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Creation and annihilation operators

creation and annihilation operators, creation and annihilation operators harmonic potential
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]

Contents

  • 1 Ladder operators for the quantum harmonic oscillator
    • 11 Explicit eigenfunctions
    • 12 Matrix representation
  • 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

See also: Quantum harmonic oscillator § Ladder operator method

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,

− ℏ 2 2 m d 2 d x 2 + 1 2 m ω 2 x 2 ψ x = E ψ x }}}}}+}m\omega ^x^\right\psi x=E\psi x}

Make a coordinate substitution to nondimensionalize the differential equation

x   =   ℏ m ω q }}q}

The Schrödinger equation for the oscillator becomes

ℏ ω 2 − d 2 d q 2 + q 2 ψ q = E ψ q }\left-}}}+q^\right\psi q=E\psi q}

Note that the quantity ℏ ω = h ν is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as

− d 2 d q 2 + q 2 = − d d q + q d d q + q + d d q q − q d d q }}}+q^=\left-}+q\right\left}+q\right+}q-q}}

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

d d q q − q d d q f q = d d q q f q − q d f q d q = f q }q-q}\rightfq=}qfq-q}=fq}

which implies,

d d q q − q d d q = 1 , }q-q}=1,}

coinciding with the usual canonical commutation relation − i [ q , p ] = 1 , in position space representation: p := − i d d q }}

Therefore,

− d 2 d q 2 + q 2 = − d d q + q d d q + q + 1 }}}+q^=\left-}+q\right\left}+q\right+1}

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

ℏ ω [ 1 2 − d d q + q 1 2 d d q + q + 1 2 ] ψ q = E ψ q }}\left-}+q\right}}\left}+q\right+}\right]\psi q=E\psi q}

If one defines

a †   =   1 2 − d d q + q \ =\ }}\left-}+q\right}

as the "creation operator" or the "raising operator" and

a     =   1 2       d d q + q }}\left\ \ \ \!}+q\right}

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

ℏ ω a † a + 1 2 ψ q = E ψ q a+}\right\psi q=E\psi q}

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 p = − i d d q }} , where p is the nondimensionalized momentum operator one has

[ q , p ] = i

and

a = 1 2 q + i p = 1 2 q + d d q }}q+ip=}}\leftq+}\right} a † = 1 2 q − i p = 1 2 q − d d q =}}q-ip=}}\leftq-}\right}

Note that these imply

[ a , a † ] = 1 2 [ q + i p , q − i p ] = 1 2 [ q , − i p ] + [ i p , q ] = − i 2 [ q , p ] + [ q , p ] = 1 ]=}=}+=}+=1}

The operators a and a† 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

H ^ = ℏ ω a a † − 1 2 = ℏ ω a † a + 1 2 ∗ }=\hbar \omega \lefta\,a^-}\right=\hbar \omega \lefta^\,a+}\right\qquad \qquad }

One may compute the commutation relations between the a and a† operators and the Hamiltonian:[5]

[ H ^ , a ] = [ ℏ ω a a † − 1 2 , a ] = ℏ ω [ a a † , a ] = ℏ ω a [ a † , a ] + [ a , a ] a † = − ℏ ω a },a]==\hbar \omega =\hbar \omega a+a^=-\hbar \omega a} [ H ^ , a † ] = ℏ ω a † },a^]=\hbar \omega \,a^}

These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows

Assuming that ψ n } is an eigenstate of the Hamiltonian H ^ ψ n = E n ψ n }\psi _=E_\,\psi _} Using these commutation relations, it follows that[5]

H ^ a ψ n = E n − ℏ ω a ψ n }\,a\psi _=E_-\hbar \omega \,a\psi _} H ^ a † ψ n = E n + ℏ ω a † ψ n }\,a^\psi _=E_+\hbar \omega \,a^\psi _}

This shows that a ψ n } and a † ψ n \psi _} are also eigenstates of the Hamiltonian, with eigenvalues E n − ℏ ω -\hbar \omega } and E n + ℏ ω +\hbar \omega } respectively This identifies the operators a and a† as "lowering" and "raising" operators between adjacent eigenstates The energy difference between adjacent eigenstates is ΔΕ=ħω

The ground state can be found by assuming that the lowering operator possesses a nontrivial kernel: a ψ 0 = 0 =0} with ψ 0 ≠ 0 \neq 0} Application of the above formula for the Hamiltonian yields

