(1) Experimental evidence shows the particles of microscopic systems moves according to the laws of wave motion, and not according to the Newton laws of motion obeyed by the particles of macroscopic system. (2) Planck’s thermal radiation : discrete energy level. (3) de Broglie wave: connect particle momentum and wavelength by Planck constant (4) A theory is needed to treat more complicated cases: Schroedinger’s theory of quantum mechanics. 5.1 Introduction Chapter 5 Schroedinger theory of quantum mechanics The Schroedinger equation is a partial differential equation has a solution. The equation may include

Ex : Chapter 5 Schroedinger theory of quantum mechanics 5.2 Plausibility argument leading to Schroedinger equation The reasonable assumption concerning about the wave equations: (1) de Broglie-Einstein relation: (2) total energy: (3) linear wave function : (4) potential energy :

Chapter 5 Schroedinger theory of quantum mechanics Schroedinger wave equation

5.3 Born’s interpretation of wave functions Max Born (1926): complex wave function probability density P(x,t)dx is the probability that the particle with wave function Ψ(x,t) will be found at a coordinate between x and x+dx. Chapter 5 Schroedinger theory of quantum mechanics Classical wave theory: Wave function is a real function.

Chapter 5 Schroedinger theory of quantum mechanics Ex: (1) Evaluate the probability density for the simple harmonic oscillator lowest energy state wave function (2) Evaluate the probability density of S.H.O. in classical mechanics. Q.M. C.M. In C.M., no uncertainty principle is an error.

Ex: Normalize the wave function of S.H.O. expressed as Chapter 5 Schroedinger theory of quantum mechanics

5.4 Expectation values

Chapter 5 Schroedinger theory of quantum mechanics Momentum and Energy operators:

Chapter 5 Schroedinger theory of quantum mechanics Momentum expectation value Energy expectation value

Ex: Consider a particle of mass m which can move freely along the x axis between two walls at x=-a/2 and x=+a/2, and the particle can not penetrate the two walls. Try to find the wave function of the particle and energy. Chapter 5 Schroedinger theory of quantum mechanics

eigenfunctioneigenvalue

Chapter 5 Schroedinger theory of quantum mechanics Eigenvalue equation Hamiltonian or total energy operator Uncertainty principle

5.5 The time-independent Schroedinger equation Chapter 5 Schroedinger theory of quantum mechanics time-independent Schroedinger equation eigenfunction wave function

Chapter 5 Schroedinger theory of quantum mechanics 5.6 Required properties of eigenfunctions must be finite must be finite must be single valued must be single valued must be continuous must be continuous (1)Physical measurable quantities, e.g., p, x, are all finite and single- valued, so are finite and single-valued. (2) is finite, it is necessary is continuous. (3)For finite V(x), E and, must be continuous.

Chapter 5 Schroedinger theory of quantum mechanics Ex: When a particle is in a state such that a measurement of its total energy can lead (1) only to a single result, the eigenvalue E, it is described by the wave function (2) two results, the eigenvalue wave function is What are their probability density? oscillating frequency of probability density independent of time