Table of Contents
- Introduction
- Physical Setup of the Finite Square Well
- Mathematical Formulation
- Schrödinger Equation in Different Regions
- Bound State Conditions
- Even and Odd Solutions
- Transcendental Equations for Energy Levels
- Number of Bound States
- Comparison with Infinite Square Well
- Normalized Wavefunctions
- Probability Density and Penetration into Barriers
- Quantum Tunneling and Decay
- Energy Quantization and Graphical Solutions
- Semi-Classical Approximation
- Real-World Applications
- Conclusion
1. Introduction
The finite square well is a foundational quantum mechanical model used to illustrate bound states, tunneling, and the emergence of discrete energy levels within a potential that is not infinite. It represents a more realistic version of the infinite potential well and is widely applicable in solid-state physics and quantum devices.
2. Physical Setup of the Finite Square Well
The potential \( V(x) \) is defined as:
\[
V(x) = \begin{cases}
-V_0, & \text{for } |x| \le a \
0, & \text{for } |x| > a
\end{cases}
\]
- Depth: \( V_0 > 0 \)
- Width: \( 2a \)
- Particle mass: \( m \)
3. Mathematical Formulation
We solve the time-independent Schrödinger equation:
\[
-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)
\]
We consider bound states: \( -V_0 < E < 0 \)
4. Schrödinger Equation in Different Regions
Let:
- Region I: \( x < -a \)
- Region II: \( |x| \le a \)
- Region III: \( x > a \)
Region I and III:
\[
\frac{d^2 \psi}{dx^2} = \kappa^2 \psi, \quad \kappa = \sqrt{\frac{-2mE}{\hbar^2}}
\]
\[
\psi(x) \propto e^{-\kappa |x|}
\]
Region II:
\[
\frac{d^2 \psi}{dx^2} = -k^2 \psi, \quad k = \sqrt{\frac{2m(E + V_0)}{\hbar^2}}
\]
\[
\psi(x) \propto \cos(kx) \text{ or } \sin(kx)
\]
5. Bound State Conditions
The wavefunction must be:
- Continuous at \( x = \pm a \)
- Smooth (derivative continuous)
- Normalizable (finite total probability)
6. Even and Odd Solutions
Symmetric potential allows classification:
- Even parity: \( \psi(x) = \psi(-x) \)
\[
\psi(x) = \begin{cases}
A \cos(kx), & |x| \le a \
B e^{-\kappa x}, & x > a
\end{cases}
\] - Odd parity: \( \psi(x) = -\psi(-x) \)
\[
\psi(x) = \begin{cases}
A \sin(kx), & |x| \le a \
B e^{-\kappa x}, & x > a
\end{cases}
\]
7. Transcendental Equations for Energy Levels
Apply boundary conditions to obtain:
Even states:
\[
k \tan(ka) = \kappa
\]
Odd states:
\[
k \cot(ka) = -\kappa
\]
Use:
\[
k = \sqrt{\frac{2m(E + V_0)}{\hbar^2}}, \quad \kappa = \sqrt{\frac{-2mE}{\hbar^2}}
\]
These transcendental equations must be solved graphically or numerically.
8. Number of Bound States
Finite number of bound states, depending on well parameters:
\[
\lambda = \frac{a}{\hbar} \sqrt{2mV_0}
\]
Each solution to the transcendental equation gives one bound state. As \( V_0 \to \infty \), the number of bound states increases.
9. Comparison with Infinite Square Well
| Property | Infinite Well | Finite Well |
|---|---|---|
| Wall Height | \( \infty \) | Finite |
| Energy Levels | \( E_n \propto n^2 \) | Levels determined numerically |
| Penetration | None | Exponential decay into classically forbidden regions |
| Number of States | Infinite | Finite |
10. Normalized Wavefunctions
Wavefunctions are continuous and piecewise-defined. Normalization involves integrating over both well and tail regions.
\[
\int_{-\infty}^\infty |\psi(x)|^2 dx = 1
\]
11. Probability Density and Penetration into Barriers
- \( |\psi(x)|^2 \) decays exponentially in regions \( |x| > a \)
- Probability of finding the particle outside the well is non-zero
- Demonstrates quantum tunneling
12. Quantum Tunneling and Decay
A particle confined in a well can leak into classically forbidden regions — a feature absent in classical mechanics. This is the basis of:
- Nuclear alpha decay
- Electron tunneling in semiconductors
- Quantum scanning devices
13. Energy Quantization and Graphical Solutions
Plot left and right-hand sides of:
\[
k \tan(ka) = \kappa \quad \text{and} \quad k \cot(ka) = -\kappa
\]
Intersections determine allowed \( k \), and hence allowed energies \( E_n \).
14. Semi-Classical Approximation
In large wells or for high energies:
- Energy levels become closer together
- Bound state energies approach those of the infinite well
- WKB approximation gives accurate estimates
15. Real-World Applications
- Modeling quantum dots, wells, and barriers
- Explaining tunneling and resonance in electronics
- Foundations of quantum well lasers, MOSFETs, and heterostructures
16. Conclusion
The finite square well introduces the important concept of bound quantum states in finite potentials and demonstrates the tunneling effect. Its blend of analytic and numerical richness makes it an essential tool for building quantum intuition, with wide applications in physics, chemistry, and nanotechnology.
