One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.
For a free falling particle moving in a media which has quadratic velocity force effect on the particle, two equivalent constants of motion, with units of energy, two Lagrangians, and two Hamiltonians are deduced. These quantities describe the dynamics of the same classical system. However, their quantization and the associated statistical mechanics (for an ensemble of particles) describe two completely different quantum and statistical systems. This is shown at first order in the dissipative parameter.
It is well known that to quantize any dynamical system it is necessary for a generator of the classical motion to exist in the form of a Hamiltonian function. It is shown, by using the example of a damped harmonic oscillator, that a particular class of inequivalent classical Hamiltonians exist which make quantization of the system ambiguous. Hence it is conclued that it is not sufficient for the Hamiltonian to merely generate the motion, but it must also be necessarily related via a canonical transformation to the total energy of the system.
Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable (
x ,
v^) , with the commutation relation [
x ,
v^] =
iℏ/
m , 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 [
x ,
p^] =
iℏ. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.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. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.