文档介绍：The Principles of Quantum
Mechanics. I. Classical Waves and
the Schriidinger Equation
494 14 The Principles of Quantum Mechanics. I. Classical Waves and the Schr6dinger Equation
Classical Mechanics
Classical mechanics is based on the laws of motion discovered by Sir Isaac Newton,
and is also called Newtonian mechanics. Appendix D presents a brief survey of
classical mechanics, which is now known to be accurate only for objects of relatively
large mass and for relatively high energies.
The Classical Mechanical Analysis of the Harmonic Oscillator
The harmonic oscillator is a model system that represents a mass suspended from a
stationary object by a spring as shown in Figure . Let the vertical coordinate z of the
mass equal zero at its equilibrium position and be positive if the mass is above this
position and negative below it. The force on a mass suspended by a spring is described
Hooke's law is named for Robert for fairly small values of z by Hooke's law:
Hooke, 1635-1703, one of Newton's
contemporaries and rivals. F z = -kz (-1)
where k is called the force constant. The larger the force constant, the stiffer the spring.
The harmonic oscillator obeys Hooke's law exactly for all values of the z coordinate.
The mass of the spring suspending the oscillator is assumed to be negligible. From
Newton's second law, Eq. (D-l) of Appendix D, the force on an object equals its mass
times its acceleration. This gives the equation of motion of the harmonic oscillator:
d2z
-kz - m dt----T (-2)
This differential equation is called linear because the variable z enters only to the first
power and is called second order because its highest-order derivative is the second
derivative.
The general solution of a differential equation is a family of functions that includes
nearly every solution of the equation. The general solution for Eq. (-2) must
contain two arbitrary constants, since this