Mathematical background

The quantum state of a physical system is described by a wave function (in turn—an element of a ray in projective Hilbert space). This can be expressed as a vector using Dirac or bra–ket notation :

${\displaystyle |\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle .}$

The kets ${\displaystyle |\phi _{1}\rangle ,|\phi _{2}\rangle ,|\phi _{3}\rangle ,\dots }$ specify the different quantum "alternatives" available—a particular quantum state. They form an orthonormal eigenbasis, formally

${\displaystyle \langle \phi _{i}|\phi _{j}\rangle =\delta _{ij}}$

where ${\displaystyle \delta _{ij}}$ represents the Kronecker delta.

An observable (i.e. measurable parameter of the system) is associated with each eigenbasis, with each quantum alternative having a specific eigenvalue of the observable. A "measurable parameter of the system" could be the usual position ${\displaystyle r}$ and the momentum ${\displaystyle p}$ of, say, a particle, but also its energy ${\displaystyle E}$, ${\displaystyle z}$ components of spin (${\displaystyle s_{z}}$), orbital (${\displaystyle L_{z}}$) and total angular (${\displaystyle J_{z}}$) momenta, etc. In the basis representation these are respectively ${\displaystyle |\mathbf {r} ,t\rangle =|x,t\rangle +|y,t\rangle +|z,t\rangle ,|\mathbf {p} ,t\rangle =|p_{x},t\rangle +|p_{y},t\rangle +|p_{z},t\rangle ,|E\rangle ,|s_{z}\rangle ,|L_{z}\rangle ,|J_{z}\rangle ,\dots }$.

The coefficients ${\displaystyle c_{1},c_{2},c_{3}...}$ are the probability amplitudes corresponding to each basis ${\displaystyle |\phi _{1}\rangle ,|\phi _{2}\rangle ,|\phi _{3}\rangle ,\dots }$. These are complex numbers. The moduli square of ${\displaystyle c_{i}}$, that is ${\displaystyle |c_{i}|^{2}={c}_{i}^{*}c_{i}}$ (where ${\displaystyle *}$ denotes complex conjugate), is the probability of measuring the system to be in the state ${\displaystyle |\phi _{i}\rangle }$.

For simplicity in the following, all wave functions are assumed to be normalized; the total probability of measuring all possible states is one:

${\displaystyle \langle \psi |\psi \rangle =\sum _{i}|c_{i}|^{2}=1.}$

The process of collapse

With these definitions it is easy to describe the process of collapse. For any observable, the wave function is initially some linear combination of the eigenbasis ${\displaystyle \{|\phi _{i}\rangle \}}$ of that observable. When an external agency (an observer, experimenter) measures the observable associated with the eigenbasis ${\displaystyle \{|\phi _{i}\rangle \}}$, the wave function collapses from the full ${\displaystyle |\psi \rangle }$ to just one of the basis eigenstates, ${\displaystyle |\phi _{i}\rangle }$, that is:

${\displaystyle |\psi \rangle \rightarrow |\phi _{i}\rangle .}$

The probability of collapsing to a given eigenstate ${\displaystyle |\phi _{k}\rangle }$ is the Born probability, ${\displaystyle P_{k}=|c_{k}|^{2}}$. Immediately post-measurement, other elements of the wave function vector, ${\displaystyle c_{i\neq k}}$, have "collapsed" to zero, and ${\displaystyle |c_{i}|^{2}=1}$.^{[note 1]}

More generally, collapse is defined for an operator ${\displaystyle {\hat {Q}}}$ with eigenbasis ${\displaystyle \{|\phi _{i}\rangle \}}$. If the system is in state ${\displaystyle |\psi \rangle }$, and ${\displaystyle {\hat {Q}}}$ is measured, the probability of collapsing the system to eigenstate ${\displaystyle |\phi _{i}\rangle }$ and measuring the eigenvalue ${\displaystyle q_{i}}$ of ${\displaystyle |\phi _{i}\rangle }$ with respect to ${\displaystyle {\hat {Q}}}$ would be ${\displaystyle |\langle \psi |\phi _{i}\rangle |^{2}}$. Note that this is not the probability that the particle is in state ${\displaystyle |\phi _{i}\rangle }$; it is in state ${\displaystyle |\psi \rangle }$ until cast to an eigenstate of ${\displaystyle {\hat {Q}}}$.

However, we never observe collapse to a single eigenstate of a continuous-spectrum operator (e.g. position, momentum, or a scattering Hamiltonian), because such eigenfunctions are non-normalizable. In these cases, the wave function will partially collapse to a linear combination of "close" eigenstates (necessarily involving a spread in eigenvalues) that embodies the imprecision of the measurement apparatus. The more precise the measurement, the tighter the range. Calculation of probability proceeds identically, except with an integral over the expansion coefficient ${\displaystyle c(q,t)dq}$.^[8] This phenomenon is unrelated to the uncertainty principle, although increasingly precise measurements of one operator (e.g. position) will naturally homogenize the expansion coefficient of wave function with respect to another, incompatible operator (e.g. momentum), lowering the probability of measuring any particular value of the latter.

Quantum decoherence

Quantum decoherence explains why a system interacting with an environment transitions from being a pure state, exhibiting superpositions, to a mixed state, an incoherent combination of classical alternatives.^[5] This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible, as the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Decoherence is thus very important for explaining the classical limit of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them.^[2][9][5]