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{{Quantum mechanics|cTopic=Fundamental concepts}}
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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.
== Hermes Ceinture  well ==


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>.
  <li>[http://mariettakaramanli.fr/spip.php?article385/ http://mariettakaramanli.fr/spip.php?article385/]</li>
 
 
==One-particle systems==
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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:
 
 
</ul>
<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]]

Latest revision as of 20:43, 28 December 2014

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