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{{Quantum mechanics|cTopic=Fundamental concepts}}
A '''first quantization''' of a physical system is a semi-[[classical mechanics|classical]] treatment of [[quantum mechanics]], in which particles or physical objects are treated using quantum [[wave function]]s 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==


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The starting point is the notion of [[quantum states]] and the [[observable]]s of the system under consideration. [[Quantum mechanics|Quantum theory]] postulates that all [[quantum states]] are represented  by [[Quantum state vector|state vector]]s in a [[Hilbert space]], and that all observables are represented by [[Hermitian operator]]s acting on that space.<ref name="dirac">{{cite book|last=Dirac|first=P. A. M.|authorlink=Paul Dirac|title=Principles of Quantum Mechanics|publisher=Oxford University Press|location=USA|isbn=0-19-852011-5|year=1982}}</ref> Parallel state vectors represent the same physical state, and therefore one mostly deals with [[Normalisable wave function|normalized]] state vectors. Any given Hermitan operator <math>\hat{A}</math> has a number of [[eigenstates]] <math>|\psi_\alpha\rangle</math> that are left invariant by the action of the operator up to a real scale factor <math>\alpha</math>, i. e., <math>\hat{A}|\psi_\alpha\rangle=\alpha|\psi_\alpha\rangle</math>. 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 [[Basis (linear algebra)|complete basis]] set of the Hilbert space.


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In general the [[eigenstates]] <math>|\psi_\alpha\rangle</math> and <math>|\psi_\beta\rangle</math> of two different Hermitian operators <math>\hat{A}</math> and <math>\hat{B}</math> are not the same. By measurement of the type <math>\hat{B}</math> the quantum state can be [[Preparation (physics)|prepared]] to be in an eigenstate <math>|\psi_\beta\rangle</math>. This state can also be expressed as a superposition of eigenstates <math>|\psi_\alpha\rangle</math> as <math>|\psi_\beta\rangle=\sum_\alpha|\psi_\alpha\rangle C_{\alpha\beta}</math>. If one measures the dynamical variable associated with the operator <math>\hat{A}</math> 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 <math>|\psi_\alpha\rangle</math> as the outcome is given as the absolute square <math>|C_{\alpha\beta}|^2</math> 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.


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


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


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Each state vector <math>|\psi\rangle</math> is associated with an adjoint state vector <math>(|\psi\rangle)^\dagger =  \langle \psi |</math> and can form [[inner product]]s, "[[Dirac notation|bra(c)ket]]s" <math> \langle \psi |\phi\rangle</math> between [[adjoint]]{{dn|date=December 2013}} "bra" states <math>\langle \psi|</math> and "ket" states <math> |\phi\rangle</math>. The standard geometrical [[Inner product space|terminology]] is used; e.g. the norm squared of <math>|\psi\rangle</math>  is given by <math>\langle \psi |\psi\rangle</math> and <math>|\psi\rangle</math> and <math>|\phi\rangle</math> are said to be [[orthogonal]] if <math>\langle \psi |\phi\rangle = 0</math>. If <math>{|\psi_\alpha\rangle}</math> is an [[orthonormal basis]] of the Hilbert space, the above-mentioned expansion coefficient <math>C_{\alpha \beta}</math> is found forming inner products: <math>C_{\alpha \beta}=\langle \psi_\alpha |\psi_\beta\rangle</math>. A further connection between the direct and the adjoint Hilbert space is given by the relation <math>\langle \psi |\phi\rangle = \langle \phi |\psi\rangle^*</math>, which also leads to the definition of adjoint operators. For a given operator <math>\hat{A}</math> the adjoint operator <math>\hat{A}^\dagger</math>  is defined by demanding <math>\langle \psi |\hat{A}|\phi\rangle = \langle \phi |\hat{A}^\dagger|\psi\rangle^*</math> for any <math>|\psi\rangle</math> and <math>|\phi\rangle</math>.


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


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<math>\sum_\nu|\nu\rangle\langle \nu|=\bold{\hat 1}</math>
 
All the properties of the particle could be known using this vector basis.
 
==Many-particle systems==
 
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]], [[Electric charge|charge]] and [[spin (physics)|spin]], is necessary an extension of single-particle state function <math>\psi(\bold{r})</math> to the N-particle state function <math>\psi(\bold{r}_1,\bold{r}_2,...,\bold{r}_N)</math>.<ref name="Merzbacher">{{cite book|last=Merzbacher|first=E.|authorlink=|title=Quamtum mechanics|publisher= John Wiley & sons|location=New York|isbn=0471887021|year=1970}}</ref> 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:
 
<math>\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)</math> (bosons),
 
<math>\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)</math> (fermions).
 
Where we have interchanged two coordinates <math>(\bold{r}_j, \bold{r}_k)</math> 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==
<references/>
 
== See also ==
* [[Quantization (physics)|Quantization]]
* [[Canonical quantization]]
* [[Second quantization]]
 
{{DEFAULTSORT:First Quantization}}
[[Category:Quantum mechanics]]

Revision as of 21:50, 3 February 2014

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Theoretical background

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 A^ has a number of eigenstates |ψα that are left invariant by the action of the operator up to a real scale factor α, i. e., A^|ψα=α|ψα. 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 |ψα and |ψβ of two different Hermitian operators A^ and B^ are not the same. By measurement of the type B^ the quantum state can be prepared to be in an eigenstate |ψβ. This state can also be expressed as a superposition of eigenstates |ψα as |ψβ=α|ψαCαβ. If one measures the dynamical variable associated with the operator 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 |ψα as the outcome is given as the absolute square |Cαβ|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 |ψ(t) is governed by the central operator in quantum mechanics, the Hamiltonian H^ (the operator associated with the total energy of the system), through Schrödinger's equation:

it|ψ(t)=H^|ψ(t)

Each state vector |ψ is associated with an adjoint state vector (|ψ)=ψ| and can form inner products, "bra(c)kets" ψ|ϕ between adjointTemplate:Dn "bra" states ψ| and "ket" states |ϕ. The standard geometrical terminology is used; e.g. the norm squared of |ψ is given by ψ|ψ and |ψ and |ϕ are said to be orthogonal if ψ|ϕ=0. If |ψα is an orthonormal basis of the Hilbert space, the above-mentioned expansion coefficient Cαβ is found forming inner products: Cαβ=ψα|ψβ. A further connection between the direct and the adjoint Hilbert space is given by the relation ψ|ϕ=ϕ|ψ*, which also leads to the definition of adjoint operators. For a given operator A^ the adjoint operator A^ is defined by demanding ψ|A^|ϕ=ϕ|A^|ψ* for any |ψ and |ϕ.

One-particle systems

In general, the one-particle state could be described by a complete set of quantum numbers denoted by ν. 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 |ν and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using ψν(r)=r|ν. All eigenvectors of an Hermitian operator form a complete basis, so one can construct any state |ψ=ν|νν|ψ obtaining the completeness relation:

ν|νν|=1^

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

Many-particle systems

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 ψ(r) to the N-particle state function ψ(r1,r2,...,rN).[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:

ψ(r1,...,rj,...,rk,...,rN)=+ψ(r1,...,rk,...,rj,...,rN) (bosons),

ψ(r1,...,rj,...,rk,...,rN)=ψ(r1,...,rk,...,rj,...,rN) (fermions).

Where we have interchanged two coordinates (rj,rk) 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

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  2. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

See also