AcademyQuantum Wave Functions
Academy
Measurement in Quantum Mechanics
Level 1 - Physics topic page in Quantum Wave Functions.
Principle
A measurement yields an eigenvalue with probability set by the state's projection onto the corresponding eigenstate.
Notation
\(\hat A\)
observable operator
varies
\(a_n\)
eigenvalue of the observable
varies
\(|a_n\rangle\)
eigenstate of the observable
1
\(c_n\)
probability amplitude in an eigenbasis
1
\(P_n\)
probability of result n
1
\(\langle A\rangle\)
expectation value of A
unit of A
Method
Derivation 1: Eigenvalue model
An ideal measurement of an observable returns one of its eigenvalues.
Eigenvalue equation
\[\hat A|a_n\rangle=a_n|a_n\rangle\]
State expansion
\[|\psi\rangle=\sum_n c_n|a_n\rangle\]
Derivation 2: Born rule
The squared magnitude of each coefficient gives the corresponding outcome probability.
Outcome probability
\[P_n=|c_n|^2\]
Normalized amplitudes
\[\sum_n |c_n|^2=1\]
Derivation 3: Expectation value
The expectation value is the probability-weighted average of repeated measurement outcomes.
Discrete expectation
\[\langle A\rangle=\sum_n a_n|c_n|^2\]
Operator expectation
\[\langle A\rangle=\langle\psi|\hat A|\psi\rangle\]
Rules
For an ideal projective measurement:
Born probability
\[P_n=|\langle a_n|\psi\rangle|^2\]
Expectation value
\[\langle A\rangle=\langle\psi|\hat A|\psi\rangle\]
Uncertainty
\[\Delta A=\sqrt{\langle A^2\rangle-\langle A\rangle^2}\]
Examples
Question
A state is
\[|\psi\rangle=(3/5)|a_1\rangle+(4/5)|a_2\rangle\]
What are the two measurement probabilities?Answer
\[P_1=|3/5|^2=9/25\]
and \[P_2=|4/5|^2=16/25\]
Checks
- Probabilities come from amplitudes in the measurement basis.
- The expectation value need not be an allowed single-shot measurement result.
- Repeating the same ideal measurement immediately gives the same result.
- Noncommuting observables cannot generally have sharp values in the same state.