First quantization

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A first quantization of a physical system is a semi-classical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well or a bulk electromagnetic field or gravitational field) is treated classically. First quantization is appropriate for studying a single quantum-mechanical system being controlled by a laboratory apparatus that is itself large enough that classical mechanics is applicable to most of the apparatus.

Theoretical background[edit]

The starting point is the notion of quantum states and the observables of the system under consideration. Quantum theory postulates that all quantum states are represented by state vectors in a Hilbert space, and that all observables are represented by Hermitian operators acting on that space.[1] Parallel state vectors represent the same physical state, and therefore one mostly deals with normalized state vectors. Any given Hermitan operator \hat{A} has a number of eigenstates |\psi_\alpha\rangle that are left invariant by the action of the operator up to a real scale factor \alpha, i. e., \hat{A}|\psi_\alpha\rangle=\alpha|\psi_\alpha\rangle. The scale factors are denoted the eigenvalues of the operator. It is a fundamental theorem of Hilbert space theory that the set of all eigenvectors of any given Hermitian operator forms a complete basis set of the Hilbert space.

In general the eigenstates |\psi_\alpha\rangle and |\psi_\beta\rangle of two different Hermitian operators \hat{A} and \hat{B} are not the same. By measurement of the type \hat{B} the quantum state can be prepared to be in an eigenstate |\psi_\beta\rangle. This state can also be expressed as a superposition of eigenstates |\psi_\alpha\rangle as |\psi_\beta\rangle=\sum_\alpha|\psi_\alpha\rangle C_{\alpha\beta}. If one measures the dynamical variable associated with the operator \hat{A} in this state, one cannot in general predict the outcome with certainty. It is only described in probabilistic terms. The probability of having any given |\psi_\alpha\rangle as the outcome is given as the absolute square |C_{\alpha\beta}|^2 of the associated expansion coefficient. This non-causal element of quantum theory is also known as the wave function collapse. However, between collapse events the time evolution of quantum states is perfectly deterministic.

The time evolution of a state vector |\psi (t)\rangle is governed by the central operator in quantum mechanics, the Hamiltonian \hat{H} (the operator associated with the total energy of the system), through Schrödinger's equation:

i \hbar \frac{\partial}{\partial t}|\psi (t)\rangle = \hat H |\psi (t)\rangle

Each state vector |\psi\rangle is associated with an adjoint state vector (|\psi\rangle)^\dagger =  \langle \psi | and can form inner products, "bra(c)kets"  \langle \psi |\phi\rangle between adjoint[disambiguation needed] "bra" states \langle \psi| and "ket" states  |\phi\rangle. The standard geometrical terminology is used; e.g. the norm squared of |\psi\rangle is given by \langle \psi |\psi\rangle and |\psi\rangle and |\phi\rangle are said to be orthogonal if \langle \psi |\phi\rangle = 0. If {|\psi_\alpha\rangle} is an orthonormal basis of the Hilbert space, the above-mentioned expansion coefficient C_{\alpha \beta} is found forming inner products: C_{\alpha \beta}=\langle \psi_\alpha |\psi_\beta\rangle. A further connection between the direct and the adjoint Hilbert space is given by the relation \langle \psi |\phi\rangle = \langle \phi |\psi\rangle^*, which also leads to the definition of adjoint operators. For a given operator \hat{A} the adjoint operator \hat{A}^\dagger is defined by demanding \langle \psi |\hat{A}|\phi\rangle = \langle \phi |\hat{A}^\dagger|\psi\rangle^* for any |\psi\rangle and |\phi\rangle.

One-particle systems[edit]

In general, the one-particle state could be described by a complete set of quantum numbers denoted by \nu. For example, the three quantum numbers n,l,m associated to an electron in a coulomb potential, like the hydrogen atom, form a complete set (ignoring spin). Hence, the state is called |\nu\rangle and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using \psi_\nu(\bold{r})= \langle \bold{r}|\nu\rangle. All eigenvectors of an Hermitian operator form a complete basis, so one can construct any state |\psi\rangle=\sum_\nu|\nu\rangle\langle \nu|\psi\rangle obtaining the completeness relation:

\sum_\nu|\nu\rangle\langle \nu|=\bold{\hat 1}

All the properties of the particle could be known using this vector basis.

Many-particle systems[edit]

When turning to N-particle systems, i.e., systems containing N identical particles i.e. particles characterized by the same quantum numbers such as mass, charge and spin, is necessary an extension of single-particle state function \psi(\bold{r}) to the N-particle state function \psi(\bold{r}_1,\bold{r}_2,...,\bold{r}_N).[2] A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-called bosons and fermions which obey the rules:

\psi(\bold{r}_1,...,\bold{r}_j,...,\bold{r}_k,...,\bold{r_N})=+\psi(\bold{r}_1,...,\bold{r}_k,...,\bold{r}_j,...,\bold{r}_N) (bosons),

\psi(\bold{r}_1,...,\bold{r}_j,...,\bold{r}_k,...,\bold{r_N})=-\psi(\bold{r}_1,...,\bold{r}_k,...,\bold{r}_j,...,\bold{r}_N) (fermions).

Where we have interchanged two coordinates (\bold{r}_j, \bold{r}_k) of the state function. The usual wave function is obtained using the slater determinant and the identical particles theory. Using this basis, it is possible to solve any many-particle problem.

References[edit]

  1. ^ Dirac, P. A. M. (1982). Principles of Quantum Mechanics. USA: Oxford University Press. ISBN 0-19-852011-5. 
  2. ^ Merzbacher, E. (1970). Quamtum mechanics. New York: John Wiley & sons. ISBN 0471887021. 

See also[edit]