Infinite Potential Well: Quantization in a Box

Table of Contents

1. Introduction
2. Concept of the Infinite Square Well
3. Mathematical Formulation
4. Schrödinger Equation Inside the Well
5. Boundary Conditions and Wavefunction Form
6. Quantized Energy Levels
7. Normalized Wavefunctions
8. Probability Distributions and Nodes
9. Expectation Values and Uncertainty
10. Comparison with Classical Mechanics
11. Time Evolution of Superpositions
12. 3D Infinite Potential Well
13. Applications in Nanotechnology and Quantum Dots
14. Idealization and Realistic Potentials
15. Conclusion

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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.

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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 \( (0, L) \)

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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.

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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.

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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:

\[
\psi_n(x) = A_n \sin\left( \frac{n\pi x}{L} \right)
\]

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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

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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.

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8. Probability Distributions and Nodes

• \( |\psi_n(x)|^2 \) gives probability density
• Number of nodes = \( n – 1 \)
• Higher \( n \): more oscillatory behavior

Expectation values:

\[
\langle x \rangle = \frac{L}{2}, \quad \langle x^2 \rangle = \frac{L^2}{3} – \frac{L^2}{2\pi^2 n^2}
\]

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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.

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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)

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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 \( e^{-iE_n t/\hbar} \)

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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 \( n_x^2 + n_y^2 + n_z^2 \).

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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

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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.

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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.

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