# Quantum mechanical measurement

The quantum mechanical measurement process describes the measurement of a physical quantity on an object in quantum physics . For classical physics, it always applies, but only partially for quantum physics, that the measured value is clearly established before the measurement and always has the same value in the case of repeated measurements on the same and identically prepared measurement objects.

In quantum physics, however, many physical quantities do not have a certain value before the measurement. This also applies when the condition of the measurement object is prepared with ideal accuracy. In the case of repeated measurements, the measured values ​​then inevitably spread over a whole range of values. Examples are the point in time at which a radioactive atomic nucleus emits its radiation quantum, or the place at which an electron hits the screen in a diffraction experiment with electrons. As with every measurement in classical physics, only one value can be read on the measuring device, but it has not yet been solved satisfactorily how this is selected from the many possible values. With quantum mechanics and quantum field theory , only the probability that each of the possible measured values ​​will occur can be calculated, and the possibility of a precise prediction seems to be excluded in principle. Thus, the quantum mechanical measurement process represents one of the greatest unsolved problems for the interpretation of these otherwise extremely successful theories.

## overview

When taking measurements on a macroscopic object, the same result is obtained if the preparation and measurement are repeated exactly (ideally exact, real within the scope of the measurement accuracy). This fulfills the requirement of science that its results are reproducible . Furthermore, the reaction of the measuring process on the macroscopic object can either be neglected because of its insignificance ( ideal measurement ) or specified precisely.

In the case of measurements on quantum objects, on the other hand, it is typical that identical measurement procedures on identically prepared ('prepared') objects lead to widely scattered measurement results. Examples are the point in time at which a radioactive nucleus emits its radiation quantum, but also the place at which this radiation quantum triggers the reaction with which it is detected in an extended detector. According to the predominant Copenhagen interpretation of quantum mechanics, such deviations are not due to a lack of knowledge of the exact state of the object or the measuring device, but rather lie in the nature of quantum objects and are therefore an essential characteristic of quantum physics. A quantum physical measurement and its result can only be reproduced in special cases. Only in these cases does the object remain in the state it was in before the measurement. Otherwise it will be changed in an unpredictable way - in each case to match the measured value received. But even with the non-reproducible measurements one can find reproducible values ​​if one determines mean values ​​from a sufficient number of individual measurements, e.g. B. for the service life , the reaction rate or the cross section .

## Procedure and consequences of the measurement process

In every measuring process there is a physical interaction between certain properties of the measuring object (e.g. location, impulse, magnetic moment) and the condition of the measuring apparatus (generally called "pointer position"). After the measuring process, the value of the measured variable can be read from the pointer position of the measuring device.

The quantum mechanical measuring process requires a more in-depth interpretation because of the fundamental differences to the classic measuring process. John v. In 1932 Neumann was the first to formally describe the measurement process as part of the Copenhagen Interpretation and thus developed the view that is still largely accepted today. Formal basis is that in the quantum mechanics , the states of a physical system by vectors in a Hilbert space , and the observable quantities (z. B. location , pulse , spinning , energy ) by Hermitian operators are represented (for. Example, energy, angular momentum etc ., or mass and charge of the particle). The eigenvalues ​​of the operators are the possible measured values. They are also those measured values ​​that are well-determined and give the same value for every good measurement if the object is in an eigenstate of the operator concerned.

According to von Neumann , a distinction must first be made between three steps in the typical quantum physical measurement process:

• Preparation: A population of particles (or other quantum systems ) is created, which is represented by a state of one of the particles. The measurement process refers to the measurement on one specimen.${\ displaystyle \ psi}$
• Interaction: The quantum system to be measured and the measuring device form an overall system. An interaction takes place between them, through which the overall system develops over time .
• Registration: After the interaction is complete, the measurement result is read on the measuring device. If the measurement results vary, the entire measurement is repeated until a reliable mean value can be formed.

Although these three steps also apply to measurements in classical physics, the consequences of quantum mechanical measurements are very different. Due to the concept of state in quantum mechanics, the measured value before the measurement is only specified in the special cases that the quantum object is in an intrinsic state of the measured variable. In general, however, this is only selected during the measurement from a large number of the eigenstates present in the state under consideration. An example is the measurement of the position coordinate in a diffraction experiment when the matter wave belonging to the particle hits the whole screen, but only causes a signal in one place.

