First quantization

Quantum mechanics
${\hat {H}}\|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}\|\psi (t)\rangle$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function (collapse)
Experiments Afshar Bell's inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Mach–Zehnder Popper Quantum eraser (delayed-choice) Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Overview Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Many-worlds Objective collapse Bayesian Quantum logic Relational Stochastic Scale relativity Transactional
Advanced topics Quantum chaos Quantum computing Density matrix Quantum field theory Fractional quantum mechanics Quantum information science Relativistic quantum mechanics Scattering theory Quantum statistical mechanics
Scientists Aharonov Bell Blackett Bloch Bohm Bohr Born Bose de Broglie Candlin Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Pauli Lamb Landau Laue Moseley Millikan Onnes Planck Rabi Raman Rydberg Schrödinger Sommerfeld von Neumann Weyl Wien Wigner Zeeman Zeilinger

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.

One-particle systems

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 $\psi _{\nu }({\mathbf {r}})=\langle {\mathbf {r}}|\nu \rangle$ . All eigenvectors of a 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:

$\sum _{\nu }|\nu \rangle \langle \nu |={\mathbf {{\hat 1}}}$

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 physical parameters such as mass, charge and spin, is necessary an extension of single-particle state function $\psi(\bold{r})$ to the N-particle state function $\psi(\bold{r}_1,\bold{r}_2,...,\bold{r}_N)$ .^[1] 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

↑ Merzbacher, E. (1970). Quantum mechanics. New York: John Wiley & sons. ISBN 0471887021.

This article is issued from Wikipedia - version of the 11/3/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

First quantization

One-particle systems

Many-particle systems

See also

References