Creation and annihilation operators

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 photons of the quantum harmonic oscillator. Start with the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator, \left(-\frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2\right) \psi(x) = E \psi(x). Make a coordinate substitution to nondimensionalize the differential equation x \ = \ \sqrt{ \frac{\hbar}{m \omega}} q. 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). Note that the quantity \hbar \omega = h \nu is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as -\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 can be simplified by considering their effect on an arbitrary differentiable function f(q), \left(\frac{d}{dq} q- q \frac{d}{dq} \right)f(q) = \frac{d}{dq}(q f(q)) - q \frac{df(q)}{dq} = f(q) which implies, \frac{d}{dq} q- q \frac{d}{dq} = 1 , coinciding with the usual canonical commutation relation -i[q,p]=1 , in position space representation: p:=-i\frac{d}{dq}. Therefore, -\frac{d^2}{dq^2} + q^2 = \left(-\frac{d}{dq}+q \right) \left(\frac{d}{dq}+ q \right) + 1 and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2, \hbar \omega \left[\frac{1}{\sqrt{2}} \left(-\frac{d}{dq}+q \right)\frac{1}{\sqrt{2}} \left(\frac{d}{dq}+ q \right) + \frac{1}{2} \right] \psi(q) = E \psi(q). If one defines a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right) as the "creation operator" or the "raising operator" and a \ \ = \ \frac{1}{\sqrt{2}} \left(\frac{d}{dq} + q\right) as the "annihilation operator" or the "lowering operator", the Schrödinger equation for the oscillator reduces to \hbar \omega \left( a^\dagger a + \frac{1}{2} \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 \frac{d}{dq}, where p is the nondimensionalized momentum operator, one has [q, p] = i \, and \begin{align} a &= \frac{1}{\sqrt{2}}(q + i p) = \frac{1}{\sqrt{2}}\left( q + \frac{d}{dq}\right) \\[1ex] a^\dagger &= \frac{1}{\sqrt{2}}(q - i p) = \frac{1}{\sqrt{2}}\left( q - \frac{d}{dq}\right). \end{align} Note that these imply [a, a^\dagger ] = \frac{1}{2} [ q + ip , q-i p] = \frac{1}{2} ([q,-ip] + [ip, q]) = -\frac{i}{2} ([q, p] + [q, p]) = 1. The operators a\, and a^\dagger\, may be contrasted to normal operators, which commute with their adjoints. Using the commutation relations given above, the Hamiltonian operator can be expressed as \hat H = \hbar \omega \left( a \, a^\dagger - \frac{1}{2}\right) = \hbar \omega \left( a^\dagger \, a + \frac{1}{2}\right).\qquad\qquad(*) One may compute the commutation relations between the a\, and a^\dagger\, operators and the Hamiltonian: \begin{align} \left[\hat H, a \right] &= \left[\hbar \omega \left ( a a^\dagger - \tfrac{1}{2}\right ) , a\right] = \hbar \omega \left[ a a^\dagger, a\right] = \hbar \omega \left( a [a^\dagger,a] + [a,a] a^\dagger\right) = -\hbar \omega a. \\[1ex] \left[\hat H, a^\dagger \right] &= \hbar \omega \, a^\dagger . \end{align} These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows. Assuming that \psi_n is an eigenstate of the Hamiltonian \hat H \psi_n = E_n\, \psi_n. Using these commutation relations, it follows that Matrix representation The matrix expression of the creation and annihilation operators of the quantum harmonic oscillator with respect to the above orthonormal basis is \begin{align} a^\dagger &= \begin{pmatrix} 0 & 0 & 0 & 0 & \dots & 0 & \dots \\ \sqrt{1} & 0 & 0 & 0 & \dots & 0 & \dots \\ 0 & \sqrt{2} & 0 & 0 & \dots & 0 & \dots \\ 0 & 0 & \sqrt{3} & 0 & \dots & 0 & \dots \\ \vdots & \vdots & \vdots & \ddots & \ddots & \dots & \dots \\ 0 & 0 & 0 & \dots & \sqrt{n} & 0 & \dots & \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots \end{pmatrix} \\[1ex] 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} \end{align} These can be obtained via the relationships a^\dagger_{ij} = \left\langle\psi_i \right| a^\dagger \left| \psi_j\right\rangle and a_{ij} = \left\langle\psi_i \right| a \left| \psi_j\right\rangle. The eigenvectors \psi_i are those of the quantum harmonic oscillator, and are sometimes called the "number basis". == Generalized creation and annihilation operators ==

