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 (\(\mathit{x}\) , \(\hat{v}\)) , with the commutation relation [\(\mathit{x}\) , \(\hat{v}\)] = \(\mathit{i}\hbar\)/\(\mathit{m}\) , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion \(\mathit{K}\)(\(\mathit{x, v, t}\)) of the classical system, The comparison with the quantization in the space (\(\mathit{x, p}\)) is done with the usual Schrödinger’s equation for the Hamiltonian \(\mathit{H}\)(\(\mathit{x, p, t}\)), and with the commutation relation [\(\mathit{x}\) , \(\hat{p}\)] = \(\mathit{i}\hbar\). 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 (\(\mathit{x}\) , \(\hat{v}\)) quantization has fewer oscillations than the (\(\mathit{x}\) , \(\hat{p}\)) quantization, the average energy of the system is higher in (\(\mathit{x}\) , \(\hat{p}\)) quantization than on the (\(\mathit{x}\) , \(\hat{v}\)) quantization, and the Boltzmann- Shannon entropy on the (\(\mathit{x}\) , \(\hat{p}\)) quantization is higher than on the (\(\mathit{x}\) , \(\hat{v}\)) 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.
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 ((\mathit{x}) , (\hat{v})) , with the commutation relation [(\mathit{x}) , (\hat{v})] = (\mathit{i}\hbar)/(\mathit{m}) , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion (\mathit{K})((\mathit{x, v, t})) of the classical system, The comparison with the quantization in the space ((\mathit{x, p})) is done with the usual Schrödinger’s equation for the Hamiltonian (\mathit{H})((\mathit{x, p, t})), and with the commutation relation [(\mathit{x}) , (\hat{p})] = (\mathit{i}\hbar). 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 ((\mathit{x}) , (\hat{v})) quantization has fewer oscillations than the ((\mathit{x}) , (\hat{p})) quantization, the average energy of the system is higher in ((\mathit{x}) , (\hat{p})) quantization than on the ((\mathit{x}) , (\hat{v})) quantization, and the Boltzmann- Shannon entropy on the ((\mathit{x}) , (\hat{p})) quantization is higher than on the ((\mathit{x}) , (\hat{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.
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.
Quantization of the 1-D forced harmonic oscillator in the space (x,v)
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 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,p^]=iℏ. 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,v^) quantization has less 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 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.
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. We obtain the Schrödinger equation from a mathematical identity by a slight generalization of the formulation of classical statistical mechanics based on the Hamilton-Jacobi equation. This approach brings out most clearly the fact that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave.
Quantization of the 1-D harmonic oscillator with variable mass using the operators v^ and p^
For the 1-D harmonic oscillator with position depending variable mass, a Hamiltonian and constant of motion are given through a consistent approach. Then, the quantization of this system is carried out using the operator p^, for the Hamiltonian, and the operator v^ for the constant of motion. We find that the result of both quantizations brings about different quantum dynamics.
The time-dependent Schrodinger problem of an oscillator .5kx*x together with the additional potential -x f(t), where f(t) is a function of time (often periodic), has already been solved in the literature (1). The solution involves a variable transformation y= x - b(t), where b(t) is a function of time which will turn out to be related to f(t) and W(x,t)= exp(i y d/dt b(t)) Wosc( y) exp(i Eosc t) . This yields an oscillator type time dependent equation with a term q(t) Wosc(y) which can be removed by using Wosc(y) exp(i c(t)). Wosc( y) is the oscillator wavefunction with the space-time variable y and Eosc the usual oscillator energy levels. Thus, by using the variable transformation together with W(x,t)=Wosc(x)exp(iEosc t) g(x,t) (i.e. a product of factors) one can obtain an “time-independent like” Schrodinger equation in y (after performing one more transformation Wosc(y)exp(i c(t)). In a previous note (2), we argued that if one writes a wavefunction as W(x)=Wo(x)q(x), where Wo(x) is the ground state wavefunction, one obtains a time-independent Schrodinger equation for Wo(x), which contains V(x), and a second eigenvalue equation coupling Wo(x) and q(x) with an eigenvalue equal to E-Eo. In particular, it is the time flux [d/dt W(x,t)]/W(x,t) which becomes linked with q(x) and the spatial flux [d/dx W(x)]/W(x). In (3), we argued that bound state quantum mechanics contains two velocities v(x), the classical velocity and u(x)/m = [d/dx W(x)]/ W(x) and that each behave differently. For example, v(x) behaves according to Newton’s law, while u(x)/m does not (except for the oscillator ground state). In this note, we try to apply these two ideas to the case of the oscillator with the time-dependent extra potential x f(t). We see that writing W(x,t) as a product (after the variable change) leads to two sets equation, one being a time-independent like equation for the oscillator and the other linking u(x)/m to a portion of time flux [d/dt W(x,t) ]/ W(x,t) |x constant. In this case, the time flux is directly related to d/dx [W*(x,t)W(x,t)] i.e. the physical density. Thus, a change in physical density in time is related to a change in physical density in space. Thus, two physical processes, it seems, are occurring. One is the W osc (y) resonance (albeit it a strange one because y=x + b(t)) and the second flux coupling with exp(i b(t) y), part of the oscillator potential and part of the additional potential -x f(t). Given that the problem has been mathematically solved already in the literature (1), our objective is to attempt to see if there is physical meaning in terms of ideas we have presented in previous notes, namely flux, the notion of a product wavefunction leading to a separate resonance type of reaction and double resonances.