Creation and annihilation operators
The creation and annihilation operators are the core of a possible solution of the Schrödinger equation of the harmonic oscillator. These operators can also be used to easily solve problems with quantum mechanical angular momentum. Furthermore, the creation and annihilation operators find use in the quantization of fields ( the so-called second quantization or occupation number representation).
There are a variety of alternative names such as ladder operators, climbing operators, upgrades and lowering operators as well as lifting and Senkoperatoren. Instead of " creation operator " also creation operator is used sometimes. In German-speaking countries are also the operators and that change the states of an atom, called creation and annihilation operators.
The problem of the harmonic oscillator in quantum mechanics can be treated using the creation and annihilation operators method, which is also called algebraic method. It was mainly developed by Paul Dirac. For this approach we define two operators, and each drain a quantum of energy of an oscillator or add. That's why they are called annihilation and creation operators.
The circumflex ( symbol) symbolizes above, that this is an operator. This does not apply the same calculation rules as for scalars, because the sequence of operators can be, for example, generally do not swap. Below is omitted, the caret symbol in favor of clarity, unless otherwise stated. All Latin capitals, with the exception of E, are operators.
- 2.1 Details
- 3.1 details
- Convert 4.1 Hamiltonian
- 4.2 Properties of the creation and annihilation operators
- 4.3 Solution of the eigenvalue problem
- 4.4 eigenfunctions in coordinate representation
- 5.1 creation operator
- 5.2 annihilation operator
- 5.3 occupation number operator
- 5.4 Hamiltonian of the harmonic oscillator
- 7.1 reference to diagram techniques
Definition
We define the creation operator and the annihilation operator adjoint following commutation relations with the occupation number operator:
The occupation number operator is a Hermitian operator and therefore has real eigenvalues n The corresponding eigenvalue equation is, where Fock states are:
The occupation number is a non-negative integer, ie. For fermions here still will be restricted to the values 0 and 1
By applying or to the state obtained the above or the underlying condition:
The constants and are dependent on whether and satisfy the commutator or anticommutator commutation.
Details
In the following, various properties are derived from. The eigenstates are normalized.
- The occupation number operator is Hermitian, ie, self-adjoint:
- The eigenvalues are non-negative:
- The smallest eigenvalue is 0
- The eigenvalues are integers:
- Is eigenvalue, then
- Is eigenvalue, then
Bosonic climbing operators
In the bosonic case and satisfy the commutator - commutation relations:
Thus
In the bosonic case the occupation numbers can be arbitrarily large.
Details
- First is to examine whether the above conditions are met:
- With possible to construct the next opposite state. The factor is obtained from the following calculation with the commutator:
- Can be constructed with the under -lying state. The factor is derived from the following calculation:
- All eigenstates can be constructed starting from the ground state:
Climbing fermionic operators
In the fermionic case and meet the anti- commutator - commutation relations:
Thus
In the fermionic case the occupation numbers can only assume the values 0 or 1.
Details
- With and is:
- First is to examine whether the above conditions are met:
- With possible to construct the next opposite state. The factor is obtained from the following calculation with the anti - commutator:
- Can be constructed with the under -lying state. The factor is derived from the following calculation:
- All eigenstates can be constructed starting from the ground state:
Example of bosonic climbing operators: Harmonic Oscillator
The Hamiltonian of the harmonic oscillator is
Momentum operators, local operator, mass, natural frequency
The following are the stationary Schrödinger equation is solved:
Energy eigenvalue energy eigenstate
Transform the Hamiltonian
The Hamiltonian can be rewritten:
It defines two new operators:
The Hamiltonian expressed in the new operators:
Now trying to write the contents of the clip as a product, ie (the imaginary unit )
However, since u and v are operators that do not exchange here the last equal sign does not apply. At two operators to exchange with each other, the commutator is needed:
The commutator may be returned to the commutator of the original operators and:
The Hamiltonian now looks like this:
Now the two ladder operators are defined:
Often, they are also known as and - written. Note that the ladder operators are not Hermitian since.
The ladder operators expressed by the position operator and momentum operator:
Conversely, for and:
With the ladder operators to write the Hamiltonian:
Properties of the creation and annihilation operators
To determine nor the commutator of the two ladder operators:
Moreover, since true, it is at the climbing operators of the harmonic oscillator by climbing bosonic operators. To follow all the above properties for bosonic operators are climbing.
The product defines the occupation number operator:
Solution of the eigenvalue problem
The Hamiltonian can be expressed by the occupation number operator:
The eigenvalue problem can be traced back to the eigenvalue equation of the occupation number operator.
The eigenstates of are also eigenstates of since. The eigenvalues of the Hamiltonian are derived from the eigenvalues of the occupation number operator:
A particularly important property of the climbing operators is this:
If a solution of the Schrödinger equation for the energy, so is a solution to the energy and a solution for the energy. This means that you can obtain all solutions from a solution by simply applying the creation or annihilation operator on this solution. Thus, a new solution for the adjacent power level is generated, which is shifted by the energy.
