In this chapter, we present some fundamental issues about approximation methods that are often used when a quantum-mechanical system is perturbed and about the relationship between classical and quantum mechanics. In Sec. 10.1 we introduce the stationary perturbation theory, while Sec. 10.2 is devoted to time-dependent perturbations. In Sec. 10.3 we briefly examine the adiabatic theorem. In Sec. 10.4 we introduce the variation method, an approximation method that is not based on perturbation theory. In Sec. 10.5 we discuss the classical limit of the quantum-mechanical equations, whereas in Sec. 10.6 we deal with the semiclassical approximation, in particular the WKB method. On the basis of the previous approximation methods in Sec. 10.7 we present scattering theory. Finally, in Sec. 10.8 we treat a method that has a wide range of applications: the path-integral method. Stationary perturbation theory Perturbation theory is a rather general approximation method that may be applied when a small additional force (the perturbation) acts on a system (the unperturbed system), whose quantum dynamics is fully known. If the disturbance is small, it modifies both the energy levels and the stationary states. This allows us to make an expansion in power series of a perturbation parameter, which is assumed to be small. Perturbation theory may be applied both to the case where the additional force is time-independent (in which case a stationary treatment suffices – the subject of the present section) as well as to the case where it explicitly depends on time.
This study examines integrating the Schrödinger equation with classical mechanics using a virtual axis-to-dimensional expansion. One-dimensional material fluctuations are viewed in a two-dimensional plane, explaining the random nature of these fluctuations and their spatial and temporal trajectories. A quantum-consistent force field is proposed, with its strength determined by the Planck constant and inversely proportional to the distance from the stationary point. Newton's second law is applied to establish a second-order linear differential equation for material fluctuations, from which the standard one-dimensional Schrödinger equation is derived, showing their equivalence. The study extends the three-dimensional Schrödinger equation to include external forces and explains quantum phenomena like energy levels and transitions through particle trajectory changes. This approach connects classical mechanics and quantum mechanics, offering a concise and intuitive formulation with clear physical significance.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
Sec. I. This paper inverts certain quantum mechanical ideas by constructing a "probability of presence" from first principles. If
xi=fi(s,a1,a2) represents a 2-parameter family of curves in 3-dimensional space, the density of the functions
fi with respect to the parameters
a1a2 is defined by means of an expression for the normal cross section of a sheaf of curves. This density plays the role of a probability. Its derivative along a curve of the family is independent of the parameters
a1a2. This circumstance permits a special density to be constructed, called the "characteristic" or "smoothest" density, which is independent of the parameters and which can be found by solving a linear second-order partial differential equation. The extension to
n-dimensional space and ($n\ensuremath{-}1$) parameters is given. Sec. II. The concept of orbital density is the dynamical equivalent of the density of Sec. I. Dividing the orbital density by the velocity of the representative point gives the quantity we call kinematical probability
P, which satisfies the fundamental equation (2.8), and which has the same force as any other probability of presence. Equation (2.8) is used to derive from the Hamilton-Jacobi equation a "wave" equation for $\ensuremath{\surd}P$ for a system of
n planets moving around a sun under the condition that the space average of the mutual interactions of the planets is a minimum. This provides a new approach to planetary dynamics and yields an equation that is very much like Schroedinger's equation for the stationary states of
n particles, and has the same basic interpretation. Sec. III. A non-conservative system of
n particles is assumed to be in a state such that the action function can be written $W=\ensuremath{-}Et+R({x}_{i},t)$ in which
R still contains the time explicitly. Starting with the Hamilton-Jacobi equation, using Eq. (2.8), and imposing the condition that the space average of $\frac{\ensuremath{\partial}R}{\ensuremath{\partial}t}$ is a minimum, an equation [viz., (3.10)] is derived which differs from Schroedinger's equation for stationary states only in certain respects that cannot be regarded as fundamental. Despite this basic equivalence of Schroedinger's equation and our equation (3.10), conflicts in interpretation and formalism arise which are briefly discussed. Sec. IV. Complex
U' s are introduced into Eq. (3.10) in a simple way which fails to bridge the gap between our formalism and that of quantum mechanics. The kinematical probability defined by Eq. (3.10) is shown to have the "smoothest" property. To account physically for the stationary states an electromagnetic wave system is proposed which has the property that a particle moving in it may be in "phase-mechanical" resonance with it. Macroscopic examples of this kind of resonance are given. This concept represents an inversion of physical ideas of quantum mechanics corresponding to the mathematical inversion with which we began.
It is pointed out that the property of a constant energy characteristic for the circular motions of macroscopic bodies in classical mechanics does not hold when the quantum conditions for the motion are applied. This is so because any macroscopic body—lo-cated in a high-energy quantum state—is in practice forced to change this state to a state having a lower energy. The rate of the energy decrease is usually extremely small which makes its effect uneasy to detect in course of the observations, or experiments. The energy of the harmonic oscillator is thoroughly examined as an example. Here our point is that not only the energy, but also the oscillator amplitude which depends on energy, are changing with time. In result, no constant positions of the turning points of the oscillator can be specified; consequently the well-known variational procedure concerning the calculation of the action function and its properties cannot be applied.