5. Delta Function Potential
Reading time: ~22 minutes | Pages: 6
An exactly solvable model with a point interaction, illustrating bound states and scattering.
The Potential
Attractive delta function:
$$V(x) = -\alpha\delta(x), \quad \alpha > 0$$
Infinitely deep, infinitesimally narrow well with "strength" $\alpha$
Bound State ($E < 0$)
There is exactly one bound state:
$$E = -\frac{m\alpha^2}{2\hbar^2}$$
Wave function:
$$\psi(x) = \frac{\sqrt{m\alpha}}{\hbar}e^{-m\alpha|x|/\hbar^2}$$
Exponential decay on both sides, cusp at origin
Boundary Condition at Delta Function
Continuity: $\psi$ is continuous at $x = 0$
Discontinuity in derivative:
$$\frac{d\psi}{dx}\bigg|_{0^+} - \frac{d\psi}{dx}\bigg|_{0^-} = -\frac{2m\alpha}{\hbar^2}\psi(0)$$
This condition replaces solving inside the delta function
Scattering States ($E > 0$)
For $x < 0$ (incident from left):
$$\psi_L(x) = Ae^{ikx} + Be^{-ikx}$$
For $x > 0$:
$$\psi_R(x) = Ce^{ikx}$$
where $k = \sqrt{2mE}/\hbar$
Transmission and Reflection Coefficients
Reflection coefficient:
$$R = \frac{1}{1 + (2\hbar^2 E/m\alpha^2)}$$
Transmission coefficient:
$$T = \frac{1}{1 + (m\alpha^2/2\hbar^2 E)} = 1 - R$$
Note: $T \to 1$ as $E \to \infty$ (high energy particles barely notice potential)
Double Delta Function
Two delta functions at $x = \pm a$:
$$V(x) = -\alpha[\delta(x-a) + \delta(x+a)]$$
Results:
- Two bound states (symmetric and antisymmetric)
- Resonances in transmission coefficient
- Simple model for molecular bonding
Delta Function Barrier
Repulsive delta function ($V(x) = +\alpha\delta(x)$):
- No bound states
- Pure scattering: reflection and transmission
- Same $T$ and $R$ formulas as attractive case
Physical Significance
- Point interactions: Model for very short-range forces
- Contacts: Junctions in quantum wires
- Impurities: Defects in crystals
- Teaching: Illustrates bound states without complicated math