Measurement in quantum mechanics

The special case of a measurement process is the need of a physical interaction between certain properties of the measuring object (for example, position, momentum, magn. Torque) and the state (the " hand position " ) of the measuring apparatus.

The characteristic feature in a measurement process on a macroscopic scale is that each physical quantity to be measured, after the interaction of object and apparatus, a pointer position can be assigned and at a repetition of the measurement under the same conditions ( approximately ) would set the same result.

In a measuring process in microscopic dimensions, however, it may happen that the repetition of measurements under exactly the same condition lead to any different measurement results. The latter deviations can not be justified by the measurement inaccuracy of the measurement apparatus, but are caused by the statistical nature of quantum mechanics, which comes to expression in the interaction in the measuring apparatus. The analysis of the measurement process in microscopic dimensions is sometimes referred to as quantum mechanical measurement process.

• 3.1 probabilities in spin measurements
• 3.2 Spin measurements on composite systems

Preparation and measurement in quantum mechanics

The theoretical interpretation of a measurement process in quantum mechanics requires according to Heisenberg three steps:

• There is generated a population of particles which is represented by a wave function ψ ( preparation ).
• There is a temporal evolution of the physical system, consisting of a measuring object and measuring apparatus, instead of (interaction ).
• Following the time evolution of the measurement result is determined (registration).

Especially Lamb refers in this context to a unique clarification, wherein the first step of the process is to consist exactly, so the fixing of the external conditions, by which the initial situation of the observed system is defined before the measurement, for example, that the particles to beginning are in a certain region of space or have a certain magnetic moment.

On the other hand is the result of a single measurement of a real number or a property of the considered metric. In repetitions of the measurement, not the same result as before, but at best a values ​​- statistic for the parameter in question. For a given preparation, the probability of the result of measurement in the frame of quantum mechanics, it can be predicted from the time evolution of the wave function in the measuring apparatus.

It was John von Neumann, who this concept in 1932 as the first detail formalized in his mathematical textbook on quantum mechanics .. Von Neumann's aim was to both the measuring object ( called system ) thus describe also the measuring apparatus including their interaction in the framework of quantum mechanics. This concept is often referred to in the literature as von Neumann measurement process.

Von Neumann measurement process

In quantum mechanics, the observable quantities (eg position, momentum, spin or energy) are presented of a physical system by the eigenvalues ​​of Hermitian operators. The hermiticity of the operators thereby ensures that all measured variables are always real numbers. The eigenvalues ​​of the operators on the one hand take discrete values ​​, then one speaks of a so-called point spectrum, or the eigenvalues ​​correspond to a continuum of real numbers, the so-called continuous spectrum.

Preparation

In the preparation of the quantum-mechanical state of the system is specified prior to the measurement. This specification is made formally in the abstract Hilbert space of the physical quantity associated. The ( normalized ) is the Hermitian operator of the considered physical quantity and the corresponding point spectrum of eigenvalues ​​, thus forming eigenvectors of the operator basis vectors of the so- defined Hilbert space. Any condition of this Hilbert space can be uniquely represented in this basis

Wherein the components of the state vector is. For such states is the probability to get one of the possible values ​​, given by the absolute square of the components of the state vector ( Born'sches postulate ). For physical systems with a continuous eigenvalue spectrum, the procedure is similar, but mathematically demanding.

Measurement

In analogy to the considered measured parameter describes the von Neumann ( macroscopic ) measuring apparatus by a corresponding base vectors in Hilbert space. The purpose of the measuring device is to display the state of the system according to the observed interaction of the instrument. The " link positions" of the instrument are thereby indicated by the state, if the system was in the state prior to the interaction. At the start of the measurement is defined such that the instrument is in a state, the state indicating that is not measured. The interaction of the system, and measuring results according to the Schrödinger time evolution after a time t, the following scheme:

Wherein the product corresponds to the so-called tensor product of two states of the overall system. This product can be easier to understand as an AND operation. In accordance with this scheme, there is a unique mapping of the possible states of the system and the possible hand positions of the instrument. Since this scheme the state of the system observed before and not influenced by the interaction of the instrument, it is referred to as the ideal measurement.

Measurement on state mixtures

For systems which are described by a density operator, the probability of measuring the eigenvalue of the operator given by:

The system is located directly after the measurement in the state.

The measurement problem

The time evolution of the interaction between a measured object and the measuring device can give a final state of the overall system, which initially corresponds to no clear pointer position of the instrument. On the other hand, unambiguous measurement results can be read on the meter but in practice. The question of how the decision is done in this process for the display of the device is known as the measurement problem. The measurement problem is explained below using an example:

A measurement object is initially in the eigenstate. The meter is in the initial state at the beginning. After the object has undergone the interaction with the meter after time t is the unit in the state. Schematically, this process can be represented as follows

Is analogous to the situation when the object is present at the start in the state. In this case, the interaction results in the meter with the following scheme

Is now the initial state, which is to be measured is not an eigenstate of the observable to be measured, but a superposition of different eigenstates ( superposition ), the process of measurement is complicated.

