“Creation and annihilation operators”的意思、由来-开放百科全书

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.^[1] 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.

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.^[2] 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.^[3]

Ladder operators for the quantum harmonic oscillator

In the context of the quantum harmonic oscillator, one reinterprets 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,

Note that the quantity

is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as

The last two terms can be simplified by considering their effect on an arbitrary differentiable function

coinciding with the usual canonical commutation relation

, in position space representation:

and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,

as the "annihilation operator" or the "lowering operator", the Schrödinger equation for the oscillator reduces to

This is significantly simpler than the original form. Further simplifications of this equation enable one to derive all the properties listed above thus far.

The operators

and

may be contrasted to normal operators, which commute with their adjoints.^[4]

Using the commutation relations given above, the Hamiltonian operator can be expressed as

One may compute the commutation relations between the

and

operators and the Hamiltonian:^[5]

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^[5]

This shows that

and

are also eigenstates of the Hamiltonian, with eigenvalues

and

respectively. This identifies the operators

and

as "lowering" and "raising" operators between adjacent eigenstates. The energy difference between adjacent eigenstates is

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

This gives the ground state energy

, which allows one to identify the energy eigenvalue of any eigenstate

as^[5]

Furthermore, it turns out that the first-mentioned operator in (*), the number operator

plays the most important role in applications, while the second one,

can simply be replaced by

Explicit eigenfunctions

The ground state

of the quantum harmonic oscillator can be found by imposing the condition that

The normalization constant {{mvar|C}} is found to be

from

, using the Gaussian integral. Explicit formulas for all the eigenfunctions can now be found by repeated application of

.^[6]

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

and

. The eigenvectors

are those of the quantum harmonic oscillator, and are sometimes called the "number basis".

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

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

is the algebra-with-conjugation-operator (called *) abstractly generated by elements

, where

ranges freely over

, subject to the relations

The map

from

to the bosonic CCR algebra is required to be complex antilinear (this adds more relations). Its adjoint is

, and the map

is complex linear in {{mvar|H}}. Thus

embeds as a complex vector subspace of its own CCR algebra. In a representation of this algebra, the element

will be realized as an annihilation operator, and

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

is closely related to, but not identical to, a Weyl algebra.

For fermions, the (fermionic) CAR algebra over is constructed similarly, but using anticommutator relations instead, namely

The CAR algebra is finite dimensional only if

is finite dimensional. If we take a Banach space completion (only necessary in the infinite dimensional case), it becomes a

algebra. The CAR algebra is closely related to, but not identical to, a Clifford algebra.

Physically speaking,

removes (i.e. annihilates) a particle in the state

whereas

creates a particle in the state

The free field vacuum state is the state {{Dket|0}} with no particles, characterized by

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 diffuse and interact on contact, forming an inert product: . To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider

particles at a site {{mvar|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 {{math|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 {{math|dt}} is so short, the probability that two or more will leave during {{math|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

. It represents the juxtaposition (or conjunction, or tensor product) of the number states

located at the individual sites of the lattice. Recall

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 so that it applies to

. Correspondingly, define

as applying

. Thus, for example, the net effect of

is to move a particle from the to the site while multiplying with the appropriate factor.

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

at a certain rate.

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,

and

. These operators change the eigenvalues of the number operator,

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.

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

See also

References

Footnotes