 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. 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. 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. ==ladder operators for quantum harmonic oscillator== 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. first consider the simpler bosonic case of the phonons of the quantum harmonic oscillator. start with the schrÃ¶dinger 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. and the schrÃ¶dinger 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
Quantum_field_theory Quantum_mechanics