NationMaster - Encyclopedia: Creation and annihilation operators

Quantum optics operators
Ladder operators
Creation and annihilation operators
Displacement operator
Rotation operator (quantum optics)
Squeeze operator
Anti-symmetric operator
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In physics, an annihilation operator is an operator that lowers the number of particles in a given state by one. A creation operator is an operator that increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. Depending on the context, the identity of the particles in question varies; for example, in quantum chemistry and many-body theory the creation and annihilation operators often act on electrons. Annihilation and creation operators 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. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization. Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ... Simmelar to the creation and annihilation operators used for a harmonic occilator, Ladder operators are used in quantum mechanics for angular momentum to go from one energy eigenstate to another. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... The squeeze operator for a single mode is where the operators inside the exponential are the ladder operators. ... In quantum mechanics, a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In mathematics, the term adjoint applies in several situations. ... Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. ... This article is about the many-body problem in quantum mechanics. ... Properties The electron (also called negatron, commonly represented as e−) is a subatomic particle. ... Simmelar to the creation and annihilation operators used for a harmonic occilator, Ladder operators are used in quantum mechanics for angular momentum to go from one energy eigenstate to another. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... A phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Chemistry (disambiguation). ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... Second quantization refers to quantizing fields by expressing them as operator-valued distributions The most elementary, or semiclassical treatments of quantum mechanics fix the number of particles and treat the field classically, including it as a parameter in the Hamiltonian or Lagrangian or whatever. ...

The mathematics behind the creation and the annihilation operators is identical as the formulae for ladder operators that appear in the quantum harmonic oscillator. For example, the commutator of the annihilation and the creation operator associated with the same state equals one; all other commutators vanish. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

While the concept of creation and annihilation operators is well defined for free field theories, in interacting QFTs, they can only be defined in the interaction picture, which does not exist according to Haag's theorem. Classically, a free field is a field described by linear partial differential equations which has a unique solution given initial data. ... In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the SchrÃ¶dinger picture and the Heisenberg picture. ... Rudolf Haag showed in 1955 that the interaction picture cannot be rigorously defined in quantum field theory, a result now commonly cited as Haags Theorem. ...

1 Derivation of bosonic creation and annihilation operators
- 1.1 Applications
2 Matrix representation
3 Mathematical details
4 Creation and annihilation operators for reaction diffusion equations
5 Notational caveats and considerations
- 5.1 The vacuum state
6 See also

Derivation of bosonic creation and annihilation operators

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. In physics quanta is the plural of quantum. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ...

Suppose the wavefunctions are dependent on N properties. Then

For bosons: ψ(1,2,3,4,...N) = ψ(2,1,3,4,...N)

For fermions: ψ(1,2,3,4,...N) = -ψ(2,1,3,4,...N)

For now let's just consider the case of bosons because fermions are more complicated.

Start with the Schrödinger equation for the one dimensional time independent quantum harmonic oscillator For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

$left(-frac{hbar^2}{2m} frac{d^2}{d x^2} + frac{1}{2}m omega^2 x^2right) psi(x) = E psi(x)$

Make a coordinate substitution to nondimensionalize the differential equation Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ...

$x stackrel{mathrm{def}}{=} sqrt{ frac{hbar}{m omega}} q$ .

and the Schrödinger equation for the oscillator becomes

$frac{hbar omega}{2} left(-frac{d^2}{d q^2} + q^2 right) psi(q) = E psi(q)$ .

Notice that the quantity ħω = hν is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as In physics quanta is the plural of quantum. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...

$-frac{d^2}{dq^2} + q^2 = left[-frac{d}{dq}+q right] left[frac{d}{dq}+ q right] + frac {d}{dq}q - q frac {d}{dq}$

The last two terms in that equation form the commutator of q with its derivative. So let's calculate that commutator [ q, ∂/∂q ] In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

$left(q frac{d}{dq}- frac{d}{dq} q right)f(q) = q frac{df}{dq} - frac{d}{dq}(q f(q)) = -f(q)$

In other words [ q, d/dq ] = - 1 or [ d/dq, q ] = 1.

