Probability, Operators, and Measurement

Manual: General · Subject: Quantum Physics

Understand observables, eigenstates, expectation values, and the measurement postulate.

Observables as operators

Operators and eigenvalues

Physical observables such as position, momentum, and energy are represented by Hermitian operators. If $\hat A\phi = a\phi$ , then $\phi$ is an eigenstate of observable $A$ with eigenvalue $a$ .

Expectation value

The average outcome of repeated measurements in state $\psi$ is $\langle A \rangle = \int \psi^*(x)\,\hat A\,\psi(x)\,dx$ for position-space wavefunctions, assuming the state is normalized.

⚠️

Measurement postulate

A single measurement returns one eigenvalue of the corresponding operator, not the expectation value.

Momentum operator

In position representation, momentum is $\hat p = -i\hbar \frac{d}{dx}$ . This makes momentum an operator that generates spatial translations.

Commutators and compatibility

If $[\hat A,\hat B] = \hat A\hat B - \hat B\hat A = 0$ , the observables can share eigenstates and are compatible in the quantum-mechanical sense. Nonzero commutators imply uncertainty relations.

What does it mean for an observable to be represented by a Hermitian operator?

It has only positive eigenvalues Its expectation values are real It is independent of measurement It cannot be measured

What does an eigenstate of an observable produce upon measurement?

Which operator represents momentum in one dimension?

\hbar \frac{d}{dx}

-i\hbar \frac{d}{dx}

i\hbar \frac{d}{dt}

-\hbar^2 \frac{d^2}{dx^2}

What does a nonzero commutator between two observables imply?

Probabilistic predictions

Quantum theory is not about predicting single outcomes with certainty in general, but about computing the full probability distribution for possible outcomes of future measurements.

Neuroanatomy Explorer

Memory Deck

Match the Pairs

Concept Constellation