Superposition

A superposition is a linear combination of vectors in the same Hilbert space. If $\lvert u\rangle,\lvert v\rangle\in\mathcal H$, then

$$\lvert\chi\rangle =a\lvert u\rangle+b\lvert v\rangle$$

is another vector in $\mathcal H$. If it is nonzero, normalization produces a pure-state ray. The physical content lies in amplitudes and relative phases with respect to a declared set of alternatives, basis, observable, path, or subsystem structure.

In Symbols

For an orthonormal basis $\{\lvert n\rangle\}$,

$$\lvert\psi\rangle =\sum_n c_n\lvert n\rangle, \qquad c_n=\langle n\vert\psi\rangle.$$

Normalization requires

$$\sum_n\lvert c_n\rvert^2=1.$$

If that basis labels a projective measurement, the Born probability of outcome $n$ is

$$p(n)=\lvert c_n\rvert^2.$$

For two normalized but nonorthogonal vectors, the norm is not merely $\lvert a\rvert^2+\lvert b\rvert^2$:

$$\begin{aligned} \lVert a\lvert u\rangle+b\lvert v\rangle \rVert^2 &=\lvert a\rvert^2+\lvert b\rvert^2\\ &\quad +2\operatorname{Re} \left[ a^*b\langle u\vert v\rangle \right]. \end{aligned}$$

The overlap term is an interference term already present in normalization.

Canonical Home

See Superposition and Relative Phase for the canonical conceptual treatment, worked interferometer examples, phase randomization, and operational caveats.

Probability in Different Bases develops basis changes in the Born rule. Classical Mixtures versus Quantum Superpositions owns the density-matrix comparison.

Basis Dependence

The statement that a vector is “in a superposition” is incomplete unless the components are named. The same state may be one basis vector in one basis and a multi-term expansion in another.

For example,

$$\lvert+\rangle =\frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}$$

is a two-term superposition in the $\{\lvert0\rangle,\lvert1\rangle\}$ basis but a single basis vector in the $\{\lvert+\rangle,\lvert-\rangle\}$ basis.

This does not make interference arbitrary. Predictions are basis independent when states, observables, and transformations are translated consistently. The basis-dependent expansion is a description; outcome probabilities for a physically specified experiment are invariant.

Choosing a new coordinate basis is not the same physical operation as preparing a new superposition. A basis change rewrites the same vector, while a unitary gate or interaction can change the state.

Relative and Global Phase

Consider

$$\lvert\psi_\phi\rangle =\frac{1}{\sqrt2} \left( \lvert0\rangle+e^{i\phi}\lvert1\rangle \right).$$

Multiplying the entire state by $e^{i\alpha}$ changes no probability:

$$\lvert\psi_\phi\rangle \sim e^{i\alpha}\lvert\psi_\phi\rangle.$$

By contrast, changing the relative phase $\phi$ can change measurement statistics. A measurement in the $\{\lvert0\rangle,\lvert1\rangle\}$ basis gives

$$p(0)=p(1)=\frac12$$

for every $\phi$. In the basis

$$\lvert\pm\rangle =\frac{\lvert0\rangle\pm\lvert1\rangle}{\sqrt2},$$

the probabilities are

$$p(+)=\cos^2\frac{\phi}{2}, \qquad p(-)=\sin^2\frac{\phi}{2}.$$

The phase becomes observable through a measurement that recombines the alternatives.

Interference Term

Suppose an outcome $\lvert f\rangle$ can be reached from coherent alternatives with amplitudes $A_1$ and $A_2$. The total probability is

$$\begin{aligned} p(f) &=\lvert A_1+A_2\rvert^2\\ &=\lvert A_1\rvert^2+\lvert A_2\rvert^2 +2\operatorname{Re}(A_1^*A_2). \end{aligned}$$

The cross term distinguishes coherent amplitude addition from classical probability addition. Whether alternatives are coherent depends on preparation, environmental records, measurement resolution, and accessible relative phase, not merely on how an equation is written.