0 = ℏ ω a † a ψ 0 = H ^ − ℏ ω 2 ψ 0 a\psi _=\left}-}\right\,\psi _}

So ψ 0 } is an eigenfunction of the Hamiltonian

This gives the ground state energy E 0 = ℏ ω / 2 =\hbar \omega /2} , which allows one to identify the energy eigenvalue of any eigenstate ψ n } as[5]

E n = n + 1 2 ℏ ω =\leftn+}\right\hbar \omega }

Furthermore, it turns out that the first-mentioned operator in , the number operator N = a † a , a\,,} plays the most important role in applications, while the second one, aa† can simply be replaced by N +1

Consequently,

ℏ ω N + 1 2 ψ q = E ψ q   }\right\,\psi q=E\,\psi q~}

The time-evolution operator is then

U t = exp ⁡ − i t H ^ / ℏ = exp ⁡ − i t ω a † a + 1 / 2   , }/\hbar =\exp-it\omega a^a+1/2~,} = e − i t ω / 2   ∑ k = 0 ∞ e − i ω t − 1 k k ! a † k a k   ~\sum _^-1^ \over k!}a^}a^~}

Explicit eigenfunctions

The ground state   ψ 0 q q} of the quantum harmonic oscillator can be found by imposing the condition that

a   ψ 0 q = 0 q=0}

Written out as a differential equation, the wavefunction satisfies

q ψ 0 + d ψ 0 d q = 0 +}}=0}

with the solution

ψ 0 q = C exp ⁡ − q 2 2 q=C\exp \left- \over 2}\right}

The normalization constant C is found to be   1 / π 4 ]}}  from ∫ − ∞ ∞ ψ 0 ∗ ψ 0 d q = 1 ^\psi _^\psi _\,dq=1} ,  using the Gaussian integral Explicit formulas for all the eigenfunctions can now be found by repeated application of a † } to ψ 0 } [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

a † = 0 0 0 … 0 … 1 0 0 … 0 … 0 2 0 … 0 … 0 0 3 … 0 … ⋮ ⋮ ⋮ ⋱ ⋮ … 0 0 0 … n … ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ =0&0&0&\dots &0&\dots \\}&0&0&\dots &0&\dots \\0&}&0&\dots &0&\dots \\0&0&}&\dots &0&\dots \\\vdots &\vdots &\vdots &\ddots &\vdots &\dots \\0&0&0&\dots &}&\dots &\\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end}} a = 0 1 0 0 … 0 … 0 0 2 0 … 0 … 0 0 0 3 … 0 … 0 0 0 0 ⋱ ⋮ … ⋮ ⋮ ⋮ ⋮ ⋱ n … 0 0 0 0 … 0 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ 0&}&0&0&\dots &0&\dots \\0&0&}&0&\dots &0&\dots \\0&0&0&}&\dots &0&\dots \\0&0&0&0&\ddots &\vdots &\dots \\\vdots &\vdots &\vdots &\vdots &\ddots &}&\dots \\0&0&0&0&\dots &0&\ddots \\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end}}

These can be obtained via the relationships a i j † = ⟨ ψ i ∣ a † ∣ ψ j ⟩ ^=\langle \psi _\mid a^\mid \psi _\rangle } and a i j = ⟨ ψ i ∣ a ∣ ψ j ⟩ =\langle \psi _\mid a\mid \psi _\rangle } The eigenvectors ψ i } are those of the quantum harmonic oscillator, and are sometimes called the "number basis"

Generalized creation and annihilation operators

Main article: CCR and CAR algebras

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

The bosonic CCR algebra over H is the algebra-with-conjugation-operator called abstractly generated by elements af, where f ranges freely over H, subject to the relations

[ a f , a g ] = [ a † f , a † g ] = 0 f,a^g]=0} [ a f , a † g ] = ⟨ f ∣ g ⟩ , g]=\langle f\mid g\rangle ,}

in bra–ket notation

The map a : f ↦ af 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 af will be realized as an annihilation operator, and a†f 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 H is closely related to, but not identical to, a Weyl algebra

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

= = 0 =\f,a^g\}=0} = ⟨ f ∣ g ⟩ g\}=\langle f\mid g\rangle }

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, af removes ie annihilates a particle in the state | f  ⟩ whereas a†f creates a particle in the state | f  ⟩

