Creation and annihilation operators can act on states of various types of particles. For example, in quantum chemistry and many-body theory the creation and annihilation operators often act on electron states. They 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.
The operators a and a† may be contrasted to normal operators, which commute with their adjoints.
Using the commutation relations given above, the Hamiltonian operator can be expressed as
One may compute the commutation relations between the a and a† operators and the Hamiltonian:
These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows.
Assuming that is an eigenstate of the Hamiltonian . Using these commutation relations, it follows that
This shows that and are also eigenstates of the Hamiltonian, with eigenvalues and respectively. This identifies the operators a and a† as "lowering" and "raising" operators between adjacent eigenstates. The energy difference between adjacent eigenstates is ΔΕ=ħω.
The ground state can be found by assuming that the lowering operator possesses a nontrivial kernel: with . Application of the above formula for the Hamiltonian yields
So is an eigenfunction of the Hamiltonian.
This gives the ground state energy , which allows one to identify the energy eigenvalue of any eigenstate as
Furthermore, it turns out that the first-mentioned operator in (*), the number operator plays the most important role in applications, while the second one, aa† can simply be replaced by N +1.
The operators derived above are actually a specific instance of a more generalized notion of creation and annihilation operators. The more abstract form of the operators are constructed as follows. Let H be a one-particle Hilbert space (that is, any Hilbert space, viewed as representing the state of a single particle).
The (bosonic) CCR algebra over H is the algebra-with-conjugation-operator (called *) abstractly generated by elements a(f), where f ranges freely over H, subject to the relations
The map a : f ↦ a(f) from H to the bosonic CCR algebra is required to be complex antilinear (this adds more relations). Its adjoint is a†(f), and the map f ↦ a†(f) is complex linear in H. Thus H embeds as a complex vector subspace of its own CCR algebra. In a representation of this algebra, the element a(f) will be realized as an annihilation operator, and a†(f) as a creation operator.
In general, the CCR algebra is infinite dimensional. If we take a Banach space completion, it becomes a C* algebra. The CCR algebra over H is closely related to, but not identical to, a Weyl algebra.
For fermions, the (fermionic) CAR algebra over H is constructed similarly, but using anticommutator relations instead, namely
The CAR algebra is finite dimensional only if H is finite dimensional. If we take a Banach space completion (only necessary in the infinite dimensional case), it becomes a C* algebra. The CAR algebra is closely related to, but not identical to, a Clifford algebra.
Physically speaking, a(f) removes (i.e. annihilates) a particle in the state | f whereas a†(f) creates a particle in the state | f.
If | f is normalized so that f | f = 1, then N = a(f) a†(f) gives the number of particles in the state | f.
Creation and annihilation operators for reaction-diffusion equationsEdit
The annihilation and creation operator description has also been useful to analyze classical reaction diffusion equations, such as the situation when a gas of molecules A diffuse and interact on contact, forming an inert product: A + A → ∅ . To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider particles at a site i on a one dimensional lattice. Each particle moves to the right or left with a certain probability, and each pair of particles at the same site annihilates each other with a certain other probability.
The probability that one particle leaves the site during the short time period dt is proportional to , let us say a probability to hop left and to hop right. All particles will stay put with a probability . (Since dt is so short, the probability that two or more will leave during dt is very small and will be ignored.)
We can now describe the occupation of particles on the lattice as a `ket' of the form | ..., n−1, n0, n1, ... . It represents the juxtaposition (or conjunction, or tensor product) of the number states ..., | n−1, | n0, | n1, ... located at the individual sites of the lattice. Recall
for all n ≥ 0, while
This definition of the operators will now be changed to accommodate the "non-quantum" nature of this problem and we shall use the following definition:
note that even though the behavior of the operators on the kets has been modified, these operators still obey the commutation relation
Now define ai so that it applies a to | ni. Correspondingly, define ai† as applying to | ni. Thus, for example, the net effect of ai−1†ai is to move a particle from the ith to the (i − 1)th site while multiplying with the appropriate factor.
This allows writing the pure diffusive behavior of the particles as
where the sum is over i.
The reaction term can be deduced by noting that particles can interact in different ways, so that the probability that a pair annihilates is , yielding a term
where number state n is replaced by number state n − 2 at site i at a certain rate.
Thus the state evolves by
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.
Creation and annihilation operators in quantum field theoriesEdit
by one, in analogy to the harmonic oscillator. The indices (such as ) represent quantum numbers that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a tuple of quantum numbers is used to label states in the hydrogen atom.
The commutation relations of creation and annihilation operators in a multiple-boson system are,
Therefore, exchanging disjoint (i.e. ) operators in a product of creation or annihilation operators will reverse the sign in fermion systems, but not in boson systems.
If the states labelled by i are an orthonormal basis of a Hilbert space H, then the result of this construction coincides with the CCR algebra and CAR algebra construction in the previous section but one. If they represent "eigenvectors" corresponding to the continuous spectrum of some operator, as for unbound particles in QFT, then the interpretation is more subtle.
^A normal operator has a representation A= B + i C, where B,C are self-adjoint and commute, i.e. . By contrast, a has the representation where are self-adjoint but . Then B and C have a common set of eigenfunctions (and are simultaneously diagonalizable), whereas p and q famously don't and aren't.