Schrödinger's equation (time-dependent and independent)

Schrödinger's equation (time-dependent and independent)

1. Time-Dependent Schrödinger Equation (TDSE) — The Fundamental Equation

For particle mass $m$ in potential $V(r,t)$:

In 1D:

It is first order in time, second in space, linear, complex. It determines how $ψ$ evolves; $|ψ|²$ evolves via continuity equation. For free particle $V=0$, solution is plane wave $ψ=A e^{i(kx-ωt)}$ with $E=ħω=p²/2m$.

2. Time-Independent Schrödinger Equation (TISE) — Stationary States

If $V=V(r)$ independent of $t$, separate $ψ(r,t)= φ(r) e^{-iEt/ħ}$ ⇒ $iħ ∂ψ/∂t =Eψ$ and:

In 1D:

This is eigenvalue equation: $φ$ is eigenfunction, $E$ eigenenergy. Solutions that satisfy boundary conditions give quantized $E_n$.

3. Meaning

• TDSE: how state evolves (like $F=m a$).
• TISE: finds allowed stationary energies and standing wave shapes for time-independent potentials; full solution then $ψ=φ e^{-iEt/ħ}$ with $|ψ|²$ time-independent.

4. Worked Illustration

Free particle V=0: TISE -ħ²/2m d²φ/dx² =E φ ⇒ φ=A e^{ikx}+B e^{-ikx}, k=√(2mE)/ħ, E continuous ≥0

Infinite well V=0 inside 0<x<L, ∞ outside: TISE gives -ħ²/2m φ''=E φ with φ=0 at 0,L ⇒ φ_n=√(2/L) sin(nπx/L), E_n=n²h²/(8mL²) (next topic)

5. Exam Points

• Write TDSE $iħ∂ψ/∂t = [-ħ²/2m ∇²+V]ψ$ — 3-5 marks, very frequent.
• Derive TISE by separation $ψ=φ e^{-iEt/ħ}$ — 5 marks, frequent.
• Explain stationary state $|ψ|²$ time-independent — 3 marks.
• Distinguish time-dependent vs independent and when to use — 3-5 marks.