Creation and annihilation operators
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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.
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.
- 1 Ladder operators for the quantum harmonic oscillator
- 2 Generalized creation and annihilation operators
- 3 Creation and annihilation operators for reaction-diffusion equations
- 4 Creation and annihilation operators in quantum field theories
- 5 See also
- 6 References
- 7 Footnotes
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.
Make a coordinate substitution to nondimensionalize the differential equation
and the Schrödinger equation for the oscillator becomes
The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q),
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,
If we define
- as the "creation operator" or the "raising operator" and
- as the "annihilation operator" or the "lowering operator"
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 , where "p" is the nondimensionalized momentum operator then we have
Note that these imply that
The operators and may be contrasted with normal operators, which commute with their adjoints. A normal operator has a representation where are self-adjoint and commute, i.e. . By contrast, has the representation where are self-adjoint but . Then and have a common set of eigenfunctions (and are simultaneously diagonalizable), whereas p and q famously don't and aren't.
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 and operators and the Hamiltonian:
These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator. 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 and as "lowering" and "raising" operators between the 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 . Using the formula above for the Hamiltonian, one obtains
so is an eigenfunction of the Hamiltonian. This gives the ground state energy . This 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 a most important role in applications, while the second one, can simply be replaced by So one simply gets
The ground state of the quantum harmonic oscillator can be found by imposing the condition that
Written out as a differential equation, the wavefunction satisfies
which has the solution
The normalization constant C is found to be from , using the Gaussian integral. Explicit formulas for all the eigenfunctions can now be found by repeated application of to . This, and further operator formalism, can be found in Glimm and Jaffe, Quantum Physics, pp. 12–20.
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 H be a one-particle Hilbert space (that is, any Hilbert space, viewed as representing the state of a single particle).
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.
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 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 particles at a site on a 1-d 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 is proportional to , let us say a probability to hop left and to hop right. All particles will stay put with a probability . (Since is so short, the probability that two or more will leave during 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. A slight modification[clarification needed] of the annihilation and creation operators is needed so that
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 in an abstract place and then applies a to it. Then ai† = ιi a†, where ιi inserts an abstract state at the ith site. Thus, for example, the net effect of ai−1†ai is to move an eigenstate from the ith to the (i − 1)th site while multiplying with the appropriate factor.
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 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 theories
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.
- Bogoliubov transformations – arises in the theory of quantum optics.
- Optical phase space
- Fock space
- Canonical commutation relations
- Feynman, Richard P. (1998) . Statistical Mechanics: A Set of Lectures (2nd ed.). Reading, Massachusetts: Addison-Wesley. ISBN 978-0-201-36076-9.