Creation and annihilation operators Totally Explained

Everything about Creation And Annihilation Operators totally explained

In physics, an annihilation operator is an operator 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's the adjoint of the annihilation operator. Depending on the context, the identity of the particles in question varies; for example, 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 for the quantum harmonic oscillator. In the latter case, the raising operator is interpreted as a creation operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent phonons. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.
The mathematics behind the creation and the annihilation operators is identical as the formulae for ladder operators that appear in the quantum harmonic oscillator. For example, the commutator of the annihilation and the creation operator associated with the same state equals one; all other commutators vanish.
While the concept of creation and annihilation operators is well defined for free field theories, in interacting QFTs, they can only be defined in the interaction picture, which doesn't exist according to Haag's theorem.

Derivation of bosonic creation and annihilation operators

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 ť For bosons: ψ(1,2,3,4,...N) = ψ(2,1,3,4,...N)

   For fermions: ψ(1,2,3,4,...N) = -ψ(2,1,3,4,...N)
   For now let's just consider the case of bosons because fermions are more complicated.
   Start with the Schrödinger equation for the one dimensional time independent quantum harmonic oscillator ť

left(-frac)|psi
angle

Other kinds of interactions can be included in a similar manner.
This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.

Notational caveats and considerations

In quantum mechanics, Dirac bra-ket notation is often used. However, there's some ambiguity in this notation, particularly when there's the need to differentiate between these things:

The lowest energy state
The vacuum state
The zero ket

Often, these are all interchangeably notated as |0>, or even | >. As a result, it's 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 property that when the annihilation operator b is applied to it, it satisfies b|0> = 0| > = 0
   The intermediate step is rarely indicated as it's considered necessary only when more conceptual/mathematical rigour is needed.
   In this example, the lowest energy state is denoted as |0>. It is labeled as the "zero state", but it's important to emphasize that any state can be labeled as the "zero" state. The zero state is often used as a reference state to other quantum states. Therefore, the |0> state need not be the state with the absolutely lowest energy. In the case of the harmonic oscillator, it's due to the particulars of the mathematics that the ground state is chosen to be |0>. The vacuum state is the state where no quanta is available to be extracted. This special null state is denoted by | >. This vacuum state is also known as the "zero ket" because there are zero particles in the state. Unfortunately, the lowest energy state |0> is also known as the "zero ket" for the different reason that the state is labeled as "zero". Care must be taken that the four concepts listed above are not mixed together.
   Sometimes, the terms "null state" and "empty state" are used interchangeably for |> and |0>. The meaning for this usage is again dependent on the context.

The vacuum state

The vacuum state is a conceptual state which has no particles. The state is usually denoted as |0>, not the "empty ket" | >. Interestingly enough, no actual function represents the |0> state, but for notational purposes, we define the vacuum state as being normalized such that <0|0> = 1 and that |0> is orthogonal to all other states of the form |N>, where N is any indexing of quantum states for a particular system.

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