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).
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).
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.
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.
Copenhagen physicists said; The electronics that we use today-radio, televisions, computers, lasers, MRI machines GPS-communications- depend on calculations using Schrodinger’s equation plus a lot of other Mathematics and Measurements, our idea of “shut up and calculate” has worked very well for us technologically. My comment’s these technological achievements do not justify that the equation itself as (I will try show) has any true scientific basis. In 1925, Erwin Schrodinger wrote excitingly to Einstein. “I have been intensely concerned these days with Louis de Broglie’s theory [1]. It is extraordinary, exciting, but still has some very grave difficulties”. So regardless of you are consider Schrodinger’s equation is a fundamental equation (has no derivation from simples forms) or derived one, you cannot deny first that Schrodinger’s equation is a wave equation and second it is shows how (de Broglie hypothesis of) matter wave change with time. i.e. Schrodinger equation describe the evaluation of the physical system in which the quantum effects, the “Wave-particle” duality are significant.
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.
The father of quantum mechanics, Erwin Schrodinger, was one of the most important figures in the development of quantum theory. He is perhaps best known for his contribution of the wave equation, which would later result in his winning of the Nobel Prize for Physics in 1933. The Schrodinger wave equation describes the quantum mechanical behaviour of particles and explores how the Schrodinger wave functions of a system change over time. This project is concerned about exploring the one-dimensional case of the Schrodinger wave equation in a harmonic oscillator system. We will give the solutions, called eigenfunctions, of the equation that satisfy certain conditions. Furthermore, we will show that this happens only for particular values called eigenvalues.