Therefore

$frac{1}{2} hbar omega left( -frac{d^2}{dq^2} + q^2 right) = hbar omega left[ frac{-d/ dq + q}{sqrt{2}}right] left[frac{d / dq + q}{sqrt{2}}right] + frac{1}{2} hbar omega$

If we define

$a^dagger stackrel{mathrm{def}}{=} frac{1}{sqrt{2}} left(-frac{d}{dq} + qright)$ as the "creation operator" or the "raising operator" and

$a stackrel{mathrm{def}}{=} frac{1}{sqrt{2}} left(+frac{d}{dq} + qright)$ as the "annihilation operator" or the "lowering operator"

the Hamiltonian becomes

$H = hbar omega left( a^dagger a + frac{1}{2} right)$ .

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

Letting p = - i d/dq, where p is the nondimensionalized momentum operator This article is about momentum in physics. ...

$a^dagger = frac{1}{sqrt{2}}(q - i p)$

$a = frac{1}{sqrt{2}}(q + i p)$ .

Applications

The ground state $ψ 0 (q)$ of the quantum harmonic oscillator can be found by imposing the condition that The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

$hat{a} psi_0(q) = 0.$

Written out as a differential equation, the wavefunction satisfies

$q psi_0 + frac{dpsi_0}{dq} = 0$

which has the solution

$psi_0(q) = C exp(-{q^2 over 2}).$

The normalization constant C can be found to be : $1over sqrt pi$ by noting that the Gaussian integral of $psi_0^* psi_0$ over all q must equal 1 for the wavefunction. The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...

Matrix representation

The matrix counterparts of the creation and annihilation operators obtained from the quantum harmonic oscillator model are

$a=begin{pmatrix} 0 & sqrt{1} & 0 & 0 & dots & 0 & dots 0 & 0 & sqrt{2} & 0 & dots & 0 & dots 0 & 0 & 0 & sqrt{3} & dots & 0 & dots 0 & 0 & 0 & 0 & ddots & vdots & dots vdots & vdots & vdots & vdots & ddots & sqrt{n} & dots 0 & 0 & 0 & 0 & dots & 0 & ddots vdots & vdots & vdots & vdots & vdots & vdots & ddots end{pmatrix}$

$a^{dagger}=left(begin{array}{cccccc} 0 & 0 & 0 & dots & dots sqrt{1} & 0 & 0 & dots & dots 0 & sqrt{2} & 0 & dots & dots 0 & 0 & sqrt{3} & dots & dots vdots & vdots & vdots 0 & 0 & 0 & sqrt{n+1} & 0dots vdots & vdots & vdots & vdots & vdotsend{array}right)$

Substituting backwards, the laddering operators are recovered. They can be obtained via the relationships $a^dagger_{ij} = langlepsi_i | hat{a}^dagger | psi_jrangle$ and $a_{ij} = langlepsi_i | hat{a} | psi_jrangle$ . The wavefunctions are those of the quantum harmonic oscillator, and are sometimes called the "number basis".

Mathematical details

The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators. The more abstract form of the operators satisfy the properties below.

Let H be the one-particle Hilbert space. To get the bosonic CCR algebra, look at the algebra generated by a(f) for any f in H. The operator a(f) is called an annihilation operator and the map a(.) is antilinear. Its adjoint is a^†(f) which is linear in H. The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... In quantum field theory, if V is a real vector space equipped with a nonsingular real antisymmetric bilinear form (,) (i. ... In mathematics, a real linear transformation f from a complex vector space V to another is said to be antilinear (or conjugate-linear or semilinear) if for all c, d in C and all x, y in V. See also: complex conjugate, sesquilinear form ... In mathematics, the term adjoint applies in several situations. ... For other uses, see Linear (disambiguation). ...

For a boson,

$[a(f),a(g)]=[a^dagger(f),a^dagger(g)]=0$

$[a(f),a^dagger(g)]=langle f|g rangle$ ,

where we are using bra-ket notation. Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

For a fermion, the anticommutators are For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if...

${a(f),a(g)}={a^dagger(f),a^dagger(g)}=0$

${a(f),a^dagger(g)}=langle f|g rangle$ .

A CAR algebra. In quantum field theory, if V is a real vector space equipped with a nonsingular real antisymmetric bilinear form (,) (i. ...

Physically speaking, a(f) removes (i.e. annihilates) a particle in the state |f> wheareas a^†(f) creates a particle in the state |f>.

