Creation and annihilation operators - Mathematical details

An annihilation operator is the operator in that lowers the number of particles in a given state by one. A creation operator is an operator that increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. Depending on the context, the identity of the particles in question varies, i.e. in quantum chemistry , and many-body theory the creation and annihilation operators often act on electrons. Annihilation and creation operators can also refer specifically to the ladder operators ...

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• Creation and annihilation operators - Derivation of bosonic creation and annihilation operators
• Creation and annihilation operators - Mathematical details
• Creation and annihilation operators - Notational caveats and considerations
• Creation and annihilation operators - The vacuum state
• Creation and annihilation operators - Energy spectra

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The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators. The more abstract (and hence more applicable) form of the operators satisfy the properties below. Let H be the one-particle Hilbert space. To get the bosonic CCR algebra, look at the algebra generated by a(f) for any f in H. The operator a(f) is called an annihilation operator and the map a(.) is antilinear. Its adjoint is a†(f) which is linear in H. For a boson, ...

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Creation and annihilation operators, Creation and annihilation operators - Derivation of bosonic creation and annihilation operators, Creation and annihilation operators - Mathematical details, Creation and annihilation operators - Notational caveats and considerations, Creation and annihilation operators - The vacuum state, Creation and annihilation operators - Energy spectra

Read more here: » Creation and annihilation operators: Encyclopedia II - Creation and annihilation operators - Mathematical details

In quantum mechanics, Dirac bra-ket notation is often used. However, there is some ambiguity in this notation, particularly when there is the need to differentiate between these things: The lowest energy state The zero state The vacuum state The zero ket Often, these are all interchangeably notated as |0>, or even | >. As a result, it is necessary to read carefully, and consider the context in which the notation is used. For example, in the quantum harmonic oscillator, the ground state has the proper ...

See also:

Creation and annihilation operators, Creation and annihilation operators - Derivation of bosonic creation and annihilation operators, Creation and annihilation operators - Mathematical details, Creation and annihilation operators - Notational caveats and considerations, Creation and annihilation operators - The vacuum state, Creation and annihilation operators - Energy spectra

Read more here: » Creation and annihilation operators: Encyclopedia II - Creation and annihilation operators - Notational caveats and considerations

In the context of the quantum harmonic oscillator, we reinterpret the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system. Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties. Suppose the wavefunctions are dependent on N properties. Then For bosons: ψ(1,2,3,4,...N) = ψ(2,1,3,4,. ...

See also:

Creation and annihilation operators, Creation and annihilation operators - Derivation of bosonic creation and annihilation operators, Creation and annihilation operators - Mathematical details, Creation and annihilation operators - Notational caveats and considerations, Creation and annihilation operators - The vacuum state, Creation and annihilation operators - Energy spectra

Read more here: » Creation and annihilation operators: Encyclopedia II - Creation and annihilation operators - Derivation of bosonic creation and annihilation operators