Energy Eigen values and Eigen functions

Energy Eigen values and Eigen functions

1. What They Are (TISE Language)

Time-independent Schrödinger equation:

is an eigenvalue equation: $φ$ is eigenfunction (wavefunction), $E$ is eigenvalue (allowed energy), $\hat{H} = -ħ²/2m ∇² + V$ is the Hamiltonian operator.

• Eigenfunctions $φ_n$ form complete orthonormal set: $∫ φ_m* φ_n dV = δ_{mn}$, any state $ψ = Σ c_n φ_n$.
• Eigenvalues $E_n$ are discrete for bound states (infinite box, harmonic oscillator, hydrogen) and continuous for free particle.
• Each $E_n$ corresponds to one (or more if degenerate) $φ_n$; ground state is lowest $E_1$, no nodes; higher $n$ more nodes, higher energy.

2. For Particle in a Box (example)

$\hat{H} = -ħ²/2m d²/dx²$ inside. Solving gave $φ_n =√(2/L) sin(nπx/L)$, $E_n=n²h²/8mL²$ — these are eigen pairs. Orthonormal: $∫_0^L φ_m φ_n dx =δ_{mn}$.

3. General Properties

• $E_n$ real, $φ_n$ can be chosen real for 1D bound states.
• Non-degenerate in 1D bound; in 3D, degeneracy when symmetry (e.g., cubic box $L_x=L_y=L_z$ ⇒ $E$ depends on $n_x²+n_y²+n_z²$, permutations same $E$).
• Superposition: any $ψ$ is $Σ c_n φ_n e^{-iE_n t/ħ}$; probability $|c_n|²$ to measure $E_n$.

4. Hydrogen Preview

For Coulomb $V=-e²/4π ε0 r$, TISE in spherical gives

, eigenfunctions spherical harmonics × radial; degeneracy $n²$.

5. Worked Checks

(a) Show φ_n normalized: ∫_0^L (2/L) sin²(nπx/L)dx=1 ✓

(b) Particle in box: orthonormal? ∫ φ1 φ2 =0 (different n orthogonal) ✓

(c) Need to find expectation ⟨x⟩ for φ1: ⟨x⟩=L/2 (symmetry), for φ2 also L/2 but variance larger.

(d) Degeneracy: cubic L= a: E for (1,1,2),(1,2,1),(2,1,1) all same 6h²/8ma² ⇒ threefold degenerate.

6. Exam Points

• Define eigenvalue equation $\hat{H}φ=Eφ$ — 3-5 marks, very frequent.
• State that boundary conditions quantize $E_n$ — 3 marks.
• Give $E_n, φ_n$ for box as example — 5-7 marks, very frequent.
• Mention orthonormality and degeneracy — 3-5 marks, frequent.