The free field vacuum state is the state | 0  ⟩ with no particles, characterized by

a f | 0 ⟩ = 0

If | f  ⟩ is normalized so that ⟨  f | f  ⟩ = 1, then N = a†f af 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 n i } particles at a site 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 dt is proportional to n i d t \,dt} , let us say a probability α n i d t dt} to hop left and α n i d t \,dt} to hop right All n i } particles will stay put with a probability 1 − 2 α n i d t \,dt} Since dt is so short, the probability that two or more will leave during 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 | , 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 Recall

a ∣ n ⟩ = n   | n − 1 ⟩ }\ |n-1\rangle }

a † } and

a † ∣ n ⟩ = n + 1 ∣ n + 1 ⟩ , \mid \!n\rangle =}\mid n+1\rangle ,}

for all n  ≥ 0, while

[ a , a † ] = 1 ]=1}

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:

a ∣ n ⟩ = n   | n − 1 ⟩

a † ∣ n ⟩ =   ∣ n + 1 ⟩ \mid \!n\rangle =\ \mid n+1\rangle }

note that even though the behavior of the operators on the kets has been modified, these operators still obey the commutation relation

[ a , a † ] = 1 ]=1}

Now define ai so that it applies a to | ni  ⟩ Correspondingly, define ai†  as applying a † } to | ni  ⟩ Thus, for example, the net effect of ai−1†ai is to move a particle from the ith to the i − 1th site while multiplying with the appropriate factor

This allows writing the pure diffusive behavior of the particles as

∂ t ∣ ψ ⟩ = − α ∑ 2 a i † a i − a i − 1 † a i − a i + 1 † a i ∣ ψ ⟩ = − α ∑ a i † − a i − 1 † a i − a i − 1 ∣ ψ ⟩ , \mid \!\psi \rangle =-\alpha \sum 2a_^a_-a_^a_-a_^a_\mid \!\psi \rangle =-\alpha \sum a_^-a_^a_-a_\mid \!\psi \rangle ,}

where the sum is over i

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 , yielding a term

λ ∑ a i a i − a i † a i † a i a i a_-a_^a_^a_a_}

where number state n is replaced by number state n − 2 at site i at a certain rate

Thus the state evolves by

∂ t ∣ ψ ⟩ = − α ∑ a i † − a i − 1 † a i − a i − 1 ∣ ψ ⟩ + λ ∑ a i 2 − a i † 2 a i 2 ∣ ψ ⟩ \mid \!\psi \rangle =-\alpha \sum a_^-a_^a_-a_\mid \!\psi \rangle +\lambda \sum a_^-a_^a_^\mid \!\psi \rangle }

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

Main articles: Second quantization and Quantum field theory § Second quantization

In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, a i † ^} and a i ^} These operators change the eigenvalues of the number operator,

N = ∑ i n i = ∑ i a i † a i n_=\sum _a_^a_^} ,

by one, in analogy to the harmonic oscillator The indices such as i 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 n , l , m , s is used to label states in the hydrogen atom

The commutation relations of creation and annihilation operators in a multiple-boson system are,

[ a i , a j † ] ≡ a i a j † − a j † a i = δ i j , ^,a_^]\equiv a_^a_^-a_^a_^=\delta _,} [ a i † , a j † ] = [ a i , a j ] = 0 , ^,a_^]==0,}

where [     ,     ] is the commutator and δ i j } is the Kronecker delta

For fermions, the commutator is replaced by the anticommutator } ,

≡ a i a j † + a j † a i = δ i j , ^,a_^\}\equiv a_^a_^+a_^a_^=\delta _,} = = 0 ^,a_^\}=\^,a_^\}=0}

Therefore, exchanging disjoint ie i ≠ j 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

See also

  • Segal–Bargmann space
  • Bogoliubov transformations – arises in the theory of quantum optics
  • Optical phase space
  • Fock space
  • Canonical commutation relations

References

  • Feynman, Richard P 1998 Statistical Mechanics: A Set of Lectures 2nd ed Reading, Massachusetts: Addison-Wesley ISBN 978-0-201-36076-9 
  • Albert Messiah, 1966 Quantum Mechanics Vol I, English translation from French by G M Temmer North Holland, John Wiley & Sons Ch XII online

Footnotes

  1. ^ Feynman 1998, p 151
  2. ^ Feynman 1998, p 167
  3. ^ Feynman 1998, pp 174–5
  4. ^ A normal operator has a representation A= B + i C, where B,C are self-adjoint and commute, ie B C = C B By contrast, a has the representation a = q + i p where p , q are self-adjoint but [ p , q ] = 1 Then B and C have a common set of eigenfunctions and are simultaneously diagonalizable, whereas p and q famously don't and aren't
  5. ^ a b c Branson, Jim "Quantum Physics at UCSD" Retrieved 16 May 2012 
  6. ^ This, and further operator formalism, can be found in Glimm and Jaffe, Quantum Physics, pp 12–20

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