Infinite Square Well

Overview

Infinite square well: idealized quantum system with infinitely high potential walls confining a particle. Model: particle trapped in one-dimensional region with zero potential energy inside well and infinite potential outside. Significance: demonstrates quantized energy levels, wavefunction behavior, and boundary constraints fundamental to quantum mechanics.

"The infinite square well is the simplest example illustrating quantization in quantum mechanics." -- David J. Griffiths

Potential Definition

Mathematical Expression

Potential function V(x) defined piecewise:

V(x) = { 0, 0 < x < L ∞, otherwise}

Physical Meaning

Particle confined strictly within interval [0, L]. Outside: infinite potential prohibits penetration. Model: perfect confinement, no tunneling.

Significance

Simplifies Schrödinger equation. Boundary conditions become Dirichlet type (wavefunction zero at walls). Enables analytical solutions.

Schrödinger Equation Formulation

Time-Independent Equation

One-dimensional time-independent Schrödinger equation (TISE) inside well:

- (ħ² / 2m) (d²ψ/dx²) = Eψ, 0 < x < L

Regions Outside Well

V(x) = ∞, wavefunction ψ(x) = 0 due to infinite potential barrier.

Variables and Constants

ħ: reduced Planck constant, m: particle mass, E: energy eigenvalue, ψ: wavefunction.

Boundary Conditions

Wavefunction at Walls

ψ(0) = 0, ψ(L) = 0 due to infinite potential barriers.

Physical Reasoning

Probability of finding particle outside well is zero. Continuity and finiteness of ψ enforced inside well.

Implications

Quantization of allowed wavefunctions. Discrete energy levels arise from these constraints.

Energy Eigenvalues

Quantized Energy Formula

Allowed energy levels given by:

E_n = (n² π² ħ²) / (2m L²), n = 1, 2, 3, ...

Energy Level Characteristics

Discrete, non-degenerate, increasing quadratically with quantum number n.

Physical Interpretation

Lowest energy E_1 nonzero: zero-point energy prevents particle from rest.

Eigenfunctions and Wavefunctions

Analytical Form

Normalized eigenfunctions inside well:

ψ_n(x) = sqrt(2/L) * sin(n π x / L), 0 < x < L

Boundary Conditions Satisfaction

ψ_n(0) = 0, ψ_n(L) = 0 satisfied by sine functions.

Orthogonality and Completeness

Eigenfunctions orthogonal: ∫₀ᴸ ψ_m(x) ψ_n(x) dx = δ_mn. Form complete basis in Hilbert space.

Normalization

Normalization Condition

Total probability must equal one:

∫₀ᴸ |ψ_n(x)|² dx = 1

Normalization Constant

Constant sqrt(2/L) ensures integral condition met.

Verification

Integral of sin² over [0,L] equals L/2; constant corrects to unity.

Physical Interpretation

Probability Density

|ψ_n(x)|² gives likelihood of particle position within well.

Energy Quantization

Discrete energies reflect quantum confinement and wave nature.

Zero-Point Energy

Lowest energy state nonzero, unlike classical particle at rest.

Applications

Quantum Dots and Nanostructures

Model electrons confined in semiconductor nanostructures.

Pedagogical Tool

Introduces core quantum concepts: quantization, boundary effects.

Approximate Analytical Solutions

Basis for more complex quantum systems via perturbation and numerical methods.

Extensions and Generalizations

Finite Square Well

Finite potential barriers allow tunneling and evanescent waves.

Higher Dimensions

2D and 3D infinite wells model particles in boxes, quantum dots.

Multiple Wells and Superlattices

Arrays of wells produce band structures and minibands.

Limitations

Idealization

Infinite potential walls unphysical; no real system has infinite barriers.

No Tunneling

Ignores penetration into classically forbidden regions.

One Dimensionality

Cannot capture full spatial complexity of real quantum systems.

Numerical Solutions

Discretization Methods

Finite difference and matrix diagonalization solve generalized wells.

Eigenvalue Solvers

Numerical routines find energies and wavefunctions for complex potentials.

Computational Tools

MATLAB, Python (NumPy, SciPy), and others implement infinite well simulations.

References

• Griffiths, D.J., Introduction to Quantum Mechanics, 2nd Ed., Pearson, 2005, pp. 50-70.
• Shankar, R., Principles of Quantum Mechanics, 2nd Ed., Springer, 1994, pp. 120-135.
• Liboff, R.L., Quantum Mechanics, 4th Ed., Addison-Wesley, 2003, pp. 75-90.
• Sakurai, J.J., Modern Quantum Mechanics, Revised Ed., Addison-Wesley, 1994, pp. 45-60.
• Cohen-Tannoudji, C., Diu, B., Laloe, F., Quantum Mechanics, Wiley, 1977, pp. 105-125.