Since the quantum objects develop continuously outside of a measurement process according to an equation of motion (such as the Schrödinger equation), no finite changes are possible in infinitesimal times. Therefore, a second measurement immediately following the first measurement for the same measured variable must also provide the same result. In order for the theory to ensure this, it must presuppose that the quantum object has been converted into the eigenstate of the measurand that has the found measured value as its eigenvalue. All components of the state prepared for the measurement that belong to other eigenvalues ​​must be deleted during the measurement. This process is irreversible, because the surviving eigenstate cannot be used to specify which other components the quantum system had previously. This process is referred to as the collapse of the wave function or state reduction, but its course has not yet been clarified.

A crucial difference compared to a “classic” description of the state is sometimes overlooked: the wave function contains probabilities <100% for the individual eigenstates before the measurement. It therefore does not really describe the system, so to speak, 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).

## Preparation of the measurement object

### General

The preparation of a quantum object is a process by which the object is brought into a certain state, such as that described by the vector of the Hilbert space (e.g. an electron with a certain momentum and a certain direction of spin ). More important in practice is the case that a whole group of states is present (e.g. all possible directions of spin for a given momentum). Then it is a question of a mixture of states, which is better described with a density operator (see below). ${\ displaystyle | \ psi \ rangle}$

For the mathematical description of the measurement process, any state vector is represented as a linear combination of the eigenvectors of the operator assigned to the measurand : ${\ displaystyle | \ psi \ rangle}$ ${\ displaystyle | \ phi _ {n} \ rangle}$${\ displaystyle {\ hat {Q}}}$

${\ displaystyle | \ psi \ rangle = \ sum _ {n} \, c_ {n} \, | \ phi _ {n} \ rangle,}$

If they are normalized , as usual, the coefficients are clearly defined by and it is ${\ displaystyle | \ phi _ {n} \ rangle}$${\ displaystyle c_ {n}}$${\ displaystyle | \ psi \ rangle}$

${\ displaystyle \ sum _ {n} \, | c_ {n} | ^ {2} = 1 \ ,.}$

The following applies for the eigenstates and the eigenvalues , which are the possible measurement results in principle . ${\ displaystyle | \ phi _ {n} \ rangle}$${\ displaystyle q_ {n}}$${\ displaystyle {\ hat {Q}} | \ phi _ {n} \ rangle = q_ {n} | \ phi _ {n} \ rangle}$

Here this is written for a finite or at most countably infinite set of relevant eigen-states. In the case of a continuum , an integral should be used instead of the sum.

### Preparing by measuring

To prepare a state, one measures material that is in the form of a (different) state or as a mixture of states. A pure state is represented as a linear combination of orthogonal components, one of which is the desired state. The measurement then reduces the present to the target state. With a suitable mixture of states, the measurement only serves to sort out the objects that are in the desired state. A gap that fades out a part of a wide beam primarily causes a position measurement in the direction across the beam. As part of a spectrograph, it is used to measure frequency and energy. A polarization filter can be used in both functions, namely on pure as well as mixed states.

## Interaction creates entanglement with the measuring apparatus

The (macroscopic) measuring apparatus with its various “pointer positions” is also described by basis vectors in a corresponding Hilbert space. Each base state corresponds to a specific pointer position . The measuring device is designed in such a way that when measuring, the object changes to its own state. Before starting a measurement, let the measuring device be in any state and the object in its own state . Then the overall system consisting of the measurement object and the measuring device initially has the status ${\ displaystyle | M_ {n} \ rangle}$${\ displaystyle n}$${\ displaystyle | \ phi _ {n} \ rangle}$${\ displaystyle | M_ {n} \ rangle}$${\ displaystyle | M _ {\ text {before}} \ rangle}$${\ displaystyle | \ phi _ {n_ {0}} \ rangle}$

