Eigenvalues & Eigenfunctions — Schrödinger Equation and the Particle-in-a-Box
1. Introduction — eigenvalues & eigenfunctions
When we solve the time-independent Schrödinger equation under appropriate boundary conditions, only particular energy values E_n allow nontrivial solutions. These allowed energies are called eigenvalues, and the associated wavefunctions ψ_n(x) are called eigenfunctions.
Mathematically this is an eigenvalue problem: an operator (the Hamiltonian) acting on a function gives back the same function multiplied by a scalar (the eigenvalue).
2. Time-independent Schrödinger equation (TISE)
The TISE for a single non-relativistic particle in one dimension is
−(ħ² / 2m) (d²ψ/dx²) + V(x) ψ(x) = E ψ(x)
Here ħ is reduced Planck's constant, m the particle mass, V(x) the potential energy and E the (total) energy eigenvalue.
3. Free particle solution (plane waves)
For a free particle, V(x)=0. The TISE reduces to
d²ψ/dx² + k² ψ = 0, where k² = 2mE / ħ²
The general solution is a linear combination of plane waves:
ψ(x) = A e^{i k x} + B e^{−i k x}
A e^{i k x} represents a right-moving wave and B e^{−i k x} a left-moving wave. For truly free particles extending to infinity these plane waves are not square-integrable (not normalizable) — they are useful as momentum eigenstates and in scattering theory. Real physical states are wave packets formed from superpositions of plane waves.
4. Particle in a 1D infinite potential well (particle in a box)
Consider a one-dimensional box of width L with infinite potential walls at x=0 and x=L:
V(x) = 0 for 0 < x < L V(x) = ∞ otherwise
Inside the box the TISE is the same as the free particle equation:
d²ψ/dx² + k² ψ = 0, 0 < x < L
Boundary conditions: the wavefunction must vanish at the infinite walls:
ψ(0) = 0, ψ(L) = 0
5. Derivation of energy eigenvalues
General solution inside the box:
ψ(x) = A sin(k x) + B cos(k x)
Apply boundary at x=0 → ψ(0)=0 gives B = 0. So ψ(x)=A sin(kx).
Apply boundary at x=L → ψ(L)=0 ⇒ A sin(kL)=0. For nontrivial A, we require
sin(k L) = 0 ⇒ k L = n π, n = 1,2,3,...
Thus allowed wave numbers are k_n = n π / L. Using k² = 2mE / ħ²:
E_n = (ħ² k_n²) / (2m) = (ħ² π² n²) / (2m L²) = (n² h²) / (8 m L²)
So the energy eigenvalues are discrete and scale like n². Note: n=0 is excluded because it yields ψ≡0 (the trivial solution).
6. Eigenfunctions & normalization
The (unnormalized) eigenfunction is
ψ_n(x) = A sin(n π x / L)
Normalization requires ∫₀ᴸ |ψ_n(x)|² dx = 1. Compute:
∫₀ᴸ A² sin²(n π x / L) dx = A² (L/2) ⇒ A = √(2/L)
Therefore the normalized eigenfunctions are
ψ_n(x) = √(2/L) sin(n π x / L), n = 1,2,3,...
These functions are orthonormal:
∫₀ᴸ ψ_m(x) ψ_n(x) dx = δ_{mn}
where δ_{mn} is the Kronecker delta (1 if m=n, 0 otherwise). Orthogonality follows from trigonometric integrals.
7. Orthogonality, completeness & parity
The set {ψ_n(x)} forms a complete basis for square-integrable functions on [0,L] that satisfy the boundary conditions. Any acceptable wavefunction can be expanded as
Ψ(x,t=0) = ∑_{n=1}^∞ c_n ψ_n(x)
with coefficients c_n = ∫₀ᴸ ψ_n(x) Ψ(x,0) dx. The parity of ψ_n alternates: ψ_n has n−1 nodes inside the box; the ground state n=1 has no internal node and is maximum at center.
8. Physical interpretation & probability densities
The probability density is |ψ_n(x)|². For n=1 the density peaks at the center; for n=2 there is a node at the center (zero probability). The quantization means the particle cannot have arbitrary energy while confined — only the discrete E_n are allowed.
Figure: qualitative sketch of the first three eigenfunctions inside the infinite well (nodes increase with n). Probability densities |ψ_n|² are the square of these shapes.
9. Worked examples (numerical)
Example 1 — Electron in a 1 nm box
Let L = 1 nm = 1×10⁻⁹ m, electron mass m = 9.11×10⁻³¹ kg. Ground state energy E1:
E1 = (h²) / (8 m L²) ≈ (6.626×10⁻³⁴)² / (8 × 9.11×10⁻³¹ × (1×10⁻⁹)²) ≈ 6.02×10⁻²⁰ J ≈ 0.376 eV
So the ground-state energy is about 0.38 eV; excited levels scale as n² (E2 ≈ 4E1 ≈ 1.50 eV, etc.).
Example 2 — Particle in atomic-scale box L = 0.1 nm
For L = 1×10⁻¹⁰ m, E1 ≈ 37.6 eV (order-of-magnitude), showing much larger quantization energy when confinement length is atomic scale.
10. Extensions: 3D box, degeneracy, finite wells
3D box: For a rectangular box of sides Lx,Ly,Lz, eigenfunctions separate and
E_{n_x n_y n_z} = (ħ² π² / 2m) (n_x² / L_x² + n_y² / L_y² + n_z² / L_z²)
Degeneracy arises when different triplets give the same E. For a cubic box Lx=Ly=Lz degeneracy increases with n.
Finite potential well: If walls are finite, wavefunctions penetrate (evanescent tails) into classically forbidden regions; energy quantization still occurs but energies are lower and a finite number of bound states exist.
11. Applications & why this matters
- Quantum dots: Semiconductor nanoparticles behave like particles in a box; their optical properties depend on size via quantized energy levels.
- Nanostructures: Electron confinement in wells, wires and dots underpins modern nanoelectronics.
- Foundations: The particle-in-a-box is a solvable model that illustrates quantization, boundary conditions, orthogonality and normalization — core concepts in quantum mechanics.
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