The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This chapter expresses the whole of quantum mechanics in terms of a small set of postulates, which embraces the behaviour of all known forms of matter. It introduces the postulates and illustrates how they are used, including how to apply them to problems of chemical interest, such as atomic and molecular structure and the properties of molecules. It also establishes the full significance of the Hψ=Eψequation and provides a foundation for its application. The chapter includes the solutions of the Schrödinger equation for a particle and the operator, which is a symbol for an instruction to carry out some action on a function. The principal mathematical difference between classical mechanics and quantum mechanics is that whereas in the former physical observables are represented by functions, in quantum mechanics they are represented by mathematical operators.
Abstract This chapter introduces Schrödinger’s equation and the action integral from which the differential equation is derived by the principle of stationary action. This principle is central to almost all of physics. The discretization of the integral with finite elements and interpolation with polynomials scaled by interpolation coefficients at the ends of the elements (nodes) allows us to integrate out the spatial dependence leaving the action as a binomial expression in the coefficients. Now the action is varied with respect to these coefficients to obtain a discretized matrix version of the Schrödinger’s equation. This typically leads to matrix equation or a generalized eigenvalue problem to solve. In simple terms, this is the essence of the finite element method. The boundary conditions for the wavefunction and the derivative of the wavefunction can be very easily imposed at the nodes prior to the solution of the matrix equations. The probability current has to be continuous at interfaces in the interior of the physical domain. The form of the current is obtained by a gauge-variational method in an unambiguous manner. Going from the differential equation to the action integral, and vice versa, are treated in great detail so that the reader can gain confidence in working with variational principles.
Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation,
(−2mℏ2∇2+v)Ψ=EΨ,ℏ=2πh, or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression
H=−2mℏ2∇2+V; h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and
2m−ℏ2∇2 the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
The quantum mechanical, time-dependent wave equation in generalized coordinates is obtained by a slight modification of Schrödinger's original derivation. If the classical Hamiltonian for a system is known, the variational technique may be applied to the Hamilton-Jacobi equation to yield an Euler-Lagrange equation. This latter is then the wave equation for the system. The method, which is generally easy to apply, is illustrated by applications to a single particle in curvilinear coordinates and to a symmetric top molecule.