LagrangeEuler said:In one dimensional problems in QM only in case of the potential ##V(x)=\frac{m\omega^2x^2}{2}## creation and annihilation operator is defined. Why? Why we couldn't define same similar operators in cases of other potentials?
Bill_K said:Stepping operators can also be used to solve the Coulomb problem. Also the n-dimensional harmonic oscillator. In principle such operators always exist, but only in a few problems are they simple enough to write down in useful form.
Creation and annihilation operators are mathematical operators used in quantum mechanics to describe the creation and destruction of particles. They are represented by the symbols a and a† respectively.
Creation operators act on a state to increase the number of particles present, while annihilation operators reduce the number of particles in a state. They can be used to create and destroy particles in a system, and their combination allows for the calculation of probabilities and expectation values.
Creation and annihilation operators are related to each other through their commutation relation [a, a†] = 1, where [x,y] represents the commutator of operators x and y. This relation allows for the creation and annihilation operators to be used in conjunction with each other in calculations.
In quantum field theory, creation and annihilation operators are used to quantize fields and describe the creation and destruction of particles in a field. They are essential in the formulation of the second quantization method, which allows for the description of multiple particles in a system.
Creation and annihilation operators have many applications, including in quantum optics, quantum computing, and particle physics. They are also used in the study of Bose-Einstein condensates and superconductivity. Their mathematical properties make them useful in a wide range of quantum mechanical systems.