In 1926, the Austrian physicist Erwin Schrodinger derived an equation of waves from the Hamilton variational principle inspired by the analogy between Mechanics and Optics. This equation explained much of quantum phenomenology was known at that time. Physical laws must be invariant under certain symmetries, represented by the transformations of the mathematical object that define these laws. In particular in quantum mechanics one can ask for the transformations of states and observables. These transformations must fulfill what constitutes the essence of the so-called Wigner's theorem, that is, that the transformed observables must possess the same possible sets of eigenvalues as the old ones and that the transformations of the states must give the same probabilities. The temporal evolution of the states must always be given by unit operators that conserve scalar products.
In 1926, the Austrian physicist Erwin Schrodinger derived an equation of waves from the Hamilton variational principle inspired by the analogy between Mechanics and Optics. This equation explained much of quantum phenomenology was known at that time. Physical laws must be invariant under certain symmetries, represented by the transformations of the mathematical object that define these laws. In particular in quantum mechanics one can ask for the transformations of states and observables. These transformations must fulfill what constitutes the essence of the so-called Wigner's theorem, that is, that the transformed observables must possess the same possible sets of eigenvalues as the old ones and that the transformations of the states must give the same probabilities. The temporal evolution of the states must always be given by unit operators that conserve scalar products.
Till 1925 the idea of what the quantum theory would be was unknown. During 1925 three different but equivalent versions of quantum theory were proposed – Schrödinger-proposed wave mechanics; Heisenberg-developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie’s matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner (hence called wave function and denoted it as ψ). This function is generally complex and assumed to contain information about a system. Schrödinger set up a linear and time-dependent wave-like equation, called Schrödinger wave equation, to describe the wave aspect of a particle taking account of de Broglie’s relation for wavelength. Physically, |ψ(X, t)|2, where ψ(X, t) is the solution of the Schrödinger equation, is interpreted as position probability density. That is, |ψ(X, t)|2 is the probability density of observing a particle at position X at time t. ψ does not give exact outcomes of observations but helps us to know all possible events and their probabilities . Further, the probability interpretation allows us to find the average or expected result of a set of measurements on a quantum system.
During 1925 three different but equivalent versions of quantum theory were proposed, Schrodinger proposed wave mechanics; Heisenberg developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie's matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner. Schrodinger set up a linear and time-dependent wave-like equation, called Schrodinger wave equation, to describe the wave aspect of a particle taking account of de Broglie's relation for wavelength. This chapter discusses the Schrodinger equation and explores the general properties of the solution of the Schrodinger equation. It verifies the Ehrenfest's theorem in certain interesting systems and solves equation for linear harmonic oscillator and shows that their solutions are identical to the solutions of classical equation of motion.
In the context of a new analysis of the notebooks of Erwin Schrödinger, the paper deals with the question of the relation between Schrödinger's creation of wave mechanics and the contemporary efforts by Werner Heisenberg and his colleagues to establish a new quantum mechanics. How can one explain, from a broader historical and epistemological perspective, the astonishing simultaneity and complementarity of these discoveries? The paper argues that neither the physical problems with which both approaches deal nor what ultimately turned out to be their common mathematical ground are sufficient to explain their complementarity. Instead, their closeness is explained by analyzing their common roots in classical mechanics and its transformation in the light of the most fundamental new quantum law, the relation between energy and frequency found by Planck. It is shown, in particular, that for both approaches a bridge between quantum and classical aspects involving this relation was crucial. In the case of Heisenberg, this bridge was given by Bohr’s correspondence principle. In the case of Schrödinger it was constituted by Hamilton’s optical-mechanical analogy.
This chapter is primarily about ‘‘wave mechanics’’ since that is the most convenient way to introduce undergraduates to quantum mechanics using calculus. An equivalent form called ‘‘matrix mechanics’’ will be discussed briefly in a later chapter. Consider again the 1923 paper by De Broglie and the experiments that validated the particle-wave duality in Chapter 10. One might well ask that if there really is some ‘‘wave’’ that describes the behavior of particles, then is there an equation that the wave obeys? Even today it is difficult to say what the ‘‘wave’’ is, but it may help to find an equation it obeys. Step back a moment to some basic calculus: d2 dx2 sin (ax) ¼ d dx [a cos (ax)] ¼ (a2) sin (ax): Perhaps you did not notice the pattern before but we can put this into a general form as (Operator)(Eigenfunction) ¼ (Eigenvalue)(Eigenfunction): The word ‘‘eigen’’ in German means ‘‘characteristic, unique, peculiar, special . . . ’’ and only certain functions satisfy this condition called an eigenfunction equation. An analogy that has been successful in explaining this to undergraduates is to consider an apple tree with ripe apples on it. If you hit the branches with a stout stick some apples will fall off the tree but the tree will still be there. The operator is the act or operation of hitting the tree with the stick, the tree is the eigenfunction and the apples are the eigenvalue(s). The eigen word comes from German because this relationship was first linked to the De Broglie wave idea by Erwin Schrödinger in 1926 in a series of papers that are among the most important in modern science [1]. Schrödinger (1887-1961) was an Austrian physicist who received the Nobel Prize for his work in 1933 (Figure 11.1).
Finding that in the formula of expansion of a function into a series of wave-like functions the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schrödinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.
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