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An annihilation operator is used to remove a particle from the initial state and a creation operator is used to add a particle to the final state. The term "ladder operator" or "raising and lowering operators" is also sometimes used in mathematics, in the context of the theory of Lie algebras and in particular the affine Lie algebras.
Each additional boson then corresponds to a decrease of ħ in the spin projection. Thus, the spin raising and lowering operators + = + and =, so that [+,] =, correspond (in the sense detailed below) to the bosonic annihilation and creation operators, respectively. The precise relations between the operators must be chosen to ensure the correct ...
Another type of operator in quantum field theory, discovered in the early 1970s, is known as the anti-symmetric operator.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.
In the latter case, the creation operator is interpreted as a raising operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent phonons. Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the cluster decomposition theorem.
Rotating a spin-2 particle 180° can bring it back to the same quantum state, and a spin-4 particle should be rotated 90° to bring it back to the same quantum state. The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated 180°, and a spin-0 particle can be imagined as sphere, which looks the same ...
The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation a ψ 0 = 0 {\displaystyle a\psi _{0}=0} .
The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators.It was proposed by Pascual Jordan and Eugene Wigner [1] for one-dimensional lattice models, but now two-dimensional analogues of the transformation have also been created.
The operator + will increase the z-projection of the spin at position i back to its low-energy orientation, but the operator will lower the z-projection of the spin at position j. The combined effect of the two operators is therefore to propagate the rotated spin to a new position, which is a hint that the correct eigenstate is a spin wave ...