Wave function collapse

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In certain interpretations of quantum mechanics, wave function collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. It is also called collapse of the state vector or reduction of the wave packet. The reality of wave function collapse has always been debated, i.e., whether it is a fundamental physical phenomenon in its own right (which may yet emerge from a theory of everything) or just an epiphenomenon of another process, such as quantum decoherence. In recent decades the quantum decoherence view has gained popularity.

1 Outline
2 History and Context
3 Notes
4 See also

[edit] Outline

The state or wave function of physical system, at some time, can be expressed in Dirac or bra-ket notation as:

$|\psi \rang = \sum_i |i\rang \psi_i$

where the $|i\rang$ s specify the different quantum "alternatives" available (technically, they form an orthonormal eigenvector basis which implies $\lang i|j \rang = \delta_{ij}$ ). An observable or measurable parameter of the system is associated with each eigenbasis, with each quantum alternative having a specific value or eigenvalue, $e i$ , of the observable.

The $\psi_i = \lang i|\psi \rang$ are the probability amplitude coefficients, which are complex numbers. For simplicity we shall assume that our wave function is normalised: $\lang \psi|\psi\rang = 1$ , which implies that $\lang \psi|\psi \rang = \sum_i |\psi_i|^2 = 1$ .

With these definitions it is easy to describe the process of collapse:

When an external agency measures the observable associated with the eigenbasis then the state of the wave function changes from $|\psi \rang$ to just one of the $|i\rang$ s with Born probability $| ψ i | 2$ . This is called collapse because all the other terms in the expansion of the wave function have vanished or collapsed into nothing.

If a more general measurement is made to detect if the system in a state $|\phi \rang$ then the system makes a "jump" or quantum leap from the original state $|\psi \rang$ to the final state $|\phi \rang$ with probability of $|\lang\psi|\phi\rang|^2$ . Quantum leaps and wave function collapse are therefore opposite sides of the same coin.

[edit] History and Context

By the time John von Neumann wrote his famous treatise Mathematische Grundlagen der Quantenmechanik in 1932^[1], the phenomenon of "wave function collapse" was accommodated into the mathematical formulation of quantum mechanics by postulating that there were two processes of wave function change:

The probabilistic, non-unitary, non-local, discontinuous change brought about by observation and measurement, as outlined above.
The deterministic, unitary, continuous time evolution of an isolated system that obeys Schrödinger's equation (or nowadays some relativistic, local equivalent).

In general, quantum systems exist in superpositions of those basis states that most closely correspond to classical descriptions, and -- when not being measured or observed, evolve according to the time dependent Schrödinger equation, relativistic quantum field theory or some form of quantum gravity or string theory, which is process (2) mentioned above. However, when the wave function collapses -- process (1) -- from an observer's perspective the state seems to "leap" or "jump" to just one of the basis states and uniquely acquire the value of the property being measured, $e i$ , that is associated with that particular basis state. After the collapse, the system begins to evolve again according to the Schrödinger equation or some equivalent wave equation.

Hence, in experiments such as the double-slit experiment each individual photon arrives at a discrete point on the screen, but as more and more photons are accumulated, they form an interference pattern overall.

The existence of the wave function collapse is required in

the Copenhagen interpretation
the so-called transactional interpretation
in a "spiritual interpretation" in which consciousness causes collapse.

On the other hand, the collapse is considered as redundant or just an optional approximation in

interpretations based on consistent histories
the many-worlds interpretation
the Bohm interpretation
the Ensemble Interpretation

The cluster of phenomena described by the expression wave function collapse is a fundamental problem in the interpretation of quantum mechanics known as the measurement problem. The problem is not really confronted by the Copenhagen interpretation which simply postulates that this is a special characteristic of the "measurement" process. The Everett many-worlds interpretation deals with it by discarding the collapse-process, thus reformulating the relation between measurement apparatus and system in such a way that the linear laws of quantum mechanics are universally valid, that is, the only process according to which a quantum system evolves is governed by the Schrödinger equation or some relativistic equivalent. Often tied in with the many-worlds interpretation, but not limited to it, is the physical process of decoherence, which causes an apparent collapse. Decoherence is also important for the interpretation based on Consistent Histories.

Note that a general description of the evolution of quantum mechanical systems is possible by using density operators and quantum operations. In this formalism (which is closely related to the C*-algebraic formalism) the collapse of the wave function corresponds to a non-unitary quantum operation.

Note also that the physical significance ascribed to the wave function varies from interpretation to interpretation, and 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. One of the paradoxes of quantum theory is that wave function seems to be more than just information (otherwise interference effects are hard to explain) and often less than real, since the collapse seems to take place faster-than-light and triggered by observers.

[edit] Notes

^ "the “collapse” or “reduction” of the wave function. This was introduced by Heisenberg in his uncertainty paper [3] and later postulated by von Neumann as a dynamical process independent of the Schrodinger equation"Kiefer, C. On the interpretation of quantum theory – from Copenhagen to the present day

Categories: Quantum measurement | Fundamental physics concepts

Wave function collapse

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