BENTHAM SCIENCE PUBLISHERS eBooks

Conservation of Stationary Quantum States Stationary Schrödinger Equation

P. Enders

Published Feb 12, 2019• 0 citations

Abstract

In this chapter, the stationary Schrödinger equations in configuration and momentum configuration spaces are derived from the non-classical expression of the total energy (7.15) found in the foregoing chapter. The new parameter in the Fourier transform is identified as ℏ ≡ ℎ/2. The roles of boundary conditions, probability concepts and independent dynamical variables are shortly discussed.

DOI: 10.2174/9781681084497119010010

Related papers

Conservation and Change of Stationary States

Peter Enders

The time-independent Schrödinger equation represents a stationary-state equation. The stationary wave functions obtained so far are time independent. Their time-dependence is obtained by means of rather general arguments. Then, stationarystate functions are found. The next step in Newton’s and Euler’s representations of classical mechanics is (to derive) the equation of change of stationary-states. Here, Euler’s principles of stationary-state change are generalized to quantum-mechanical systems. This enables us to derive the quantum-mechanical equation of change of stationary-states. The time-independent Schrödinger equation, i.e, the equation of motion will follow in the next chapter.

BENTHAM SCIENCE PUBLISHERS eBooksFeb 12, 20190 citations

Non-Classical Selection of the Physical Solutions

P. Enders

The energy parameter in the stationary Schrödinger equation is primarily continuous. Schrödinger has selected the discrete quantum states by means of boundary conditions. This, however, is mathematics for classical systems like strings and pipes. And, more important, it hides the intrinsic discrete structure of that equation. The latter is represented by the recursion relations between solutions to different values of the energy parameter. These follows from Whittaker’s integral expressions of the solutions and hold true for all solutions, not only for Schrödinger’s eigensolutions. The physically relevant solutions are distinguished by certain mathematical properties as well as physical criteria, in particular, by their compliance with the absence of perpetua mobilia. This non-classical approach is exemplified by means of the model system harmonic oscillator. For this, the chapter starts with a sketch of the classical harmonic oscillator, including quite general topics such as the separation of internal and external system parameters and Huygens’ principle, which will reappear in the quantum realm.

BENTHAM SCIENCE PUBLISHERS eBooksFeb 12, 20190 citations

Wave mechanics

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol, as defined in Eq. (3.6.6). The following steps are the local and global conservation laws associated to the Schrödinger equation, the probabilistic interpretation of the wave function, the spreading of wave packets and transformation properties of wave functions. The solution of the Schrödinger equation for a given initial condition is then studied. For this purpose, one first expands the initial condition in terms of eigenfunctions of the Hamiltonian operator. This is then ‘propagated’ to determine the evolution of the wave function. Thus, one is led to consider the Green kernel of the Schrödinger equation, and the boundary conditions on the wave function that lead to complete knowledge of stationary states. A dynamics involving an isometric non-unitary operator is also studied. The chapter ends with an elementary introduction to harmonic oscillators and to the JWKB method in wave mechanics, plus examples on the Bohr–Sommerfeld quantization. From classical to wave mechanics The correspondence between differential operators and polynomials in k relies on the notion of the symbol defined in Eq. (3.6.6). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is non-trivial because the map from symbols to operators contains some ambiguity. Such a transition is always possible if polynomials in p have constant coefficients. When this is not the case, i.e. if the coefficients are functions on the configuration space, there is an obvious ambiguity in the association of an operator with xp or px , for instance.

Cambridge University Press eBooksMar 11, 20040 citations

Simple applications

Peter Holland

Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]

Cambridge University Press eBooksMay 13, 19930 citations

Equation of Motion

P. Enders

Having got the quantum-mechanical equation of change of stationary states, it is rather straightforward to derive the quantum-mechanical equation of motion. As anticipated in the foregoing chapter, this is the time-dependent Schrödinger equation. Moreover, Huygens’ principle is shown to apply to QM, as obtained by Feynman in another way.

BENTHAM SCIENCE PUBLISHERS eBooksFeb 12, 20190 citations

Quantization and molecular energy levels

John M. Brown

This chapter discusses the physical properties of atoms and molecules, which are quantized. This means that they can only take certain discrete values rather than a full range of continuously varying values. The chapter examines the quantized energy levels of a molecule. These are called stationary states, which describe an energy that is independent of time. The chapter also outlines the procedure for constructing a quantum mechanical Hamiltonian, which includes writing down the complete and classical expression for the energy of the system. The chapter reviews the general approach to the determination of energy levels of a physical system. This involves two stages: First, the appropriate Hamiltonian is defined and, in a second step, the eigenfunctions and eigenvalues of the Hamiltonian are determined. The chapter also discusses the Schrödinger equation, which can only be solved for one-body systems.

Oxford University Press eBooksJul 23, 19980 citations