States, Operators & Measurement

All of quantum mechanics fits in five sentences. A system's state is a unit vector $|\psi ⟩$ in a complex vector space. Observables are Hermitian operators on that space. Measuring an observable returns one of its eigenvalues, with probability given by the squared amplitude of $|\psi ⟩$ along the matching eigenvector. The measurement leaves the system in that eigenvector. And between measurements the state rotates smoothly under $i\hslash {\partial }_t|\psi ⟩=H|\psi ⟩$. Everything else — atoms, chemistry, lasers, the rest of this course — is those five sentences applied with stamina. This page makes each one concrete on the smallest system that has any physics in it: two states.

A state is a direction, not a place

Take a spin-1/2 particle and pick a measurement axis, conventionally $z$. There are exactly two outcomes, so the state space is two-dimensional, spanned by $|\uparrow ⟩$ and $|\downarrow ⟩$. The most general state is

two real numbers, one point on a sphere. Notice what the state is not: it is not "the particle is up with some chance and down with some chance." A coin on a table is that. The complex phase $\phi$ is the difference — it carries physics the coin has no slot for, and every interference effect lives in it. Make it concrete: set $\theta =50^{\circ }$, $\phi =60^{\circ }$. The amplitudes are $(0.9063,0.2113+0.3660i)$, and their squared magnitudes sum to 1 — that normalization is the entire "unit vector" clause.

The Born rule: amplitudes squared

Measure $S_z$ in that state. The outcome is one of the two eigenvalues $\pm \hslash /2$ — never anything in between, no matter how the state points — and the probability of $+\hslash /2$ is the squared amplitude on $|\uparrow ⟩$:

A single measurement tells you almost nothing — you get one bit. The probabilities are what you reconstruct by preparing the same state a thousand times and counting. That distinction, between the state (which you prepare) and the statistics (which you observe), is the part of the postulates most worth internalizing early.

Observables are Hermitian matrices; expectations are sandwiches

On two states, every observable is a 2×2 Hermitian matrix. The spin components are (in units of $\hslash$):

Hermitian is not decoration: it guarantees real eigenvalues (outcomes are real numbers) and orthogonal eigenvectors (distinct outcomes are perfectly distinguishable). The average outcome over many repetitions is the sandwich $⟨A⟩=⟨\psi |A|\psi ⟩$. For our concrete state, computing all three (the code below does it):

Those three numbers are the Cartesian coordinates of the point $(\theta ,\phi )$ on a sphere of radius 1/2 — the state is a direction, and the expectation values are how you read it out. No single measurement returns 0.1915; every $S_x$ measurement returns $\pm 1/2$, and 0.1915 is what the average converges to.

What does not commute cannot be pinned at once

The spin matrices refuse to commute: $[S_x,S_y]=i{S}_z$. The general uncertainty relation turns that algebra into a bound on statistics,

and the bound is checkable arithmetic, not philosophy. For our state: $\Delta S_x\Delta S_y=0.1728$ against $|⟨S_z⟩|/2=0.1607$ — satisfied with 0.012 to spare. Sharpen your knowledge of $S_x$ and the spread in $S_y$ must grow, because the operators share no eigenvectors. Every "uncertainty principle" headline is this one line of linear algebra wearing a trench coat.

import numpy as np
# state: cos(t/2)|up> + e^{i phi} sin(t/2)|down>, theta = 50 deg, phi = 60 deg
t, phi = np.radians(50), np.radians(60)
psi = np.array([np.cos(t/2), np.exp(1j*phi)*np.sin(t/2)])

sx = np.array([[0, 1], [1, 0]])/2          # spin operators, hbar = 1
sy = np.array([[0, -1j], [1j, 0]])/2
sz = np.array([[1, 0], [0, -1]])/2

ev = lambda A: np.real(psi.conj() @ A @ psi)          # <psi|A|psi>
dS = lambda A: np.sqrt(ev(A @ A) - ev(A)**2)          # spread

print(ev(sx), ev(sy), ev(sz))     # 0.1915  0.3317  0.3214
print(abs(psi[0])**2)             # 0.8214 = P(measure Sz -> +1/2)
print(dS(sx)*dS(sy), abs(ev(sz))/2)   # 0.1728 >= 0.1607  (uncertainty)

Collapse, and why repeat measurements agree

Measurement changes the state: get $+\hslash /2$ and the state is now $|\uparrow ⟩$, whatever it was before. Measure $S_z$ again immediately and you get $+\hslash /2$ with probability 1 — reproducibility is built into the postulate, because $|\uparrow ⟩$ is an eigenvector. But measure $S_x$ next and you are back to a coin flip: $|\uparrow ⟩$ has equal amplitudes along both $S_x$ eigenvectors. The order of measurements matters exactly when the operators fail to commute — the same $[S_x,S_y]\ne 0$ from above, now as an experimental protocol rather than an inequality.

Where to next: the dynamics postulate — how $H$ rotates the state between measurements — gets its full workout on two-level systems, where the whole Schrödinger equation is a 2×2 matrix you can diagonalize by hand. Bound states in a potential start at the particle in a box. And when light enters the story, the operator that matters is the dipole.