Wave function collapse

For the constraint-solving algorithm, see Wave function collapse (algorithm).

In quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an observation, and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation.

Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation throughout this apparent collapse. More importantly, this is not enough to explain actual wave function collapse, as decoherence does not reduce it to a single eigenstate.

Historically, Werner Heisenberg was the first to use the idea of wave function reduction to explain quantum measurement.

Description

For an explanation of the notation used, see Bra–ket notation. For details on this formalism, see Quantum state.

In quantum mechanics each measurable physical quantity of a quantum system is called an observable which, for example, could be the position $r$ and the momentum $p$ but also energy $E$, $z$ components of spin ($s_{z}$), and so on. The observable acts as a linear function on the states of the system; its eigenvectors correspond to the quantum state (i.e. eigenstate) and the eigenvalues to the possible values of the observable. The collection of eigenstates/eigenvalue pairs represent all possible values of the observable. Writing $\phi _{i}$ for an eigenstate and $c_{i}$ for the corresponding observed value, any arbitrary state of the quantum system can be expressed as a vector using bra–ket notation: $$|\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle .$$ The kets $\{|\phi _{i}\rangle \}$ specify the different available quantum "alternatives", i.e., particular quantum states.

The wave function is a specific representation of a quantum state. Wave functions can therefore always be expressed as eigenstates of an observable though the converse is not necessarily true.

Collapse

To account for the experimental result that repeated measurements of a quantum system give the same results, the theory postulates a "collapse" or "reduction of the state vector" upon observation, abruptly converting an arbitrary state into a single component eigenstate of the observable:

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

where the arrow represents a measurement of the observable corresponding to the $\phi$ basis. For any single event, only one eigenvalue is measured, chosen randomly from among the possible values.

As examples, individual counts in a double slit experiment with electrons appear at random locations on the detector; after many counts are summed the distribution shows a wave interference pattern. In a Stern-Gerlach experiment with silver atoms, each particle appears in one of two areas unpredictably, but the final conclusion has equal numbers of events in each area.

This statistical aspect of quantum measurements differs fundamentally from classical mechanics. In quantum mechanics the only information we have about a system is its wave function and measurements of the wavefunction can only give statistical information.

Meaning of the expansion coefficients

For the wavefunction written as a linear expansion of eigenstates $|\phi _{i}\rangle$as: $$|\psi \rangle =\sum _{i}c_{i}|\phi _{i}\rangle ,$$ the corresponding complex coefficients $\{c_{i}\}$ are called probability amplitudes. According to the Born’s statistical interpretation, the square modulus $|c_{i}|^{2}$ is the probability that a measurement yields result corresponding the eigenstate $|\phi _{i}\rangle$. The total probability of measuring all possible states sums to one:

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

Quantum decoherence

Main article: 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. This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible in the same sense as in the second law of thermodynamics: 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.

Physical approaches to collapse

Quantum theory offers no dynamical description of the "collapse" of the wave function. Viewed as a statistical theory, no description is expected. As Fuchs and Peres put it, "collapse is something that happens in our description of the system, not to the system itself".

Various interpretations of quantum mechanics attempt to provide a physical model for collapse. Three treatments of collapse can be found among the common interpretations. The first group includes hidden variable theories like de Broglie–Bohm theory; here random outcomes only result from unknown values of hidden variables. Results from tests of Bell's theorem shows that these variables would need to be non-local. The second group models measurement as quantum entanglement between the quantum state and the measurement apparatus. This results in a simulation of classical statistics called quantum decoherence. This group includes the many worlds interpretation and consistent histories models. The third group postulates additional, but as yet undetected, physical basis for the randomness; this group includes for example the objective collapse interpretations. While models in all groups have contributed to better understanding of quantum theory, no explanation explanation collapse for individual events has emerged as more useful than statistical prediction.

The cluster of phenomena described by the expression wave function collapse is a fundamental problem in the interpretation of quantum mechanics, and is known as the measurement problem.

The significance ascribed to the wave function varies from interpretation to interpretation, and varies even within an interpretation (such as the Copenhagen Interpretation). If the wave function merely encodes an observer's knowledge of the universe then the wave function collapse corresponds to the receipt of new information. This is somewhat analogous to the situation in classical physics, except that the classical "wave function" does not necessarily obey a wave equation. If the wave function is physically real, in some sense and to some extent, then the collapse of the wave function is also seen as a real process, to the same extent.

History

The concept of wavefunction collapse was introduced by Werner Heisenberg in his 1927 paper on the uncertainty principle, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", and incorporated into the mathematical formulation of quantum mechanics by John von Neumann, in his 1932 treatise Mathematische Grundlagen der Quantenmechanik. Heisenberg did not try to specify exactly what the collapse of the wavefunction meant. However, he emphasized that it should not be understood as a physical process. Niels Bohr also repeatedly cautioned that we must give up a "pictorial representation", and perhaps also interpreted collapse as a formal, not physical, process.

Consistent with Heisenberg, von Neumann postulated that there were two processes of wave function change:

1. The probabilistic, non-unitary, non-local, discontinuous change brought about by observation and measurement, as outlined above.
2. The deterministic, unitary, continuous time evolution of an isolated system that obeys the Schrödinger equation (or a relativistic equivalent, i.e. the Dirac equation).

In general, quantum systems exist in superpositions of those basis states that most closely correspond to classical descriptions, and, in the absence of measurement, evolve according to the Schrödinger equation. However, when a measurement is made, the wave function collapses—from an observer's perspective—to just one of the basis states, and the property being measured uniquely acquires the eigenvalue of that particular state, $\lambda _{i}$. After the collapse, the system again evolves according to the Schrödinger equation.

By explicitly dealing with the interaction of object and measuring instrument, von Neumann has attempted to create consistency of the two processes of wave function change.

He was able to prove the possibility of a quantum mechanical measurement scheme consistent with wave function collapse. However, he did not prove the necessity of such a collapse. Although von Neumann's projection postulate is often presented as a normative description of quantum measurement, it was conceived by taking into account experimental evidence available during the 1930s (in particular the Compton–Simon experiment was paradigmatic), but many important present-day measurement procedures do not satisfy it (so-called measurements of the second kind).

See also

• Arrow of time
• Interpretations of quantum mechanics
• Quantum decoherence
• Quantum interference
• Quantum Zeno effect
• Schrödinger's cat
• Stern–Gerlach experiment
• Wave function collapse (algorithm)

External links

• Quotations related to Wave function collapse at Wikiquote