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In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, for systems where the total
number of particles In thermodynamics, the particle number (symbol ) of a thermodynamic system is the number of constituent particles in that system. The particle number is a fundamental thermodynamic property which is conjugate to the chemical potential. Unlike m ...
may not be preserved, the number operator is the
observable In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
that counts the number of particles. The following is in
bra–ket notation Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically de ...
: The number operator acts on Fock space. Let , \Psi\rangle_\nu=, \phi_1,\phi_2,\cdots,\phi_n\rangle_\nu be a Fock state, composed of single-particle states , \phi_i\rangle drawn from a basis of the underlying Hilbert space of the Fock space. Given the corresponding
creation and annihilation operators Creation operators and annihilation operators are Operator (mathematics), mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilatio ...
a^(\phi_i) and a(\phi_i)\, we define the number operator by \hat \ \stackrel\ a^(\phi_i)a(\phi_i) and we have \hat, \Psi\rangle_\nu=N_i, \Psi\rangle_\nu where N_i is the number of particles in state , \phi_i\rangle. The above equality can be proven by noting that \begin a(\phi_i) , \phi_1,\phi_2,\cdots,\phi_,\phi_i,\phi_,\cdots,\phi_n\rangle_\nu &=& \sqrt , \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_n\rangle_\nu \\ a^(\phi_i) , \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_n\rangle_\nu &=& \sqrt , \phi_1,\phi_2,\cdots,\phi_,\phi_,\phi_,\cdots,\phi_n\rangle_\nu \end then \begin \hat, \Psi\rangle_\nu &=& a^(\phi_i)a(\phi_i) \left, \phi_1,\phi_2,\cdots,\phi_,\phi_i,\phi_,\cdots,\phi_n\right\rangle_\nu \\ ex&=& \sqrt a^(\phi_i) \left, \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_n\right\rangle_\nu \\ ex&=& \sqrt \sqrt \left, \phi_1,\phi_2,\cdots,\phi_,\phi_,\phi_,\cdots,\phi_n\right\rangle_\nu \\ ex&=& N_i, \Psi\rangle_\nu \\ ex\end


See also

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Harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...
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Quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
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Second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
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Quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
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Thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
* (-1)F


References

* {{cite book, author=Bruus, Henrik, author2=Flensberg, Karsten, title=Many-body Quantum Theory in Condensed Matter Physics: An Introduction, publisher=Oxford University Press, year=2004, isbn=0-19-856633-6
Second quantization notes by Fradkin
Quantum operators