Thanks to representation theory and C*-algebras the operators derived above are actually a specific instance of a more generalized notion of creation and annihilation operators in the context of CCR and CAR algebras. Mathematically and even more generally ladder operators can be understood in the context of a root system of a semisimple Lie group and the associated semisimple Lie algebra without the need of realizing the representation as operators on a functional Hilbert space. In the Hilbert space representation case 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). Bosonic CCR algebra The (bosonic) CCR algebra over H is the algebra-with-conjugation-operator (called *) abstractly generated by elements a(f), where f\,ranges freely over H, subject to the relations \begin{align} \left[a(f), a(g)\right] &= \left[a^\dagger(f), a^\dagger(g)\right] = 0 \\[1ex] \left[a(f), a^\dagger(g)\right] &= \langle f\mid g \rangle, \end{align} in bra–ket notation. The map a: f \to a(f) from H to the bosonic CCR algebra is required to be complex antilinear (this adds more relations). Its adjoint is a^\dagger(f), and the map f\to a^\dagger(f) is complex linear in . 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^\dagger(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. Fermionic CAR algebra For fermions, the (fermionic) CAR algebra over H is constructed similarly, but using anticommutator relations instead, namely \begin{align} \{a(f),a(g)\} &= \{a^\dagger(f),a^\dagger(g)\} = 0 \\[1ex] \{a(f),a^\dagger(g)\} &= \langle f\mid g \rangle. \end{align} 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, but not identical to, a Clifford algebra. Physically speaking, a(f) removes (i.e. annihilates) a particle in the state |f\rangle whereas a^\dagger(f) creates a particle in the state |f\rangle. The free field vacuum state is the state \left\vert0\right\rangle with no particles, characterized by a(f) \left| 0\right\rangle=0. If |f\rangle is normalized so that \langle f|f\rangle = 1, then N=a^\dagger(f)a(f) gives the number of particles in the state |f\rangle. == In 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 \empty. To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider n_{i} particles at a site 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 ' is proportional to n_i \, dt, let us say a probability \alpha n_{i}dt to hop left and \alpha n_i \, dt to hop right. All n_i particles will stay put with a probability 1-2\alpha n_i \, dt. (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 |\dots, n_{-1}, n_0, n_1, \dots\rangle. It represents the juxtaposition (or conjunction, or tensor product) of the number states \dots, |n_{-1}\rangle, |n_{0}\rangle, |n_{1}\rangle, \dots located at the individual sites of the lattice. Recall that a\left| n \right\rangle = \sqrt{n} \left|n-1\right\rangle and a^\dagger \left| n\right\rangle= \sqrt{n+1}\left| n+1\right\rangle, for all , while [a,a^{\dagger}] = \mathbf 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: \begin{align} a \left|n\right\rangle &= (n) \left|n{-}1\right\rangle \\[1ex] a^\dagger \left|n\right\rangle &= \left| n{+}1\right\rangle \end{align} note that even though the behavior of the operators on the kets has been modified, these operators still obey the commutation relation [a,a^{\dagger}]=\mathbf 1 Now define a_i so that it applies a to |n_i\rangle. Correspondingly, define a^\dagger_i as applying a^\dagger to |n_i\rangle. Thus, for example, the net effect of a_{i-1} a^\dagger_i is to move a particle from the to the -th site while multiplying with the appropriate factor. This allows writing the pure diffusive behavior of the particles as \partial_{t}\left| \psi\right\rangle = -\alpha \sum_i \left(2a_i^\dagger a_i-a_{i-1}^\dagger a_i-a_{i+1}^\dagger a_i\right) \left|\psi\right\rangle = -\alpha\sum_i \left(a_i^\dagger-a_{i-1}^\dagger\right)(a_i-a_{i-1}) \left|\psi\right\rangle. 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 \lambda n(n-1)dt, yielding a term \lambda \sum_i (a_i a_i-a_i^\dagger a_i^\dagger a_i a_i) where number state is replaced by number state at site i at a certain rate. Thus the state evolves by \partial_t\left|\psi\right\rangle = -\alpha\sum_i \left(a_i^\dagger-a_{i-1}^\dagger\right) \left(a_i-a_{i-1}\right) \left|\psi\right\rangle + \lambda\sum_i \left(a_i^2-a_i^{\dagger 2}a_i^2\right) \left|\psi\right\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. ==In quantum field theories==

In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, a^\dagger_i and a^{\,}_i. These operators change the eigenvalues of the number operator, N = \sum_i n_i = \sum_i a^\dagger_i a^{\,}_i, 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, \ell, 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, \begin{align} \left[a^{\,}_i, a^\dagger_j\right] &\equiv a^{\,}_i a^\dagger_j - a^\dagger_ja^{\,}_i = \delta_{i j}, \\[1ex] \left[a^\dagger_i, a^\dagger_j\right] &= [a^{\,}_i, a^{\,}_j] = 0, \end{align} where [\cdot , \cdot ] is the commutator and \delta_{i j} is the Kronecker delta. For fermions, the commutator is replaced by the anticommutator {{nowrap|\{\cdot , \cdot \},}} \begin{align} \{a^{\,}_i, a^\dagger_j\} &\equiv a^{\,}_i a^\dagger_j +a^\dagger_j a^{\,}_i = \delta_{i j}, \\[1ex] \{a^\dagger_i, a^\dagger_j\} &= \{a^{\,}_i, a^{\,}_j\} = 0. \end{align} Therefore, exchanging disjoint (i.e. i \ne 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. Normalization conventions While Zee obtains the momentum space normalization [\hat a_{\mathbf p},\hat a_{\mathbf q}^\dagger] = \delta(\mathbf{p} - \mathbf{q}) via the symmetric convention for Fourier transforms, Tong and Peskin & Schroeder use the common asymmetric convention to obtain [\hat a_{\mathbf p},\hat a_{\mathbf q}^\dagger] = (2\pi)^3\delta(\mathbf{p} - \mathbf{q}). Each derives [\hat \phi(\mathbf x), \hat \pi(\mathbf x')] = i\delta(\mathbf x - \mathbf x'). Srednicki additionally merges the Lorentz-invariant measure into his asymmetric Fourier measure, \tilde{dk}=\frac{d^3k}{(2\pi)^3 2\omega}, yielding [\hat a_{\mathbf k},\hat a_{\mathbf k'}^\dagger] = (2\pi)^3 2\omega\,\delta(\mathbf{k} - \mathbf{k}'). ==See also==