Creation operators and annihilation operators are
mathematical operators that have widespread applications in
quantum mechanics, notably in the study of
s and many-particle systems.
An annihilation operator (usually denoted
) lowers the number of particles in a given state by one. A creation operator (usually denoted
) 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
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
, the use of these operators instead of
wavefunctions is known as
second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
. They were introduced by
Paul Dirac.
Creation and annihilation operators can act on states of various types of particles. For example, in
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
and
many-body theory the creation and annihilation operators often act on
electron states. They can also refer specifically to the
ladder operators
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
for the
. 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. Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the
cluster decomposition theorem.
The mathematics for the creation and annihilation operators for
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
is the same as for the
ladder operators
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
of the
.
For example, the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
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
, one reinterprets the ladder operators as creation and annihilation operators, adding or subtracting fixed
quanta
Quanta is the plural of quantum.
Quanta may also refer to:
Organisations
* Quanta Computer, a Taiwan-based manufacturer of electronic and computer equipment
* Quanta Display Inc., a Taiwanese TFT-LCD panel manufacturer acquired by AU Optronic ...
of energy to the oscillator system.
Creation/annihilation operators are different for
bosons (integer spin) and
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s (half-integer spin). This is because their
wavefunctions have different
symmetry properties.
First consider the simpler bosonic case of the photons of the quantum harmonic oscillator.
Start with the
Schrödinger equation for the one-dimensional time independent
,
:
Make a coordinate substitution to
nondimensionalize the differential equation
:
The Schrödinger equation for the oscillator becomes
:
Note that the quantity
is the same energy as that found for light
quanta
Quanta is the plural of quantum.
Quanta may also refer to:
Organisations
* Quanta Computer, a Taiwan-based manufacturer of electronic and computer equipment
* Quanta Display Inc., a Taiwanese TFT-LCD panel manufacturer acquired by AU Optronic ...
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
:
which implies,
:
coinciding with the usual canonical commutation relation
, in position space representation:
.
Therefore,
:
and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,
:
If one defines
:
as the "creation operator" or the "raising operator" and
:
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.
Letting
, where
is the nondimensionalized
momentum operator
In quantum mechanics, the momentum operator is the operator (physics), operator associated with the momentum (physics), linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case o ...
one has
:
and
:
:
Note that these imply
:
The operators
and
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
and
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
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
. Applying the Hamiltonian to the ground state,
:
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,
can simply be replaced by
.
Consequently,
:
The
time-evolution operator is then
:
:
Explicit eigenfunctions
The ground state
of the
can be found by imposing the condition that
:
Written out as a differential equation, the wavefunction satisfies
:
with the solution
:
The normalization constant is found to be