AcademyQuantum Wave Functions
Academy
Wave Functions
Level 1 - Physics topic page in Quantum Wave Functions.
Principle
A wave function is a probability amplitude; its squared magnitude gives probability density, not material displacement.
Notation
\(\psi(x,t)\)
one-dimensional wave function
m^{-1/2}
\(|\psi|^2\)
probability density
\(\mathrm{m^{-1}}\)
\(A\)
normalization constant
varies
\(\langle x\rangle\)
expectation value of position
\(\mathrm{m}\)
\(\Delta x\)
standard uncertainty in position
\(\mathrm{m}\)
\(\phi\)
phase of a complex wave function
\(\mathrm{rad}\)
Method
Derivation 1: Probability from amplitude
The wave function itself is not a directly measured density. Probability comes from its modulus squared.
Probability density
\[\rho(x,t)=|\psi(x,t)|^2\]
Interval probability
\[P_{[a,b]}=\int_a^b |\psi(x,t)|^2\,dx\]
Derivation 2: Normalization
If the particle must be somewhere on the allowed line, the total probability is one.
Normalizable state
\[\int_{-\infty}^{\infty}|\psi(x,t)|^2\,dx=1\]
Find a constant
\[|A|^2\int |f(x)|^2\,dx=1\]
Derivation 3: Averages
Expectation values are probability-weighted averages over many identically prepared systems.
Mean position
\[\langle x\rangle=\int_{-\infty}^{\infty}x|\psi|^2\,dx\]
Position spread
\[(\Delta x)^2=\langle x^2\rangle-\langle x\rangle^2\]
Rules
These formulas apply to a normalized one-dimensional state.
Normalization
\[\int_{-\infty}^{\infty}|\psi|^2\,dx=1\]
Interval probability
\[P_{[a,b]}=\int_a^b|\psi|^2\,dx\]
Position expectation
\[\langle x\rangle=\int_{-\infty}^{\infty}x|\psi|^2\,dx\]
Examples
Question
A particle is described by
\[\psi=A\sin(\pi x/L)\]
on \[0\le x\le L\]
zero elsewhere. Find \(A\).Answer
Use
\[1=|A|^2\int_0^L\sin^2(\pi x/L)\,dx=|A|^2L/2\]
so \[A=\sqrt{2/L}\]
Checks
- Normalize before calculating probabilities or expectation values.
- Use \(|\\psi|^2\), not \(\psi\), as the probability density.
- A global phase is unobservable; relative phase can affect interference.
- A physically bound state must be square-integrable.