Time-independent and time-dependent forms of Schrodinger equations are presented in one and three dimensions along with their interpretations. Also, the time-independent version is shown to follow from the time-dependent version. For better conceptualization of wave motion, the general equation of wave motion is discussed and solved, explaining the physical significance of variables involved. Born interpretation of the wave function as a measure of the probability of finding the particle in a region of space at a given instant is described. The requirements to be fulfilled by the wave function for representing a physical world system are outlined.
Time-independent and time-dependent forms of Schrodinger equations are presented in one and three dimensions along with their interpretations. Also, the time-independent version is shown to follow from the time-dependent version. For better conceptualization of wave motion, the general equation of wave motion is discussed and solved, explaining the physical significance of variables involved. Born interpretation of the wave function as a measure of the probability of finding the particle in a region of space at a given instant is described. The requirements to be fulfilled by the wave function for representing a physical world system are outlined.
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.
The sixth chapter begins with the equation of wave motion in one dimension and arrived at the three-dimensional Laplacian operator. After this, time-independent and time-dependent Schrödinger’s equations have been derived in a simple way. It also deals with the interpretation of wave function and mentions the conditions of acceptable wave functions taking into consideration the graphical presentation. This chapter also gives idea about the condition of normalisation and orthogonality and determination of normalisation constant. The condition of orthogonality has also been discussed in detail, which gives an important point regarding how two wave functions become orthogonal. It also speaks about orthonormality. The ideas of Eigen function and Eigen value have also been introduced, and degeneracy has been defined. This chapter also incorporates the transformation of Laplacian operator into spherical polar coordinates in a graspable manner. The Ehrenfest theorem and its proof have also been included. The chapter also provides the idea of matrix representation of wave function, matrix representation of operator, properties of matrix elements, and matrix form of time-dependent Schrödinger equations. At the end of the chapter, adequate references, solved problems, and questions on concepts have also been cited.
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.
Due to the space and time dependence of the wavefunction in the time-dependent Schrödinger equation, different boundary conditions are possible. The equation is usually solved as an `initial-value problem', by fixing the value of the wavefunction in all space at a given instant. We compare this standard approach to `source boundary conditions' that fix the wave at all times in a given region, in particular at a point in one dimension. In contrast to the well known physical interpretation of the initial-value-problem approach, the interpretation of the source approach has remained unclear, since it introduces negative energy components, even for `free motion', and a time-dependent norm. This work provides physical meaning to the source method by finding the link with equivalent initial-value problems.
Research has been carried out to determine the physical meaning and derivation of the Schrodinger equation with the literature study research method. Based on the research results obtained through data processing in the form of information obtained from various literature sources, it can be concluded that the general solution of the Schrodinger equation is a wave function. In general, based on mathematical calculations by applying the technique of solving differential equations to the Schrodinger equation, it can be seen that this equation has many solutions, but not all of these solutions can be said to be functions that meet the wave function criteria. This wave function must be able to interpret the real physical criteria of a particle.
An important feature of the wave equation is that its solutions q(r, t) are uniquely specified once the initial values q(r, 0) and (del)q(r, 0)/@t are specified. As was mentioned before, if we view the wave equation as describing a continuum limit of a network of coupled oscillators, then this result is very reasonable since one must specify the initial position and velocity of an oscillator to uniquely determine its motion. It is possible to write down other “equations of motion” that exhibit wave phenomena but which only require the initial values of the dynamical variable — not its time derivative — to specify a solution. This is physically appropriate in a number of situations, the most significant of which is in quantum mechanics where the wave equation is called the Schrodinger equation. This equation describes the time development of the observable attributes of a particle via the wave function (or probability amplitude) . In quantum mechanics, the complete specification of the initial conditions of the particle’s motion is embodied in the initial value of . The price paid for this change in the allowed initial data while asking for a linear wave equation is the introduction of complex numbers into the equation for the wave. Indeed, the values taken by are complex numbers. In what follows we shall explore some of the elementary features of the wave phenomena associated with the Schrodinger equation.