Table of Contents
- Introduction
- Dirac Delta Function in Physics
- Delta Function Potential Definition
- Schrödinger Equation with Delta Potential
- Bound State Solution for Attractive Delta Potential
- Normalization and Energy of the Bound State
- Scattering from a Delta Potential
- Reflection and Transmission Coefficients
- Discontinuity in Wavefunction Derivative
- Comparison with Finite and Infinite Wells
- Delta Potentials in 3D and Higher Dimensions
- Role in Quantum Field Theory and Models
- Multiple Delta Potentials
- Applications in Semiconductors and Modeling
- Conclusion
1. Introduction
The delta function potential is a powerful yet simple model in quantum mechanics. It captures the essence of quantum binding and scattering in a system with the most minimal potential: zero everywhere except at a single point, where it is infinitely strong and narrow. Despite its simplicity, it leads to exact and insightful results.
2. Dirac Delta Function in Physics
The Dirac delta function \( \delta(x) \) is not a function in the traditional sense but a distribution:
\[
\delta(x) = 0 \ \text{for } x \neq 0, \quad \int_{-\infty}^{\infty} \delta(x) dx = 1
\]
It models idealized point-like sources or interactions.
3. Delta Function Potential Definition
The one-dimensional delta potential is:
\[
V(x) = -\alpha \delta(x), \quad \alpha > 0
\]
- Attractive: supports bound states
- Repulsive: does not support bound states, only affects scattering
4. Schrödinger Equation with Delta Potential
Time-independent Schrödinger equation:
\[
- \frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} – \alpha \delta(x)\psi(x) = E\psi(x)
\]
We solve for:
- \( E < 0 \): bound state
- \( E > 0 \): scattering states
5. Bound State Solution for Attractive Delta Potential
Assume \( E < 0 \), let:
\[
\kappa = \frac{\sqrt{2m|E|}}{\hbar}
\]
Solution:
\[
\psi(x) = A e^{-\kappa |x|}
\]
Continuity at \( x = 0 \), and derivative condition from integrating Schrödinger equation:
\[
\frac{d\psi}{dx}\bigg|{0^+} – \frac{d\psi}{dx}\bigg|{0^-} = -\frac{2m\alpha}{\hbar^2} \psi(0)
\]
Substitute \( \psi(0) = A \), gives:
\[
2\kappa A = \frac{2m\alpha}{\hbar^2} A \Rightarrow \kappa = \frac{m\alpha}{\hbar^2}
\]
6. Normalization and Energy of the Bound State
Normalized wavefunction:
\[
\psi(x) = \sqrt{\kappa} e^{-\kappa |x|}, \quad \text{with } \kappa = \frac{m\alpha}{\hbar^2}
\]
Energy:
\[
E = -\frac{m\alpha^2}{2\hbar^2}
\]
Only one bound state exists, regardless of the strength of \( \alpha \).
7. Scattering from a Delta Potential
For \( E > 0 \), use plane waves:
- Left of origin:
\[
\psi_L(x) = e^{ikx} + R e^{-ikx}
\] - Right of origin:
\[
\psi_R(x) = T e^{ikx}
\]
Apply boundary conditions:
- Continuity at \( x = 0 \)
- Discontinuity in derivative:
\[
\psi'(0^+) – \psi'(0^-) = -\frac{2m\alpha}{\hbar^2} \psi(0)
\]
8. Reflection and Transmission Coefficients
Solve to find:
\[
T = \frac{1}{1 + \left( \frac{m\alpha}{\hbar^2 k} \right)^2}, \quad R = 1 – T
\]
Transmission decreases with increasing \( \alpha \), increases with particle energy \( E = \frac{\hbar^2 k^2}{2m} \).
9. Discontinuity in Wavefunction Derivative
Delta potential causes a finite discontinuity in derivative of \( \psi(x) \), while \( \psi(x) \) itself remains continuous. This is a defining feature of delta function interactions.
10. Comparison with Finite and Infinite Wells
| Potential Type | Number of Bound States | Behavior Outside Well |
|---|---|---|
| Infinite well | Infinite | Zero |
| Finite square well | Finite (>1) | Decaying |
| Delta potential | Exactly one | Exponential decay |
Delta potential is the minimal system showing binding.
11. Delta Potentials in 3D and Higher Dimensions
In 3D, naive use of delta potentials leads to divergence. Requires regularization or renormalization. Still used as approximations for short-range interactions.
12. Role in Quantum Field Theory and Models
Delta potentials appear as:
- Contact interactions in effective field theories
- Solvable toy models for bound states and renormalization
- Tools in scattering theory and many-body systems
13. Multiple Delta Potentials
Combining deltas gives rich models:
- Two delta wells: double-well systems with splitting
- Periodic deltas: Kronig-Penney model for band theory in solids
14. Applications in Semiconductors and Modeling
- Models ultra-thin quantum wells and interface states
- Describes impurities, point defects, and atomic traps
- Basis for teaching and approximating more complex interactions
15. Conclusion
The delta function potential is deceptively simple yet rich in physical insight. It encapsulates binding, tunneling, reflection, and quantum scattering in a point-like interaction. Serving as both a pedagogical tool and a modeling technique, it exemplifies the deep consequences of quantum theory with minimal mathematical complexity.
