Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.
This operator, similar to spin in non-relativistic quantum mechanics is a ladder operator that can create two fermions of opposite spin out of a boson or a boson from two fermions.
A Fermion, named after Enrico Fermi, is a particle with a half-integer spin, such as electrons and protons.
A boson, named after S. N. Bose, is a particle with full integer spin, such as photons and W's.
First, we will review spin for non-relativistic quantum mechanics.
Spin, an intrinsic property similar to angular momentum, is defined by a spin operator S that plays a role on a system similar to the operator L for orbital angular momentum.
These formalisms also obey the usual commutation relations for angular momentum
can be represented by the matrix representations: Recalling the generalized uncertainty relation for two operators A and B,
are as follows: Therefore, like orbital angular momentum, we can only specify one coordinate at a time.
The creation of a particle and anti-particle from a boson is defined similarly but for infinite dimensions.
Therefore, the Levi-Civita symbol for infinite dimensions is introduced.
The commutation relations are simply carried over to infinite dimensions
Defining the magnetic quantum number, angular momentum projected in the z direction, is more challenging than the simple state of spin.
The problem becomes analogous to moment of inertia in classical mechanics and is generalizable to n dimensions.
To illustrate, the electromagnetic field quantized is the photon field, which can be quantized using conventional methods of canonical or path integral quantization.
Free bosonic fields obey commutation relations: To illustrate, suppose we have a system of N bosons that occupy mutually orthogonal single-particle states
Using the usual representation, we demonstrate the system by assigning a state to each particle and then imposing exchange symmetry.
This wave equation can be represented using a second quantized approach, known as second quantization.
The number of particles in each single-particle state is listed.
The creation and annihilation operators, which add and subtract particles from multi-particle states.
These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta.
However, these operators literally create and annihilate particles with a given quantum state.
also found in quantum field theory, the creation and annihilation operators
allows us to determine whether a particle was created or destroyed in a system, the spin operators
A photon can become both a positron and electron and vice versa.
Symmetric particles, or bosons, need not obey the Pauli-Exclusion Principle so therefore we can represent the spin state of the particle as follows: The annihilation spin operator, as its name implies, annihilates a photon into both an electron and positron.
Likewise, the creation spin operator creates a photon.
is applied to a fermion twice, the resulting eigenvalue is 0.
This relation satisfies the Pauli Exclusion Principle.
However, bosons are symmetric particles, which do not obey the Pauli Exclusion Principle.