Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.
Contents
- Ladder operators for the quantum harmonic oscillator
- Explicit eigenfunctions
- Matrix representation
- Generalized creation and annihilation operators
- Creation and annihilation operators for reaction diffusion equations
- Creation and annihilation operators in quantum field theories
- References
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 mathematics for the creation and annihilation operators for bosons is the same as for the ladder operators of the quantum harmonic oscillator. For example, the commutator of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish. However, for fermions the mathematics is different, involving anticommutators instead of commutators.
Ladder operators for the quantum harmonic oscillator
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.
First consider the simpler bosonic case of the phonons of the quantum harmonic oscillator.
Start with the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator
Make a coordinate substitution to nondimensionalize the differential equation
and the Schrödinger equation for the oscillator becomes
Note that the quantity
The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q),
which implies,
coinciding with the usual canonical commutation relation
Therefore
and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,
If we define
then the Schrödinger equation for the oscillator becomes
This is significantly simpler than the original form. Further simplifications of this equation enables one to derive all the properties listed above thus far.
Letting
and
Note that these imply that
The operators
Despite this, we go on. Using the commutation relations given above, the Hamiltonian operator can be expressed as
One can compute the commutation relations between the
These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator. Assuming that
This shows that
The ground state can be found by assuming that the lowering operator possesses a nontrivial kernel,
so
Furthermore, it turns out that the first-mentioned operator in (*), the number operator
Explicit eigenfunctions
The ground state
Written out as a differential equation, the wavefunction satisfies
which has the solution
The normalization constant C is found to be
Matrix representation
The matrix expression of the creation and annihilation operators of the quantum harmonic oscillator with respect to the above orthonormal basis is
These can be obtained via the relationships
Generalized creation and annihilation operators
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
where we are using bra–ket notation. 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
The free field vacuum state is the state | 0
If | f
Creation and annihilation operators for reaction-diffusion equations
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
The probability that one particle leaves the site during the short time period
We can now describe the occupation of particles on the lattice as a `ket' of the form | ..., n−1, n0, n1, ...
and
for all n ≥ 0. This modification preserves the commutation relation
Now let ai = aπi, where πi selects the ith component of ψ. That is, ai makes a copy of the state | ni
This allows us to write the pure diffusive behaviour of the particles as
where the sum is over i.
The reaction term can be deduced by noting that
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 theories
In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states,
by one, in analogy to the harmonic oscillator. The indices (such as
The commutation relations of creation and annihilation operators in a multiple-boson system are,
where
For fermions, the commutator is replaced by the anticommutator
Therefore, exchanging disjoint (i.e.
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.