LESSON NOTES · 02

2. State, measurement, and operators

Position in the course: Lesson 2 of 10. Complete the preceding derivation and use the explained exercises to check understanding.

Prerequisites: Complex numbers, vectors, inner products

1. A state is a normalized vector

A pure state \(|\psi\rangle\) is a normalized vector, not a list of pre-existing classical values. Global phase has no observable effect. A projective measurement of a self-adjoint observable \(\hat A\) has real outcomes. If \(P_a\) projects onto an eigenspace, \(\langle\psi|P_a|\psi\rangle\) is its probability. For a nondegenerate discrete orthonormal basis, the squared amplitude is the probability.

\[ |\psi\rangle=\sum_n c_n|a_n\rangle,\quad\sum_n|c_n|^2=1,\quad p(a_n)=|c_n|^2. \]

2. Expectation is not a single measurement outcome

The expectation is an ensemble average over identically prepared states. The variance measures spread in that state. It is not simply an instrument defect. Hats denote operators; \(\hbar\) is Planck’s constant divided by \(2\pi\).

\[ \langle A\rangle=\langle\psi|\hat A|\psi\rangle=\sum_n|c_n|^2a_n,\qquad(\Delta A)^2=\langle(\hat A-\langle A\rangle)^2\rangle. \]

3. Canonical operators and their domains

Position acts by multiplication and momentum by differentiation. Apply their commutator to a sufficiently well-behaved test function in the common domain: the terms with \(x\partial_x\psi\) cancel, leaving \(i\hbar\psi\). Boundary conditions and operator domains matter; formal integration by parts does not make every differential expression self-adjoint.

\[ \hat x\psi=x\psi,\quad\hat p\psi=-i\hbar\partial_x\psi,\quad[\hat x,\hat p]\psi=-i\hbar x\psi^{\prime}+i\hbar(x\psi)^{\prime}=i\hbar\psi. \]

4. Derive the uncertainty inequality

Set \(|u\rangle=(\hat A-\langle A\rangle)|\psi\rangle\) and similarly \(|v\rangle\) for \(B\). Cauchy–Schwarz bounds the product of their norms by the overlap. Keeping only the imaginary part gives the commutator bound. Substitute the canonical commutator to obtain the position–momentum relation. These steps assume the required products and expectations exist.

\[ (\Delta A)^2(\Delta B)^2\geq|\langle u|v\rangle|^2\geq[\operatorname{Im}\langle u|v\rangle]^2=\tfrac14|\langle[\hat A,\hat B]\rangle|^2,\quad\Delta x\Delta p\geq\hbar/2. \]

State, measurement, and operators — schematic under the stated model assumptions

Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.

5. Worked two-level measurement

Take the teaching observable \(A=\operatorname{diag}(0,2)\) in stated energy units and state \((\sqrt3/2,e^{i\theta}/2)^T\). The two probabilities are \(3/4\) and \(1/4\); mean \(1/2\), second moment \(1\), standard deviation \(\sqrt3/2\). Changing the relative phase leaves these probabilities unchanged but changes outcomes in a rotated measurement basis.

\[ \langle A\rangle=0\cdot\tfrac34+2\cdot\tfrac14=\tfrac12,\qquad\Delta A=\sqrt{1-(1/2)^2}=\sqrt3/2. \]

6. Check yourself

Can a measurement give \(1/2\) here?

Answer and reasoning

No, outcomes are only \(0\) or \(2\).

Does \(\psi\to-\psi\) change observables?

Answer and reasoning

No, it is global phase.

Does \([A,B]\ne0\) guarantee a nonzero lower bound in every state?

Answer and reasoning

No, its expectation may vanish.

7. Pure states, mixtures and repeated measurements

The state \((|0\rangle+|1\rangle)/\sqrt2\) and an equal mixture of the two basis states have the same diagonal measurement probabilities but differ in other bases. Their density matrices are respectively

\[ \rho_{pure}=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad\rho_{mix}=\tfrac12\begin{pmatrix}1&0\\0&1\end{pmatrix}. \]

For either representation, \(\langle A\rangle=\operatorname{Tr}(\rho A)\). The off-diagonal elements encode coherence in the stated basis. A pure-state density operator obeys \(\rho^2=\rho\) and \(\operatorname{Tr}\rho^2=1\); the equal mixture instead has purity \(1/2\). Purity is unchanged by a unitary basis rotation even though individual matrix entries change. Classical ignorance about which preparation occurred and quantum uncertainty within a pure preparation are therefore different notions.

An ideal projective measurement giving eigenvalue \(a\) updates the state to \(P_a|\psi\rangle/\sqrt{p(a)}\) when \(p(a)>0\). Immediately repeating that same ideal measurement gives the same eigenvalue. Measuring an incompatible observable between the repetitions can change this prediction. Real apparatus noise or nonprojective measurements require a more general measurement model; the ideal postulate is not a complete theory of laboratory instruments.

Additional exercise: For the pure state above, calculate the probability of the plus state. It is one because the state is already the plus eigenvector. For the equal mixture use \(\operatorname{Tr}(\rho_{mix}|+\rangle\langle+|)=1/2\). Thus diagonal populations alone cannot distinguish coherent superposition from mixture. This distinction later matters for spin entanglement and statistical ensembles.