Collapse of the wave function

Collapse of the wave function or state reduction is a term from the Copenhagen interpretation of quantum mechanics . What is meant is the change in the description of a quantum system that results from a quantum mechanical measurement .

Explanation

In quantum mechanics, a physical system is completely described by specifying its current quantum mechanical state . This state can be developed in an operator's own basis.

In the Bra-Ket notation, the wave function is then written:

${\displaystyle |\psi \rangle =\sum \limits _{i=1}^{n}c_{i}|\varphi _{i}\rangle }$

The overall state is a superposition of all possible eigen-states with their probability amplitudes . If a measurement is carried out on such a system , the experimenters will always determine a single measured value. This is the eigenvalue of one of the eigenstates that correspond to the type of measurement. Immediately after the measurement, the system is in exactly this state of its own, because if it were repeated, it would have to reproduce the measured value just determined with certainty. Formally, this means that the superposition of eigenstates is reduced to a single one of these states through the measurement, the overall state is projected onto an eigenspace . This transition from the state of superposition to a certain eigenstate is called state reduction. If the initial state is represented as a Schrödinger wave function, the Copenhagen interpretation also speaks of the "collapse (or breakdown) of the wave function". ${\displaystyle |\psi \rangle }$ ${\displaystyle |\varphi _{i}\rangle }$ ${\displaystyle c_{i}}$

The reduction process

The collapse of the wave function occurs instantaneously, i.e. This means that even at spatially widely separated locations, there are immediate consequences for the prediction of measurements on the system. This property is known as quantum non-locality . In entangled systems , the quantum non-locality leads to statistical correlation of the measurement results, even if the locations of the measurements in an extensive entangled system are so far apart that a physical effect (information) could not be transmitted quickly enough even at the speed of light . With an inappropriate word, this is sometimes referred to as an action at a distance .

A crucial difference to a "classic" state description is sometimes overlooked: Unless the wave function describes an eigenstate before the measurement, it contains several eigenstates and a probability below 100% for each. In a sense, it does not really describe the system, but rather the incomplete knowledge of the system. Fröhner has shown that the quantum mechanical probabilities can be understood as Bayesian probabilities without contradiction . These change as the measurement changes the observer's level of information. No time is needed for this; what collapses ("collapses") is nothing physical, just the observer's lack of information. Correspondingly, Heisenberg and Styer expressed their views on this in 1960 in a discussion by letter (see quote from Fröhner).

history

The first approaches to the explanation come from Werner Heisenberg and were formulated as postulates by John von Neumann in his book Mathematical Basics of Quantum Mechanics in 1932 . The postulate of an instantaneous collapse of the wave function has been contradicting itself since its introduction. Schrödinger's cat , a popular thought experiment by Erwin Schrödinger , was supposed to reduce the idea of ​​an observer-dependent collapse of the wave function to absurdity .

Other interpretations of quantum mechanics such as the De Broglie-Bohm theory or the many-worlds interpretation manage without this concept. However, the many-worlds interpretation must allow a multitude of metrologically inaccessible "worlds" to avoid the collapse of the wave function. The decoherence plays a central role here, but it can also interpretations collapse be used to describe the time of the collapse.

See also

• Schrödinger-Newton equation

literature

The concept of quantum mechanical measurement and thus the collapse of the wave function is dealt with in many introductory and further textbooks.

• J. v. Neumann: Mathematical foundations of quantum mechanics . 2nd Edition. Springer, 1996, ISBN 978-3-540-59207-5 .
• JJ Sakurai: Modern Quantum Mechanics . Addison-Wesley, 1994, ISBN 0-201-53929-2 .
• T. Fließbach: Quantum Mechanics . 5th edition. Spectrum, Heidelberg 2008, ISBN 978-3-8274-2020-6 .

Individual evidence

1. ^ Max Born, Albert Einstein: Albert Einstein, Max Born. Correspondence 1916–1955. Nymphenburger, Munich 1955, p. 210.
   Einstein speaks of a "spooky action at a distance".
2. ^ FH Fröhner: Missing Link between Probability Theory and Quantum Mechanics: the Riesz-Fejér Theorem. In: Journal for Nature Research. 53a (1998), pp. 637-654 ( online ).
3. ^ Daniel F. Styer: The Strange World of Quantum Mechanics. Cambridge University Press, 2000, ISBN 0-521-66780-1 , p. 115.
4. ^ W. Heisenberg: About the descriptive content of quantum theoretical kinematics and mechanics . In: Journal of Physics . tape 43 , 1927, pp. 172-198 ( online ).