Comparison between time-independent and time-dependent quantum systems in the context of energy, Heisenberg uncertainty, average energy, force, average force and thermodynamic quantities
Exact solutions of time-dependent Schrödinger equation in presence of time-dependent potential is defined by point transformation and separation of variables. Energy and Heisenberg uncertainty relation are pursued for time-independent potential whereas average energy and Heisenberg uncertainty relation are defined for time-dependent potential. Forces acting on a fixed boundary wall as well as average force acting on moving boundary wall are presented along various trajectories. For high temperature, analytical forms of partition function and the corresponding thermodynamic quantities are derived following the Euler-Maclaurin summation formula over a finite as well as an infinite domain for accurate presentation. Three quantum systems are generated with the help of point transformation, separation of variables and super-symmetric quantum mechanics from one quantum system and the corresponding results are compared among all systems, where two of them are time-independent and another two are time-dependent.
Exact solutions of time-dependent Schrödinger equation in presence of time-dependent potential is defined by point transformation and separation of variables. Energy and Heisenberg uncertainty relation are pursued for time-independent potential whereas average energy and Heisenberg uncertainty relation are defined for time-dependent potential. Forces acting on a fixed boundary wall as well as average force acting on moving boundary wall are presented along various trajectories. For high temperature, analytical forms of partition function and the corresponding thermodynamic quantities are derived following the Euler-Maclaurin summation formula over a finite as well as an infinite domain for accurate presentation. Three quantum systems are generated with the help of point transformation, separation of variables and super-symmetric quantum mechanics from one quantum system and the corresponding results are compared among all systems, where two of them are time-independent and another two are time-dependent.
This article establishes the proof of the Schrödinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrödinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrödinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.
Abstract Quantum thermodynamics describes dynamic processes by means of the operators of entropy production P and time t . P and t do not commute. It exists the non-vanishing t–P commutator [ t , P ]=ik. Here the Boltzmann constant k has the physical meaning of a quantum of entropy. The t–P commutator immediately leads us to the t–P uncertainty relation ΔtΔP≥k/2. Hence the observables t and P are not sharply defined simultaneously. Similar uncertainty relations can also be expected for other pairs of conjugate variables with products of the physical meaning of an entropy. The free energy F and the reciprocal temperature (1/T) are the respective conjugate variables of an isolated system of many particles, which leads us to the F–(1/T) uncertainty relation ǀΔFǁΔ(1/T)ǀ≥k/2. It can be traced back to the t–P uncertainty relation mentioned above. In this way the Helmholtz free energy F and the temperatur T are introduced into quantum thermodynamics. The uncertainties ǀΔFǀ→0 and ǀΔTǀ→0 are negligible at low temperatures T→0, and quantum thermodynamics turns into the time-independent classical thermodynamics. Against this the uncertainties ǀΔFǀ→∞ and ǀΔTǀ→∞ grow unlimited at high temperatures T→∞, and classical thermodynamics loses its sense. In the limit of one particle the uncertainties cannot be neglected even at low temperatures. However a detailed discussion shows that the free energy f of a single particle vanishes within the whole range of temperatures T. This defines the particle entropy σ=ε/T=ak. The dimensionless entropy number a connects the particle energy ε=akT with the temperature T. The entropy number a of a single (s) independent particle can be calculated with the extended, temperature-dependent Schrödinger equation A s φ=aφ. Here A s=−(Λ 2/4π)∇2 means the dimensionless entropy operator describing the entropy number a and thus the particle entropy σ=ak. Λ is the thermal de Broglie wave length. Finally we calculate by means of quantized particle entropies σ the internal energy E, the Helmholtz free energy F, the entropy S, the chemical potential μ, and the equation of state of an ideal gas of N monatomic free particles in full agreement with classical thermodynamics. We also calculate the partition function q=V/Λ 3 of a single free particle within the volume V. Here Λ 3 is a small volume element taking into account the wave–particle dualism of a single free particle of mass m at temperature T. Extension to a system of N free particles leads us to a simple geometrical model and to the conclusion that an ideal gas of independent particles becomes instable below a critical temperature T C. T C corresponds to the critical temperature T BE of Bose–Einstein condensation.
This article establishes the proof of the Schrdinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrdinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrdinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.
The time-independent Schrodinger equation is written terms of energy, namely using V(x) the potential and E, the energy eigenvalue. In fact, the equation can be written as KE average(x) + V(x) = E and so can be expressed in terms of a classical energy conservation equation at each point. KE average, however, in such a case is complicated and seems to contain various kinds of kinetic energy motion which only yields on average the classical kinetic energy. The purpose of this note, is to try to determine the form of the time-independent Schrodinger equation without any concept of energy. Only at the end of the discussion, will comparisons be made with classical conservation of energy. In such a discussion, it is assumed that particles undergo a zitterbewegung type motion which can be modeled by exp(ikx) hbar k being the momentum and that there are forces (related to a V(x)) in the one dimensional problem. The objective is to try to understand microscopically what is happening without immediate resort to a conservation equation.
The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.
The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system.The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem.The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation.The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation.It can be expressed in terms of a complex variable that linearizes the Riccati equation.This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem.From there, also the relation between the classical and quantum dynamics of the systems can be obtained.Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown.Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations.This provides the link to a nonlinear formulation of timeindependent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law.Comparison with SUSY and the time-dependent problems concludes our analysis.