Table of Contents
- Introduction
- Concept of the Infinite Square Well
- Mathematical Formulation
- Schrödinger Equation Inside the Well
- Boundary Conditions and Wavefunction Form
- Quantized Energy Levels
- Normalized Wavefunctions
- Probability Distributions and Nodes
- Expectation Values and Uncertainty
- Comparison with Classical Mechanics
- Time Evolution of Superpositions
- 3D Infinite Potential Well
- Applications in Nanotechnology and Quantum Dots
- Idealization and Realistic Potentials
- Conclusion
1. Introduction
The infinite potential well, also known as the particle in a box, is one of the most fundamental models in quantum mechanics. It exemplifies how quantum confinement leads to energy quantization, and is a cornerstone for understanding more complex systems like atoms, molecules, and quantum wells in nanotechnology.
2. Concept of the Infinite Square Well
- A particle is confined between two impenetrable walls at \( x = 0 \) and \( x = L \)
- The potential \( V(x) \) is defined as:
\[
V(x) = \begin{cases}
0, & 0 < x < L \
\infty, & \text{otherwise}
\end{cases}
\]
- The particle is strictly confined in the region \( (0, L) \)
3. Mathematical Formulation
Inside the well, the time-independent Schrödinger equation is:
\[
\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} = E \psi(x)
\]
Outside the well, \( \psi(x) = 0 \) due to the infinite potential.
4. Schrödinger Equation Inside the Well
Define:
\[
k^2 = \frac{2mE}{\hbar^2}
\]
The general solution inside the well is:
\[
\psi(x) = A \sin(kx) + B \cos(kx)
\]
Apply boundary conditions to determine constants.
5. Boundary Conditions and Wavefunction Form
- At \( x = 0 \), \( \psi(0) = 0 \) ⇒ \( B = 0 \)
- At \( x = L \), \( \psi(L) = 0 \) ⇒ \( \sin(kL) = 0 \)
So:
\[
k = \frac{n\pi}{L}, \quad n = 1, 2, 3, \dots
\]
Thus, wavefunctions are:
\[
\psi_n(x) = A_n \sin\left( \frac{n\pi x}{L} \right)
\]
6. Quantized Energy Levels
Plug \( k \) into energy expression:
\[
E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}
\]
Key features:
- Discrete, non-zero ground state energy
- Increases with \( n^2 \)
- No degeneracy in 1D
7. Normalized Wavefunctions
Normalization condition:
\[
\int_0^L |\psi_n(x)|^2 dx = 1
\]
Gives:
\[
\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right)
\]
Each \( n \) corresponds to a unique eigenstate.
8. Probability Distributions and Nodes
- \( |\psi_n(x)|^2 \) gives probability density
- Number of nodes = \( n – 1 \)
- Higher \( n \): more oscillatory behavior
Expectation values:
\[
\langle x \rangle = \frac{L}{2}, \quad \langle x^2 \rangle = \frac{L^2}{3} – \frac{L^2}{2\pi^2 n^2}
\]
9. Expectation Values and Uncertainty
Uncertainty in position:
\[
\Delta x = \sqrt{\langle x^2 \rangle – \langle x \rangle^2}
\]
Increases with \( n \), but still bounded due to confinement.
10. Comparison with Classical Mechanics
- Classical particle has uniform probability across the well
- Quantum particle has non-uniform, oscillating probability
- As \( n \to \infty \), quantum distribution approaches classical (correspondence principle)
11. Time Evolution of Superpositions
General state:
\[
\Psi(x,t) = \sum_n c_n \psi_n(x) e^{-iE_n t/\hbar}
\]
- Leads to quantum beating, revivals, and interference
- Time-dependence arises from phase factors \( e^{-iE_n t/\hbar} \)
12. 3D Infinite Potential Well
For a cubic box of size \( L \):
\[
\psi_{n_x,n_y,n_z}(x,y,z) = \left( \frac{2}{L} \right)^{3/2} \sin\left( \frac{n_x \pi x}{L} \right) \sin\left( \frac{n_y \pi y}{L} \right) \sin\left( \frac{n_z \pi z}{L} \right)
\]
\[
E = \frac{\pi^2 \hbar^2}{2mL^2} (n_x^2 + n_y^2 + n_z^2)
\]
Degeneracy occurs for states with same \( n_x^2 + n_y^2 + n_z^2 \).
13. Applications in Nanotechnology and Quantum Dots
- Electrons in quantum wells and nanostructures behave like particles in boxes
- Infinite well approximates confinement in thin films and quantum dots
- Optical properties and energy levels are engineered using well geometry
14. Idealization and Realistic Potentials
- Infinite well is an ideal model
- Real systems have finite potential barriers
- Leads to tunneling, resonance, and bound states with leakage
Still, infinite wells give excellent qualitative insight.
15. Conclusion
The infinite potential well offers a crystal-clear example of quantum confinement, discrete energy levels, and wavefunction behavior. Though idealized, it forms the basis of quantum mechanics education and inspires real-world applications in nanophysics, photonics, and quantum computing. It captures the essence of quantization — the idea that even free particles can only exist in discrete energy states due to boundary conditions.
