Schrödinger Equation Explained – Time-Independent & Time-Dependent Forms

Schrödinger Wave Equation

The fundamental mathematical framework of wave mechanics, encompassing both time-independent and time-dependent formulations.

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Table of Contents

1. Introduction & Significance
2. Time-Independent Schrödinger Equation
   1. Derivation from the Wave Equation
   2. Application: Particle in a One-Dimensional Box
3. Time-Dependent Schrödinger Equation
4. Separation of Variables
5. Physical Interpretation of the Wave Function
6. Normalization of the Wave Function
7. Required Properties of Acceptable Wave Functions

Introduction & Significance

In 1926, Erwin Schrödinger formulated the wave equation that describes how quantum “matter waves” evolve. The Schrödinger equation plays an analogous role in quantum mechanics to Newton’s laws in classical mechanics. It provides a differential equation governing the complex wave function ψ(r,t), whose solutions encode all measurable properties of microscopic particles.

Time-Independent Schrödinger Equation

Derivation from the Wave Equation

The general wave equation in three dimensions is:

∇²u = (1/v²) ∂²u/∂t²

Replacing u by the quantum wave function ψ(r,t) and noting the de Broglie relation v→ω/k yields:

∇²ψ = (1/v²) ∂²ψ/∂t²

Assuming a separable solution ψ(r,t)=ψ(r)e^−iωt leads to:

∇²ψ(r) + (ω²/v²)ψ(r) = 0

Introducing momentum p=ħk and total energy E=p²/2m+V gives the time-independent Schrödinger equation:

−(ħ²/2m)∇²ψ(r) + V(r)ψ(r) = E ψ(r)

Application: Particle in a One-Dimensional Box

For an infinite potential well of width L (0≤x≤L, V=0 inside, V→∞ outside), the TISE becomes:

−(ħ²/2m) d²ψ/dx² = E ψ,   ψ(0)=ψ(L)=0

The normalized solutions are:

ψ_n(x) = √(2/L) sin(nπx/L),   E_n = (n²π²ħ²)/(2mL²),   n=1,2,3,...

Time-Dependent Schrödinger Equation

Starting from the separable ansatz and energy operator Ĥψ=Eψ, one obtains:

iħ ∂ψ/∂t = [−(ħ²/2m)∇² + V(r,t)] ψ(r,t)

This full equation governs non-stationary processes such as transitions and wave‐packet evolution.

Separation of Variables

Assuming V(r,t)=V(r) only, set ψ(r,t)=φ(r)T(t). Substitution yields two ODEs:

−(ħ²/2m)∇²φ + Vφ = Eφ, iħ dT/dt = E T

The spatial part is the TISE; the temporal part gives T(t)=e^−iEt/ħ.

Physical Interpretation of the Wave Function

Max Born’s statistical interpretation assigns |ψ(r,t)|² as the probability density of finding the particle at position r and time t. ψ itself is a complex probability amplitude.

Normalization of the Wave Function

The total probability must equal one:

∫ all space |ψ(r,t)|² d³r = 1

If ψ is not normalized, define the normalization constant N:

N = [∫|ψ|² d³r]¹ᐟ²,   ψ_normalized = ψ/N

Required Properties of Acceptable Wave Functions

• ψ must be finite everywhere (no infinite probabilities).
• ψ must be single-valued (unique probability at each point).
• ψ and ∂ψ/∂x must be continuous (ensures finite kinetic energy).

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