The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
An operator whose position representation is a Gaussian function is referred to as a Gaussian operator. In common with the harmonic oscillator Hamiltonian operator, the eigenvectors of the Gaussian operator are a product of Hermite polynomial functions and a Gaussian weighting factor. It is demonstrated that the thermal equilibrium density operator of a one-dimensional harmonic oscillator can be expressed in Gaussian operator form, including its eigenvalues. These can be obtained without first solving for the eigenfunctions of the harmonic oscillator Hamiltonian.
In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.
In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form
W=Sp[ρW], where
ρ is the density matrix, and
W is the universal density matrix. The density matrix
ρ for each quantum system is different, and the universal matrix
W is the same for any quantum system. Thus, the matrix
W has a fundamental character. In the work, the elements of the universal matrix
W were found explicitly and their properties were investigated. The diagonal elements of the matrix
W are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix
W contain frequency oscillations responsible for dissipations in the quantum systems.