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. MEASUREMENT PROBLEM IN QUANTUM MECHANICS

August 3, 2015
THE SYMBOLIC MATHEMATICS FOR QM DECOHERENCE

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This post gives a basic mathematical formalism to the phenomenon of decoherence, or the point at which a quantum system evolves into the classical world, apparently occurring only when measured.

To understand decoherence requires knowledge of classical phase spaces and Hilbert spaces (Hilbert space is a vector space containing quantum states in any number of dimensions, and may be expressed in integers, real or complex numbers), and a simple derivation in Dirac (bra-ket) notation to demonstrate how decoherence destroys the quantum nature of a system.

In non-relativistic quantum mechanics a N particle system is represented as a wave function

xxx

each xi being a point in 3 dimensional space, r3. This is quantum phase space. Classical phase space contains a function in 6N dimensions – 3 spatial coordinates and 3 momentum. Therefore the effective dimensionality of a system’s phase space is how many degrees of freedom, which is 6 times the total number of a system’s free particles. Position and momentum are represented in Hilbert space as operators which do not commute.

The environment selects from the original state vector individual expansions that decohere – or lose phase coherence – with each other. These decohered elements are no longer in quantum mechanical interference with each other, and are said to be quantum entangled with their environment. The environment (in this description) is the measuring device, which is read by humans.

In Dirac notation,the initial state of the system is denoted as a vector

vector psi whereby:

6.1

where the ket-vectors  1,2  denote the quantum alternative states available. A quantum state is an element in an Hilbert space.

i  specifies the different quantum alternatives states available. They form an orthonormal eigenvector basis:

6.2

An ‘observable’ – meaning any measurable parameter of the system – will have a specific eigenvalue ei for each quantum alternative state i. An observable could be, say for a particle, its position r or momentum p. Other observables, also termed ‘linear operators’, are energy E, z – components of spin sz, orbital angular momentum Lz, total angular momentum Jz etc. In the Dirac, or bra-ket representation, these are, respectively:

6.3,4,5

The coefficients c1, c2  are probability amplitudes which correspond to each quantum state kets 1,2 , and the absolute square  ci2  is the probability of measuring the system to be in the quantum state, or eigenstate, of i.

All wave functions are assumed to be normalised. The sum of all the probabilities of measuring every possible state is unity,

6.6

The probability of collapsing to a given eigenstate phi-k  is the Born probability pk. After the measurement has been made, all other elements of the wave function vector – those that did not collapse, noeq 

will all have collapsed to zero so eq.

The total combined system and environment can be described in vector terms by tensor multiplying vectors of the subsystems altogether. Let epsc  be the initial state of the environment. Prior to any interaction, the joint state is written:

6.7

where cie  means the tensor product: cxepsi.

This system can interact with its environment in one of two ways:

  • The system loses its identity and merges with the environment
  • The system is not disturbed. This is the idealised non-disturbing measurement envisaged in the EPR thought experiment.

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System absorbed by environment

6.8

and therefore:

6.9

The unitarity of time evolution demands that the total of all states remains orthonormal. Further, their scalar (sometimes called inner) products with each other vanish, since  ijd  we have:

6.10

The orthonormality of quantum environment states is required for the environment to select out those states to decohere – a process termed einselection.

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System not disturbed by environment

This is the EPR idealised undisturbed system. Here, each element of the system interacts with the environment in the following manner:

6.11

This means that the system does disturb the environment, but is itself undisturbed by the environment. This gives:

aaa

and due to the requirement for unitarity:

6.13

and due to the large number of hidden degrees of freedom in the environment, decoherence also requires:

eeapd

As before noted, this is the requirement for the environment to select out those states to decohere, or einselect – but the approximation becomes more exact when there’s an increase in the number of degrees of freedom in the environment.

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These Dirac expressions regarding decoherence occur throughout quantum mechanics of the measurement problem, specifically in von Neumann’s attempt to mathematically describe wave function collapse by the interaction of human consciousness on wave function collapse, and in the Many Worlds hypothesis.

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