12.1 Dirac notation in quantum mechanics

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), Ch. 12 opener + §12.1 (G. He, coauthor). Rewritten in original wording. Continues 11.11b. Next: 12.1b.

Quantum computing (QC) speaks Dirac. This chapter builds qubits, entanglement, and gates, then (later) Cirq / IBM hardware. Programs stay low-level — gates on bits — much like early classical machines. Sources for going deeper: Hidary; Stolze & Suter; Nielsen & Chuang; IBM Quantum; Cirq.

Problem (chapter arc): write programs that store and process information in quantum states.

Kets, bras, brackets

A state is a ket $|\psi\rangle$ — a ray in a complex Hilbert space (dimension $2^n$ for $n$ qubits). The dual bra is the adjoint $\langle\psi|=|\psi\rangle^{\dagger}$. Pick the inner-product identity and the projector $|0\rangle\langle 0|$ among lookalikes.

The wave function is the coordinate projection $\psi(x)=\langle x\mid\psi\rangle$. Outer product $|\phi\rangle\langle\psi|$ is an operator (matrix), not a scalar: it maps states to states.

Spin-½ ↔ computational basis

QC renames spin-up / spin-down:

$$|0\rangle\equiv\bigl|+\tfrac12\bigr\rangle=\begin{bmatrix}1\\0\end{bmatrix},\qquad |1\rangle\equiv\bigl|-\tfrac12\bigr\rangle=\begin{bmatrix}0\\1\end{bmatrix}.$$

(Hyperfine / spin matrices appeared in §7.6.)

Takeaway

Object	Meaning
$	\psi\rangle$
$\langle\psi	$
$\langle\phi\mid\psi\rangle$	Inner product
$	\phi\rangle\langle\psi
$	0\rangle,

Quiz

Dirac notation

1 / 4

ψ(x)=⟨x|ψ⟩ is…

Pick an answer to see if you are right.

Next: 12.1b Dirac labs.