${\ displaystyle | \ Psi _ {0} \ rangle = | \ phi _ {n_ {0}} \ rangle | M _ {\ text {before}} \ rangle}$

and after the measurement the state

${\ displaystyle | \ Psi \ rangle _ {\ text {to}} = | \ phi _ {n_ {0}} \ rangle | M_ {n_ {0}} \ rangle}$,

because the pointer is now pointing . The object itself, if it is already in a state of its own for the operator concerned, does not change in the measuring process according to von Neumann. The prerequisite is seldom given in reality, but is helpful as a model. ${\ displaystyle n_ {0}}$

In the case of interest, the system is not already in an eigenstate of the measurement operator before the measurement, but in a linear combination formed from different eigenstates . The initial state of the overall system is then . Due to the interaction, the state is initially formed according to the rules of quantum mechanics ${\ displaystyle | \ psi _ {0} \ rangle = \ sum _ {n} \, c_ {n} \, | \ phi _ {n} \ rangle}$${\ displaystyle | \ Psi _ {0} \ rangle = | \ psi _ {0} \ rangle | M _ {\ text {before}} \ rangle}$

${\ displaystyle | \ Psi \ rangle _ {\ text {to}} = \ sum _ {n} \, c_ {n} \, | \ phi _ {n} \ rangle | M_ {n} \ rangle}$,

because the measuring device reacts to every component of the object state by accepting the state . ${\ displaystyle | \ phi _ {n} \ rangle}$${\ displaystyle | M_ {n} \ rangle}$

In this state of the overall system after the interaction, all components of the initial state occur simultaneously in correlation with their associated pointer positions. The superposition of the eigenstates in the initial state of the measurement object was transferred to the macroscopic states of the measuring device through the interaction. The state is no longer to be represented as the product of a state of the system with a state of the device, but corresponds to an interlaced state of system and measuring device. ${\ displaystyle \ sum _ {n} \, c_ {n} \, | \ phi _ {n} \ rangle}$

From this entanglement, one of the components is selected at the end of the measurement process through the state reduction, and in each case with probability . The original state has now been replaced by one of the states at random . Mathematically, a mapping takes place that turns the initial state into the final state with the normalized state vector and can therefore be written as follows: ${\ displaystyle | \ phi _ {n} \ rangle | M_ {n} \ rangle}$${\ displaystyle | c_ {n} | ^ {2}}$${\ displaystyle | \ psi _ {0} \ rangle = \ sum _ {n} \, c_ {n} \, | \ phi _ {n} \ rangle}$${\ displaystyle | \ psi _ {\ text {to}} \ rangle = | \ phi _ {n _ {\ text {to}}} \ rangle}$${\ displaystyle | \ psi _ {0} \ rangle}$${\ displaystyle | \ psi _ {\ text {to}} \ rangle}$

${\ displaystyle | \ psi _ {0} \ rangle \; {\ xrightarrow {\ text {Measurement}}} \; {\ frac {P _ {\ psi _ {\ text {to}}} | \ psi \ rangle} {\ left \ | P _ {\ psi _ {\ text {to}}} | \ psi _ {\ text {to}} \ rangle \ right \ |}},}$

In it is

${\ displaystyle P _ {\ psi _ {\ text {to}}} = | \ psi _ {\ text {to}} \ rangle \ langle \ psi _ {\ text {to}} |}$

the projector on the subspace to the eigenvector . ${\ displaystyle | \ psi _ {\ text {to}} \ rangle}$

The core of the measurement problem in quantum mechanics is that no linear equation of motion can be thought that could cause this mapping, as occurs in nature with every measurement.

## Registration of the result

From the entangled superposition that has arisen from the interaction in the measuring device, exactly one of the states is formed through the measurement , with a probability ${\ displaystyle | \ Psi \ rangle _ {\ text {to}} = | \ phi _ {n} \ rangle | M_ {n} \ rangle}$

${\ displaystyle P (n) = | c_ {n} | ^ {2}}$.

This cannot be described by a development over time that follows a Schrödinger equation (or another equation of motion which, like this, is linear and maintains the norm).

