"Enchainement" and quantization: Schrodinger's equation for a linear harmonic oscillator

Gustavo V. López, Omar J. P. Bravo

The quantization of the forced harmonic oscillator is studied with the quantum variable (x, v∧), with the commutation relation , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K (x, v, t) of the classical system, The comparison with the quantization in the space (x, p) is done with the usual Schrödinger’s equation for the Hamiltonian H(x, p, t), and with the commutation relation . It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x, v∧) quantization has fewer oscillations than the (x, p∧) quantization, the average energy of the system is higher in (x, p∧) quantization than on the (x, v∧) quantization, and the Boltzmann-Shannon entropy on the (x, p∧) quantization is higher than on the (x, v∧) quantization.

Hiu Yung Wong

Arno Böhm

Gustavo V. López, Omar J. P. Bravo

The quantization of the forced harmonic oscillator is studied with the quantum variable ($x,\hat v$), with the commutation relation $[x,\hat v]=i\hbar/m$, and using a Shrödinger's like equation on these variable, and associating a linear operator to a constant of motion $K(x,v,t)$ of the classical system, The comparison with the quantization in the space ($x,p$) is done with the usual Schrödinger's equation for the Hamiltonian $H(x,p,t)$, and with the commutation relation $[x,\hat p]=i\hbar$. It is found that for the non resonant case, both forms of quantization brings about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the ($x,\hat v$) quantization has less oscillations than the ($x,\hat p$) quantization, the average energy of the system is higher in ($x,\hat p$) quantization than on the $(x,\hat v$) quantization, and the Boltzmann-Shannon entropy on the ($x,\hat p$) quantization is higher than on the ($x,\hat v$) quantization.

Arno Böhm

Steven Kenneth Kauffmann

The time-dependent Schrodinger equation with time-independent Hamiltonian matrix is a homogeneous linear oscillatory system in canonical form. We investigate whether any classical system that itself is linear, homogeneous, oscillatory and conservative is guaranteed to linearly map into a Schrodinger equation. Such oscillatory classical systems can be analyzed into their normal modes, which are mutually independent, uncoupled simple harmonic oscillators, and the equation of motion of such a system linearly maps into a Schrodinger equation whose Hamiltonian matrix is diagonal, with h times the individual simple harmonic oscillator frequencies as its diagonal entries. Therefore if the coupling-strength matrix of such an oscillatory system is presented in symmetric, positive-definite form, the Hamiltonian matrix of the Schrodinger equation it maps into is h-bar times the square root of that coupling-strength matrix. We obtain a general expression for mapping this type of oscillatory classical equation of motion into a Schrodinger equation, and apply it to the real-valued classical Klein-Gordon equation and the source-free Maxwell equations, which results in relativistic Hamiltonian operators that are strictly compatible with the correspondence principle. Once such an oscillatory classical system has been mapped into a Schrodinger equation, it is automatically in canonical form, making second quantization of that Schrodinger equation a technically simple as well as a physically very interpretable way to quantize the original classical system.