MODULE 1 In classical mechanics we define a STATE as “The specification of the position and velocity of all the particles present, at some time, and the specification of all the forces acting on the particles.” Then Newton’s (or any other) classical equations of motion allow us to determine the state of the system at any future time. In quantum theory the specification of the state of a system at a given time is provided by its wavefunction (Postulate 1). In order to predict the state of a sub-microscopic system at another time we need a quantum equivalent of Newton’s law that tells us how the wavefunction changes with time. This is the Schrödinger equation, which we present as a Postulate.

MODULE 1 POSTULATE 3 “The wavefunction  (r 1, r 2, …, t) changes in time according to where is the hamiltonian (total energy ) operator". This is the time-dependent equation that Schrödinger introduced in Here we are postulating it.

MODULE 1 In the case where we have a single particle and the sole spatial variable is the dimension x, we can substitute for the hamiltonian by the expression found earlier, thus This time-dependent Schrödinger equation can be separated into a pair of equations, one spatial and one temporal. (See Math Module A and Barrante ch 6) time independenttime dependent

MODULE 1 Integrating the temporal equation gives and so the complete wavefunction can be written as where the constant of integration has been incorporated into the normalization constant for 

MODULE 1 The time-independent Schrödinger equation has the form of a standing wave equation and if all we are interested in are spatial dependences, then this is the one for us. If the potential energy is independent of time, and the system has energy, E, we can construct the time-dependent wavefunction from the time-independent one simply by multiplication by The Euler relationship allows us to write

MODULE 1 Thus the time dependence of  is actually a phase modulation The wave function rotates in the complex plane from real to imaginary and back to real with a frequency equal to As time proceeds, the cosine will oscillate between 0 and 1, and the sine from 1 to 0. When the cosine is 0 the sine will be 1 and the second term overall will equal –i, and the total wave function will be imaginary. When the cosine is 1 the imaginary term vanishes and the total wavefunction becomes real. Thus as time goes by the total wave function will alternately change from real to imaginary, and so on.

MODULE 1 Even though the amplitude of  oscillates between real and imaginary values, the product  is real and remains constant, since These systems are stationary states. They have a specific, precise energy (E) and the potential energy is not time dependent. Their wavefunctions flicker between the real and the imaginary, but the probability density is constant with time.

MODULE 1 We have just used our recipe for constructing the hamiltonian (total energy) operator. Now we return to the spatial Schrödinger equation and put it into the form of an operator equation the quantity in the brackets is recognized as the hamiltonian operator The spatial Schrödinger equation above is a special equation. It states that the result of carrying out the (hamiltonian) operation on the wavefunction is the function itself, multiplied by a constant, in this case the energy E. When the result of an operation yields a function that is linearly proportional to the function prior to the operation, then the function is termed an EIGENFUNCTION of the operator, and the proportionality constant is called its EIGENVALUE.

MODULE 1 The generic equation (f is some function and is an operator) is an eigenvalue equation with f an eigenfunction and a (a constant) is the eigenvalue. Thus the hamiltonian operator operates on the spatial wavefunction  in an eigenvalue equation called the (time- independent) Schrödinger equation

MODULE 1 The quantity E (the eigenvalue) is the energy observable that corresponds to the total energy operator This example has demonstrated the method for calculating any desired observable, i.e., operate on the wave function with the appropriate operator, then the eigenvalue will represent the desired quantity. Eigenvalue equations play a major role in Quantum Chemistry, but Schrödinger did not invent them; they had been around in mathematics for a long time. He adopted the technique when he developed his wave mechanics.

Planck hypothesized that an atomic oscillator could exist only in discrete energy states and as it changed from one state to another it either emitted or absorbed a quantum of energy equal to the difference between the two states. Einstein postulated that light was a quantized form of radiant energy and Bohr used the quantum concept to explain stationary angular momentum states in his theory of the H atom. However the restriction of a physical system to discrete solutions is not a property of the size of particles, per se. Submicroscopic particles can undergo motion that is not subject to quantum restrictions (more on this later). MODULE 1

Moreover, some macroscopic systems under special conditions show discrete states of motion that resemble quantization. Examples are a violin/guitar/etc string, or a stretched drum skin. These are clamped at certain points where their motion away from the rest position always necessarily has zero amplitude. When the string or skin (a 2-dim string) is set into vibrational motion the equation of motion (from Newton’s second law) is a second order differential equation that can be separated into spatial and temporal eigenvalue equations.

MODULE 1 The solutions to the spatial eigenvalue equation form a set of discrete vibrational frequencies (the fundamental and its overtones). This is an example of quantization on a macro-scale that arises because the zero amplitude fixed points force boundary conditions on the spatial equation. This restricts the allowed solutions to integral values of the fundamental frequency. It is the application of particular BOUNDARY CONDITIONS that generates quantization, not the size of the entity.