Creation and annihilation operators - Derivation of bosonic creation and annihilation operators

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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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,. ...

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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 - Derivation of bosonic creation and annihilation operators

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 ...

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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 - Notational caveats and considerations

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