Superposition and Mixture

The pure state

$$\lvert+\rangle =\frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}$$

has density operator

$$\rho_{\mathrm{sup}} =\frac12 \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}$$

in the $\{\lvert0\rangle,\lvert1\rangle\}$ basis. The equal incoherent mixture has

$$\rho_{\mathrm{mix}} =\frac12 \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}.$$

Both give equal probabilities for $0$ and $1$, but they differ in another basis:

$$\Pr_{\mathrm{sup}}(+)=1, \qquad \Pr_{\mathrm{mix}}(+)=\frac12.$$

The off-diagonal terms encode coherence relative to the chosen basis. Coherence is basis dependent, while the distinction between a pure state and a genuinely mixed density operator is basis independent.

Randomizing the relative phase over many preparations removes the off-diagonal average:

$$\frac{1}{2\pi} \int_0^{2\pi} \lvert\psi_\phi\rangle \langle\psi_\phi\rvert\,d\phi =\rho_{\mathrm{mix}}.$$

This phase averaging is one route from coherent superpositions to an operational mixture.

Time Evolution

Linearity of the Schrödinger equation preserves superpositions. If

$$H\lvert n\rangle=E_n\lvert n\rangle,$$

then

$$\sum_n c_n\lvert n\rangle \longmapsto \sum_n c_ne^{-iE_nt/\hbar}\lvert n\rangle.$$

The basis populations $\lvert c_n\rvert^2$ remain fixed for a time-independent Hamiltonian in its energy basis, while relative phases evolve according to energy differences:

$$\phi_{mn}(t) =\phi_{mn}(0) -\frac{E_m-E_n}{\hbar}t.$$

Those relative phases drive oscillatory probabilities in other bases.

Wave-Mechanics Form

For wavefunctions, linearity means that if $\psi_1(x,t)$ and $\psi_2(x,t)$ solve a linear Schrödinger equation, then

$$\psi(x,t) =a\psi_1(x,t)+b\psi_2(x,t)$$

also solves it. The density is

$$\begin{aligned} \lvert\psi\rvert^2 &=\lvert a\rvert^2\lvert\psi_1\rvert^2 +\lvert b\rvert^2\lvert\psi_2\rvert^2\\ &\quad +2\operatorname{Re} \left( a^*b\,\psi_1^*\psi_2 \right). \end{aligned}$$

The final line is the spatial interference term.

Superposition and Entanglement

Superposition is a consequence of vector-space linearity. Entanglement additionally requires a declared composite factorization

$$\mathcal H_A\otimes\mathcal H_B.$$

The product state

$$\frac{\lvert0\rangle_A+\lvert1\rangle_A}{\sqrt2} \otimes \lvert0\rangle_B$$

is a superposition in a product basis but is not entangled. The Bell state

$$\frac{ \lvert0\rangle_A\lvert0\rangle_B +\lvert1\rangle_A\lvert1\rangle_B }{\sqrt2}$$

is entangled because it cannot be factored into one state of $A$ times one state of $B$.

Every entangled pure state is a superposition in many product bases, but most superpositions are not entangled.

Superselection Caveat

The Hilbert-space formalism allows linear combinations, but not every relative phase is operationally accessible. A superselection rule can forbid observables that connect different sectors, making relative phases between those sectors unmeasurable within the admitted observable algebra.

One should therefore distinguish a formally writable vector from an experimentally preparable and detectable coherent superposition under the theory’s constraints.

Common Aliases

• coherent superposition, when relative phase is controlled;
• linear combination of states;
• wave superposition, in coordinate-space language.

“Quantum parallelism” is not a synonym. It is an application-dependent description that can obscure the measurement and interference conditions.

Common Confusions

• Superposition is basis-dependent as a description, but interference probabilities for a specified experiment are physical.
• A mixture is not the same as a coherent superposition.
• Global phase is unobservable; relative phase can matter.
• Adding state vectors and adding probabilities are different operations.
• A basis change rewrites a state; it does not by itself prepare a new state.
• A superposition is not automatically entangled.
• Coherence in one basis can be absent in another.
• Equal probabilities in one measurement do not imply equal states.
• A formal linear combination can cross a superselection boundary whose relative phase is operationally inaccessible.
• Macroscopic language such as “in two places at once” is incomplete without specifying the state, alternatives, and measurement.

Related Entries

• Wavefunction
• Born Rule
• Density Matrix
• Entanglement
• Rays and Global Phase
• Change of Basis
• Entangled States
• Decoherence Preview

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 1 and 4.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, ch. 1.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, vol. 1, Wiley, 1977, chs. II and III.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010, secs. 2.1 and 2.4.