Wave function collapse: June 2011

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 $\scriptstyle |\psi\rang$ to just one of the $\scriptstyle |i\rang$ s with Born probability $\scriptstyle | \psi_i | ^ 2$ , that is:

$|\psi \rang \Rightarrow |i\rang$ .

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 that detects the system in a state $\scriptstyle | \phi \rang$ then the system makes a "jump" or quantum leap from the original state $\scriptstyle | \psi \rang$ to the final state $\scriptstyle | \phi \rang$ with probability of $\scriptstyle |\lang \psi | \phi \rang|^2$ . Quantum leaps and wave function collapse are therefore merely two sides of the same coin.

[edit]History and context

By the time John von Neumann wrote his treatise Mathematische Grundlagen der Quantenmechanik in 1932,^[3] 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, i.e. Dirac's equation).

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

, 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.

By explicitly dealing with the interaction of object and measuring instrument, von Neumann^[1] 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 has been paradigmatic), and many important present-day measurement procedures do not satisfy it (so-called measurements of the second kind).^[4]^[5]^[6]

The existence of the wave function collapse is required in

the Copenhagen interpretation
the objective collapse interpretations
the transactional interpretation
the von Neumann interpretation in which consciousness causes collapse.

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

the Bohm interpretation
the Ensemble Interpretation
the Many-Worlds Interpretation
interpretations based on Consistent Histories

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 problem is not really confronted by the Copenhagen Interpretation, which postulates that this is a special characteristic of the "measurement" process. The 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 ofdecoherence, which causes an apparent collapse. Decoherence is also important for the interpretation based on Consistent Histories.

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.

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.

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In quantum mechanics, wave function collapse (also called collapse of the state vector orreduction of the wave packet) is the phenomenon in which a wave function—initially in a superpositionof several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer. In simplified terms, it is the reduction of the physical possibilities into a single possibility as seen by an observer. It is one of two processes by which quantum systems evolve in time, according to the laws of quantum mechanics as presented by John von Neumann.^[1] The reality of wave function collapse has always been debatable, i.e., whether it is a fundamental physical phenomenon in its own right or just an epiphenomenon of another process, such as quantum decoherence.^[2] In recent decades the quantum decoherence view has gained popularity.^{[citation needed]}Collapse may be understood as an update in a probabilistic model, given the observed result.

Mathematical terminology

For an explanation of the notation used, see Bra-ket notation.

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

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

where the $\scriptstyle |i \rang$ s specify the different quantum "alternatives" available (technically, they form an orthonormal eigenvector basis, which implies $\scriptstyle \lang i | j \rang = \delta_{ij}$ ). 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 $\scriptstyle \psi_i = \lang i|\psi \rang$ are the probability amplitude coefficients, which are complex numbers. For simplicity we shall assume that our wave function is normalized, which means that

$\lang \psi|\psi \rang = \sum_i |\psi_i|^2 = 1.$

Wave function collapse

Tuesday, June 28, 2011

Wave function collapse2

[edit]History and context

[edit]

Wave function collapse1

Mathematical terminology