Since the occupation number operator has no negative eigenvalues , no negative energy eigenvalues may exist. So there is for the minimum occupation number of a solution that sits on a minimum level of energy (zero point energy):
In the state, the energy is composed of the zero-point energy and energy quanta of size. The effect of converting the system into a state with increased energy of about. This can be interpreted as generating an additional energy quantum, which makes the name creation operator course. Analog convicted of the system operator in a reduced to an energy quantum state. A quantum of energy, so it is destroyed, so annihilation operator. The eigenvalues of the operator indicate how many energy quanta are excited in an eigenstate. The occupation of a state with energy quanta explains the name of the occupation number operator.
Eigenfunctions in coordinate representation
So if we apply to the lowest state of the lowering operator, so you get the zero vector. However, this can not be reversed: By applying the zero vector is obtained not the ground state but again the zero vector. This gives an equation for the ground state:
In the coordinate representation can be represented above operator equation as a differential equation and solve: and
By applying the upgrade operator to the solution of the ground state obtained all the higher eigenfunctions:
In coordinate representation, one obtains:
Matrix representation of bosonic climbing operators
The eigenstates of the occupation number operator form a complete orthonormal system. With the help of this Hilbert space basis, a matrix representation of ladder operators is now to be determined. Note that all the indexes (not 1 ) run here from 0 to infinity. The eigenstates can be represented as vectors:
The completeness of this base provides a depiction of the unit operator:
Creation operator
Before and after the creation operator a 1 ( the identity operator ) is inserted:
The matrix element is calculated as
The creation operator represented by the basis vectors
Thus, the matrix representation of the generating operator results in respect of the occupation eigenbasis (all elements not specified are equal to 0 ):
Annihilation operator
By similar calculation we obtain for the annihilation operator:
In this case, the matrix element has been used:
Matrix representation of the annihilation operator with respect to the occupation eigenbasis:
It can be seen that the matrix is just the transpose of. This is understandable, since the two operators adjoint to each other ( = transpose complex conjugate ) are.
Occupation number operator
Matrix element of the occupation number operator with respect to the occupation eigenbasis:
Alternatively with the ladder operators:
Matrix representation of the occupation number operator with respect to the occupation eigenbasis:
Hamiltonian of the harmonic oscillator
Matrix element of the Hamiltonian for the harmonic oscillator with respect to the occupation eigenbasis or the energy eigenbasis:
Matrix representation of the Hamiltonian for the harmonic oscillator with respect to the occupation eigenbasis or the energy eigenbasis:
Since the operators and are Hermitian, it follows that the associated matrices are symmetric with respect to the natural bases.
Eigenstates of bosonic climbing operators ( " coherent states " )
The eigenstates of the annihilation operator are the coherent states. The annihilation operator ( for clarity, the symbols for the operators introduced here explicitly again) satisfies the following eigenvalue equation:
For the creation operator, this results, with a left eigenstate (Bra - eigenstate ):
The annihilation operator can - in contrast to the creation operator - has legal eigenstates ( Ket - eigenstates ). The creation operator increases the minimum number of particles of a state in the Fock space by one; Thus, the state can not be formed so that the original. In contrast, the annihilation operator reduces the maximum particle number by one; as a state in the Fock space but may include components of all particle numbers (even arbitrarily high particle numbers ), is thus not prohibited, that has eigenstates. These are the coherent states:
The " coherent state" arises as a certain linear combination of all states of fixed particle and indeed according to the formula:
This state is thus an eigenstate of the annihilation operator, namely the eigenvalue while the associated creation operator has only left- eigenstates. It is a non-zero complex number which completely defines the coherent state, and may also depend on the time explicitly. is the expectation value of the occupation number of the coherent state.
Coherent states have (such as the ground state of the harmonic oscillator ) minimum blur and remain coherent with development time. They can be used which - in general explicitly time-dependent - electromagnetic wave of a laser mode best describe (so-called Glauber states).
Creation and annihilation operators in quantum field theories
In quantum field theory and many-body physics using expressions of the form where the complex numbers, while the creation and annihilation operators represent: These increase or decrease the eigenvalues of the number operator by 1, analogous to the harmonic oscillator. The indices take into account the degrees of freedom of space-time and have ia a plurality of components. If the creation and annihilation operators depend on a kontimuierlichen variable, instead of discrete quantum numbers, to write them as field operators. The number operators are self-adjoint ( " hermitian " ) and assume all non- negative integer values: The non-trivial commutation relations are, after all, as the harmonic oscillator: [. ., ] Where is the so-called Kommutatorklammer, while the Kronecker symbol.
The above said is true for bosons, whereas for fermions we have to replace the commutator by the anticommutator, as a consequence applies in the fermionic case that the number of operators have only eigenvalues 0 and 1.
Reference to diagram techniques
Specific calculations using the creation and annihilation operators can usually by diagram techniques support (→ Feynman diagrams ). So you can, for example, illustrate three-particle interactions of the form by three lines, of which the first two enter into a vertex and there are " destroyed ", while a third line is "created" at this vertex and runs from him. This energy and momentum are explicitly taken into account in the associated rules.
Specified, the term, which describes a so-called " Konfluenzprozess " has, at low temperatures, A., less likely, than that of inverse -called " splitting process ", which belongs to the adjoint term. As corresponding to each generation operator, similar to the harmonic oscillator, the transition rate, whereas in the corresponding term of the annihilation operator missing. In this way the latter terms in the rule are more important than the former at low temperatures.