For example, if the measurement object at the top in the state, then by the interaction with the measuring device is also a superposition of states of the measuring result, that is, schematically, the picture is

In this case, can the state after the interaction is no longer represented as a product. The superposition of the states of the system are transmitted by the temporal interaction on the macroscopic states of the meter. The final state thus corresponds to a superposition of system and meter and it is no longer obvious which should correspond to the pointer position, the final state of the system (see Schrödinger's cat). Only after the registration at the end, a clear statement be made ​​as to whether either or present. After the Born rule, these two alternatives become valid with the probabilities on or off. In this case, however, initially remains unclear how the transition from the superposition of the eigenstates, ie schematically

Can be explained in physical terms. Von Neumann argues in this context, with a reduction of the quantum mechanical state ( measurement 1 type ), which is sometimes called the wave function collapse. In this context, von Neumann, in particular, point out (see Chap. 5) that the reduction of the wave function is discontinuous and instantly takes place in time and thus can not be described in the framework proposed by Schrödinger linear time trend.

Spin measurements

In the measurement of the magnetic spin in a torque source spin 1/2 particles are prepared, which move in the y direction after exiting through an inhomogeneous magnetic field which is oriented in the z direction ( the Stern-Gerlach experiment). The magnetic moment of the particles will be therein for a short time in interaction with the magnetic field, which causes the direction of the spin -dependent force on the particles. This force causes the particles in the state of spin ' up' can be deflected in the positive z - direction, and particles spin state "down" in the negative z - direction. The particles are deflected so registered at the end of detectors on the z -axis.

The coupling of the system and measuring the von Neumann measurement process is also triggered here by the product of the meter spin and the magnetic field. The relevant for the interaction part of the Hamiltonian is. A pointer states the possibility of local displacements in the upper or lower half of the z- plane are used, for example, " 1" for particles above and "-1" for particles below the coordinate origin. Schematically, this results in

The eigenvectors of the spins are in the z- direction. For certain purposes it is sometimes useful to measure the spin in the direction of any given unit vector. For this, the Stern-Gerlach apparatus is then oriented in this direction. The actual spin measurement is then possible with the help of the vector of Pauli matrices. The corresponding operator is given by the scalar product of the spin vector

The operator corresponding to this eigenvalue problem is

That is, the eigenvalues ​​of the operator are and represent the two spin directions "up" and "down " or the pointer variable with respect to the orientation of the vector. With the help of these eigenvectors can be constructed following the spin- projector

As any of the projector operator as defined own values ​​"1" and "0". The latter are interpreted sometimes as " Yes / No " statements in quantum logic.

Probabilities in spin measurements

Consider a population of particles that are oriented only in the direction of the vector, the thus prepared object represented by the state. Is in this state, then a spin measurement in the direction of a vector carried out, then the conditional probability for this to register a particle having a spin orientation in this direction is equal to

To evaluate this expression often limited to one direction vectors and both lie in the xz plane of the local area and their orientation with respect to the z- axis in both cases. We call the corresponding angles with and. For the probability of measurements with the specified orientation, the previous equation simplifies to

Often, this formula is also given in an equivalent form

The formula for the conditional probability plays an important role in connection with the Bell's Inequality and the derivation of the quantum Zeno effect inter alia.

Spin measurements on composite systems

From classical physics forth one is used to that compound systems can be decomposed into sub-systems or sub-systems. In quantum mechanics shows that composite systems may have completely different and surprising holistic properties beyond. They occur when composite quantum systems are in entangled states.

It can, for example, the quantum systems are prepared in which one photon is registered at two different locations for measurements. Analog systems are also available for electrons. Two such particles are considered to be indistinguishable. Distinguishable, however, are the locations of the objects, in which in a measurement such as a photon polarization is measured.

Formally, we consider, for example, S1 and S2 are two independent quantum systems which are respectively represented by the spin states, and. The arrows are often used in the literature when the respective spins are prepared in the z- direction ("up " or " down"). The product

Defines a polymer formed from the two part- systems the overall system, i.e., a product state of the particles. In this specific case the factored Zweiteilchenwellenfunktion and are independent subsystems, as no dependence can be detected between the systems.

An entanglement of the subsystems arises, inter alia, for the so-called singlet state, which plays an important role in the EPR thought experiment. The formal representation of this state

This condition can not be expressed as a product of two single-particle states. A measurement of the spins of the two individual particles in the z direction is with probability 1 have opposite signs, and with which the measurements are carried out independently of the order of the two components. Determining the orientation in the z direction is not a restriction of the generality, since this state has the property of the rotation invariance.

Within the EPR experiment for two fixedly predetermined direction vectors a and b, spin measurements on the two components 1 and 2 of the singlet state are performed. Here, you can go to the conditional probability ask to get a spin in the b direction with particle 2 if the measurement on particle 1 has shown the result in a- direction:

The tensor product of the operators makes it clear that each of the left occurring in this product operator is applied only to the first component of the singlet state, while the right-wing each operator is to be applied only to the second component of the Singulattzustandes. The denominator is equal to the probability of obtaining the value of the measurement at a spinning particle 1 in the A direction. It should be noted here that this probability is always equal to 1 /2 regardless of the direction in which a is in the measurement. 1 in the denominator of the operator means that the second component of the singlet in the expected value generation remains unchanged.

According to the rules of algebra spin results for the conditional probability of the simple result formally

Where and are respectively the values ​​ 1 ( "up") or -1 ( " down") can have. This probability is invariant under the interchange of the angle and the two spins.

The joint probability of the two events is equal to the product of the conditional probability and the probability of the edge, i.e.,

This allows the correlation of two spin operators calculated by ordinary expectation value over the random variables of the two spin parameters:

Where the last term representing the inner product of the normal direction vectors. The special case of equal orientation of the two measuring directions of A and B corresponds to an angle difference of 0 °. In this case, the correlation is -1. On the other hand form the orientations measured a relative angle of 90 °, i.e., they are orthogonal to each other, then the above formula gives a negligible correlation of 0