To solve, or at least to describe the measurement problem, a “reduction” of the quantum mechanical state is postulated, which is also referred to as the collapse of the wave function . It effects the transition caused by the measurement

${\ displaystyle \ sum _ {n} \, c_ {n} \, | \ phi _ {n} \ rangle | M_ {n} \ rangle \; {\ xrightarrow [{\ {\ text {process}} \} ] {\ {\ text {Mess -}} \}} \; | \ phi _ {n} \ rangle | M_ {n} \ rangle.}$

This simultaneously reduces the given probability distribution of the possible measured values ​​to a single value - the measured value. Only then can the value of the measured physical quantity be determined by reading the measuring device , and the quantum system is then with certainty in its own state . This ensures that an immediately subsequent repetition of the measurement has the same result. ${\ displaystyle P (n) = | c_ {n} | ^ {2}}$${\ displaystyle | \ phi _ {n} \ rangle}$

The state reduction is discontinuous and takes place instantaneously. When and how the reduction takes place and what its physical cause is, is still an unsolved problem today. The widely used expression that the state reduction occurs in the interaction process that is to be observed with the measuring device can be considered refuted at the latest since the implementation of delayed choice experiments and quantum erasers . Assumptions about the point in time or cause of the reduction extend up to the moment of subjective perception in the consciousness of an experimenter (e.g. with Schrödinger's cat and Wigner's friend ).

This open question has contributed significantly to the development of several interpretations of quantum mechanics that contradict the Copenhagen interpretation on this point. Mention should be made of the spontaneous reduction at stochastically distributed points in time in the GRW theory of dynamic collapse or through decoherence due to the energy-time uncertainty relation , if the self-energy is taken into account through gravity. A fundamentally different answer is offered by the many-worlds interpretation , in which as many copies of the world are created unnoticed with each measurement as there are possible measured values, so that one of the values ​​is realized in each of the worlds.

## Measurement on mixtures of states

For systems whose state is described by a density operator , the probability of finding the eigenvalue of the operator during the measurement is given by: ${\ displaystyle {\ hat {\ rho}}}$${\ displaystyle q_ {n}}$${\ displaystyle {\ hat {Q}}}$

${\ displaystyle P (n) = \ langle \ phi _ {n} | {\ hat {\ rho}} | \ phi _ {n} \ rangle = \ operatorname {Tr} ({\ hat {P}} _ { n} {\ hat {\ rho}})}$.

The operator is the projector into the subspace of the eigenstates for the eigenvalue . Immediately after the measurement, the system is in the state given by the density operator ${\ displaystyle {\ hat {P}} _ {n}}$${\ displaystyle q_ {n}}$

${\ displaystyle {\ hat {\ rho}} '= {\ frac {{\ hat {P}} _ {n} {\ hat {\ rho}} {\ hat {P}} _ {n}} {\ operatorname {Tr} ({\ hat {P}} _ {n_ {0}} {\ hat {\ rho}} {\ hat {P}} _ {n_ {0}})}}}$

given is.

## literature

• John v. Neumann: Mathematical Foundations of Quantum Mechanics , [Nachdr. the edition] Berlin, Springer, 1932. - Berlin; Heidelberg; New York: Springer, 1996.
• Jürgen Audretsch: Entangled Systems , ISBN 352740452X , 2005, in particular also for measurements on entangled systems, selective measurement and non-selective measurement, Chapter 7

## Individual evidence

1. J. v. Neumann: Mathematical Foundations of Quantum Mechanics , Springer (1932, 1968, 1996).
2. ^ W. Heisenberg: Physics and Philosophy. Hirzel, Stuttgart 1959.
3. ^ FH Fröhner: Missing Link between Probability Theory and Quantum Mechanics: the Riesz-Fejér Theorem. Zeitschrift für Naturforschung 53a (1998) pages 637–654 [1]
4. ^ Daniel F. Styer: The Strange World of Quantum Mechanics. Cambridge University Press, 2000, ISBN 0-521-66780-1 , 115
5. ^ Stuart Hameroff, Roger Penrose: Consciousness in the universe: A review of the 'Orch OR' theory . In: Physics of life reviews . tape 11 , no. 1 , 2014, p. 39-78 ( online [accessed March 13, 2019]).