symmetric finite well with oscillatory interior and decaying exterior:
Inside wave number
\[k=\frac{\sqrt{2mE}}{\hbar}\]
Outside decay constant
\[\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}\]
Penetration depth
\[\ell=\frac{1}{\kappa}\]
Examples
Question
If \(E\) approaches \(V_0\) from below, what happens to the outside tail?
Answer
The decay constant
\[\kappa=\sqrt{2m(V_0-E)}/\hbar\]
decreases, so the penetration depth \[1/\kappa\]
increases.Checks
- Bound states require normalizable tails, not growing exponentials.
- Finite wells allow nonzero probability outside the classically allowed region.
- Deeper or wider wells support more bound states.
- Parity simplifies symmetric wells into even and odd solutions.