Wave function and its characteristics

1. What the Wave Function Is

Wave function $ψ(r,t)$ (complex) is the state description in quantum mechanics: it encodes everything knowable about a particle. Born's interpretation (1926):

|ψ(r,t)|² = probability density (probability per unit volume to find particle near r at t)

so $∫_{V} |ψ|² dV$ = probability to find particle in $V$ . $ψ$ itself has no direct physical meaning; only $|ψ|²$ does.

2. Normalization and Probability

Total probability must be 1:

∫_{all space} |ψ(r,t)|² dV =1 (normalized)

If $ψ$ not normalized, normalize:

ψ_{norm}= ψ/√(∫|ψ|² dV)

Expectation value:

⟨A⟩ = ∫ ψ* \hat{A} ψ dV

where $\hat{A}$ is operator (e.g., $x$ , $p=-iħ∇$ ).

$ψ$ must be single-valued (one value per point).

3. Characteristics / Conditions for a Physically Acceptable $ψ$

Condition	Why
Finite everywhere (not ±∞)	probability cannot be infinite
Single-valued	one probability per point
Continuous ( $ψ$ itself)	else $dψ/dx$ ambiguous
Continuous derivative $dψ/dx$ (except where $V$ →∞)	ensures $p$ expectation finite; at infinite wall, derivative may jump but $ψ$ continuous and zero
Square-integrable / normalizable	$∫\|ψ\|² finite$ so can normalize to 1

These are boundary conditions that quantize energy (particle in box).

4. Superposition

If $ψ_1, ψ_2$ are allowed states, then

ψ = c_1 ψ_1 + c_2 ψ_2

is also allowed (linear Schrödinger equation) — leading to interference.

5. Worked Examples

(a) ψ = A e^{-x²/2a²} (Gaussian) normalizable: |A|² ∫ e^{-x²/a²}dx =|A|² a√π=1 ⇒ A= (a√π)^{-1/2} ✓

(b) ψ = A sin(nπx/L) for 0<x<L, zero outside: continuous at  x=0,L? ψ=0 at edges ⇒ continuous ✓

(c) ψ = x (not square-integrable over all space ⇒ not physical for free particle, but okay on finite box with normalization)

(d) Need to check: ψ = e^{ikx} not normalizable over infinite space but okay as plane wave (delta normalization) for free particle.

(e) Probability to find particle between a and b: P= ∫_{a}^{b} |ψ|² dx

6. Exam Points

State Born interpretation $|ψ|²$ as probability density — 3-5 marks, very frequent.
Normalization $∫|ψ|²=1$ — 3 marks, frequent.
List characteristics (finite, single-valued, continuous, normalizable) — 5 marks, very frequent.
Distinguish $ψ$ vs $|ψ|²$ — 3 marks.
Compute normalization constant $A$ for given $ψ$ — 5-7 marks, very frequent.

Wave function and its characteristics