The free field vacuum state is the state with no particles. In other words, Classically, a free field is a field described by linear partial differential equations which has a unique solution given initial data. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ...

$a(f)|0rangle=0$

where |0> is the vacuum state.

If |f> is normalized so that <f|f>=1, then a^†(f) a(f) gives the number of particles in the state |f>.

Note that the creation and annihilation operators are "generalized complex conjugates" of each other. Usually, the notation is chosen in such a way that the a^†(f) is the creation operator, and a(f) is the annihilation operator. The ^† reminds us that something "extra" is being added to the system. The topic can be misleadingly confusing if this is not done.

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 sitation when a gas of molecules $A$ diffuse and interact on contact, forming an inert product: $A+Arightarrowvarnothing$ . To see how this kind of reaction can be described by the annihilation and creation operator formalism consider $n i$ particles at a site $i$ on a 1-d lattice. Each particle diffuses independently, so that the probability that one of them leaves the site for short times $d t$ is proportional to $n i d t$ , say $α n i d t$ to hop left and $α n i d t$ to hop left.All $n$ particles will stay put with a probability $1 - 2α n i d t$ .

We can now describe the occupation of particles on the lattice as a `ket' of the form $|n_{1},n_{2}...rangle$ . A slight modification of the annihilation and creation operators is needed so that $a|nrangle=n|n-1rangle$ and $a^{+}|nrangle=|n+1rangle$ . This modification preserves the commutation relation $[a, a +] = 1$ , but allows us to write the pure diffusive behaviour of the particles as

$partial_{t}|psirangle=-alphasum(2a_{i}^{+}a_{i}-a_{i-1}^{+}a_{i}-a_{i+1}^{+}a_{i})|psirangle=-alphasum(a_{i}^{+}-a_{i-1}^{+})(a_{i}-a_{i-1})|psirangle$

The reaction term can be deduced by noting that $n$ particles can interact in $n (n - 1)$ different ways, so that the probability that a pair annihilates is $λ n (n - 1) d t$ and the probability that no pair annihilates is $1 - λ n (n - 1) d t$ leaving us with a term

$lambdasum(a_{i}a_{i}-a_{i}^{+}a_{i}^{+}a_{i}a_{i})$

yielding

$partial_{t}|psirangle=-alphasum(a_{i}^{+}-a_{i-1}^{+})(a_{i}-a_{i-1})|psirangle+lambdasum(a_{i}^{2}-a_{i}^{+2}a_{i}^{2})|psirangle$

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.

Notational caveats and considerations

In quantum mechanics, Dirac bra-ket notation is often used. However, there is some ambiguity in this notation, particularly when there is the need to differentiate between these things:

The lowest energy state
The zero state
The vacuum state
The zero ket

Often, these are all interchangeably notated as |0>, or even | >. As a result, it is necessary to read carefully, and consider the context in which the notation is used.

For example, in the quantum harmonic oscillator, the ground state has the property that when the annihilation operator b is applied to it, it satisfies b|0> = 0| > = 0

The intermediate step is rarely indicated as it is considered necessary only when more conceptual/mathematical rigour is needed.

In this example, the lowest energy state is denoted as |0>. It is labeled as the "zero state", but it is important to emphasize that any state can be labeled as the "zero" state. The zero state is often used as a reference state to other quantum states. Therefore, the |0> state need not be the state with the absolutely lowest energy. In the case of the harmonic oscillator, it is due to the particulars of the mathematics that the ground state is chosen to be |0>. The vacuum state is the state where no quanta is available to be extracted. This special null state is denoted by | >. This vacuum state is also known as the "zero ket" because there are zero particles in the state. Unfortunately, the lowest energy state |0> is also known as the "zero ket" for the different reason that the state is labeled as "zero". Care must be taken that the four concepts listed above are not mixed together.

Sometimes, the terms "null state" and "empty state" are used interchangeably for |> and |0>. The meaning for this usage is again dependent on the context.

The vacuum state

The vacuum state is a conceptual state which has no particles. The state is usually denoted as |0>, not the "empty ket" | >. Interestingly enough, no actual function actually represents the |0> state, but for notational purposes, we define the vacuum state as being normalized such that <0|0> = 1 and that |0> is orthogonal to all other states of the form |N>, where N is any indexing of quantum states for a particular system.

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