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Chapter 33 Applications of the Schrö dinger Equation

Main points of Chapter 33 • Schrö dinger Equation • One dimensional potential well • Barrier penetration

• The harmonic oscillation

• Orbital angular momentum and spin of the electron in the atom • Quantum theory of the hydrogen atom • Multi-electron atoms and the exclusion principle

“„Quantum mechanics‟ is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen.” --Richard P. Feynman

A professor of theoretical physics, the Nobel Prize in Physics 1965

33-1 Schrö dinger Equation • We assumed that the state of a particle is described by a ―wave function‖ (or ―probability amplitude‖):

Y(x,y,z,t)

• We need a ―wave equation‖ describing how Y(x,y,z,t) behaves. It should – be as simple as possible – make correct predictions – reduce to the usual classical laws of physics when applied to ―classical‖ objects (e.g., baseballs)

We need an equation that tells us exactly how the particle’s wave function, Y(x,y,z,t), changes in space and time…

In 1926, Erwin Schrö dinger proposed an equation that described the time- and space-dependence of the y wave function for matter waves (i.e., electrons, protons,...) The Schrö dinger Equation (SEQ)

The Schrö dinger Equation (SEQ) cannot be derived. Its validity lies in its agreement with experiment.

Development of the wave equation Wave-particle duality

If we take the form for the wave function of a free particle as a harmonic wave traveling in x-direction

the wave equation for Ex

Extend the free-particle wave equation to include the effects of external forces acting on the particle

In potential field

Hamiltonian operator momentum operator

time-dependent SEQ

Generalize for 3-dimension

If the potential is stationary can be separated, let

, the equation

E separation constant

The solution of the first equation is The second equation is

time-independent SEQ It can be written as

Wave function

the probability density |y | 2 associated with the particle

does not change with time…. it is in a ―stationary state‖

time-independent SEQ Eigenvalue equation

KE term

PE term

Total E term

E is the energy of the particle The solution depends on the form of the potential energy

“All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics...It has survived all tests and there is no reason to believe that there is any flaw in it….We all know how to use it and how to apply it to problems; and so we have learned to live with the fact that nobody can understand it.” --Murray Gell-Mann

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