Superposition

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

Intuitive
Concrete
Abstract

Intuitive

In some sense, the portrait Mona Lisa shows a superposition of cheerfulness and sadness. See Mona Lisa - ineffable smile of quantum mechanics by Slobodan Prvanovic

Concrete

In quantum mechanics the states of a system are vectors in a complex Hilbert space $\mathcal H$ . If $|\psi_1\rangle$ and $|\psi_2\rangle$ are allowed states, then so is $|\psi\rangle = c_1 |\psi_1\rangle + c_2 |\psi_2\rangle, \qquad c_1, c_2 \in \mathbb{C},$ normalised so that $\langle\psi|\psi\rangle = 1$ . This is the superposition principle. It holds for all times because the Schrödinger equation $i\hbar\,\partial_t|\psi\rangle = H|\psi\rangle$ is linear: if $|\psi_1(t)\rangle$ and $|\psi_2(t)\rangle$ are solutions, so is any linear combination with constant coefficients.

Example: a spin 1/2. With $|\uparrow\rangle$ and $|\downarrow\rangle$ the eigenstates of $S_z$ , $|\psi\rangle = \frac{1}{\sqrt 2}\left(|\uparrow\rangle + e^{i\varphi}|\downarrow\rangle\right).$ A measurement of $S_z$ gives $+\hbar/2$ or $-\hbar/2$ , each with probability $1/2$ by the Born rule. In the standard formulation the state does not assign a definite value of $S_z$ that we merely do not know. For $\varphi = 0$ it is the eigenstate of $S_x$ with eigenvalue $+\hbar/2$ , so a measurement of $S_x$ gives $+\hbar/2$ with certainty. Whether a state "is a superposition" therefore depends on the basis: every pure state is an eigenstate of some observable (for example its own projector) and a superposition of the eigenstates of others.

Interference. The relative phase between the components is measurable, the overall phase is not. For a particle passing a double slit, let $\psi_1(x)$ and $\psi_2(x)$ be the amplitudes at a point $x$ on the screen for passage through slit 1 or 2. The probability density for $\psi = \psi_1 + \psi_2$ is $|\psi(x)|^2 = |\psi_1(x)|^2 + |\psi_2(x)|^2 + 2\,\text{Re}\left[\psi_1^*(x)\,\psi_2(x)\right].$ The last term produces the fringes. A statistical mixture, where the particle went through slit 1 with probability $1/2$ and through slit 2 with probability $1/2$ , gives only the first two terms. In general a superposition $c_1|\psi_1\rangle + c_2|\psi_2\rangle$ and the corresponding mixture agree on the probabilities in the basis $\{|\psi_1\rangle, |\psi_2\rangle\}$ when $\langle\psi_1|\psi_2\rangle = 0$ , and can differ only for observables that do not commute with the projectors onto these states.

Density matrices make the difference explicit. For the spin state above $\rho = |\psi\rangle\langle\psi| = \frac{1}{2}\begin{pmatrix} 1 & e^{-i\varphi} \\ e^{i\varphi} & 1 \end{pmatrix},$ in the basis $(|\uparrow\rangle, |\downarrow\rangle)$ , while the 50/50 mixture is $\rho = \frac{1}{2}\mathbb{1}$ . The off-diagonal entries ("coherences") carry the interference. Interaction with an environment that records which component the system is in suppresses them. This is decoherence, and it is why superpositions of macroscopically different states are not seen in practice.

Abstract

Rays. Physical pure states are rays in $\mathcal H$ , that is points of the projective space $P(\mathcal H) = (\mathcal H \setminus \{0\})/\mathbb{C}^*$ . Superposition is defined on vectors, not on rays: if $[\psi_1]$ and $[\psi_2]$ are distinct rays, the set of rays $[c_1\psi_1 + c_2\psi_2]$ is the projective line through them, and which point on it one means depends on the relative phase of the chosen representatives. For a two-dimensional $\mathcal H$ this line is all of $P(\mathbb C^2) \cong S^2$ , the Bloch sphere. The invariant content of the relative phase is captured by quantities like the transition probability $|\langle\psi_1|\psi_2\rangle|^2/(\|\psi_1\|^2\|\psi_2\|^2)$ and by the Bargmann invariant $\langle\psi_1|\psi_2\rangle\langle\psi_2|\psi_3\rangle\langle\psi_3|\psi_1\rangle$ of three rays, whose phase is a geometric (Berry) phase.

Pure and mixed states. More generally, states are density operators: positive trace-class operators with $\operatorname{Tr}\rho = 1$ . They form a convex set whose extreme points are the pure states $\rho = |\psi\rangle\langle\psi|$ . Convex combination $p\rho_1 + (1-p)\rho_2$ is the operation of mixing, and it is defined directly on states. Superposition is a different operation, which uses the linear structure of $\mathcal H$ and produces another extreme point. In the algebraic formulation, states are positive normalised linear functionals on a $C^*$ -algebra $\mathfrak A$ of observables. In an irreducible representation of $\mathfrak A$ on $\mathcal H$ , a unit vector $c_1\psi_1 + c_2\psi_2$ with $\psi_1, \psi_2$ normalised defines a pure state that differs from the mixture $|c_1|^2\omega_{\psi_1} + |c_2|^2\omega_{\psi_2}$ only through the interference terms $\langle\psi_1|A|\psi_2\rangle$ for $A \in \mathfrak A$ .

Superselection rules. The superposition principle holds without restriction when the observables act irreducibly on $\mathcal H$ , that is when only multiples of the identity commute with all of them. If there is a self-adjoint operator $Q$ , not a multiple of the identity, that commutes with all observables, $\mathcal H$ splits into superselection sectors $\mathcal H = \bigoplus_q \mathcal H_q$ , the eigenspaces of $Q$ . For $\psi_q \in \mathcal H_q$ the interference terms $\langle\psi_q|A|\psi_{q'}\rangle$ vanish for all observables $A$ when $q \neq q'$ , so a vector $c\,\psi_q + c'\psi_{q'}$ gives the same expectation values as the mixture $|c|^2\rho_q + |c'|^2\rho_{q'}$ , and the relative phase has no physical meaning. A standard example is the univalence superselection rule: a rotation by $2\pi$ acts on a state of total angular momentum $j$ as $(-1)^{2j}$ , and since this rotation must act trivially on all observables, states of integer and half-integer total angular momentum cannot be coherently superposed. Superselection of electric charge is another standard example.

Composite systems. For $\mathcal H_A \otimes \mathcal H_B$ , superpositions of product states such as $\frac{1}{\sqrt 2}(|0\rangle_A|0\rangle_B + |1\rangle_A|1\rangle_B)$ are in general not product states. This is entanglement, and the reduced state $\rho_A = \operatorname{Tr}_B \rho$ of such a pure state is mixed. Decoherence is this mechanism applied to a system entangled with its environment.

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Leads to: Double Slit Experiment.