Atkins Physical Chemistry Eighth Edition

Chapter 8 Quantum Theory:

Introduction and Principles

Copyright 2006 by Peter Atkins and Julio de Paula

Peter Atkins Julio de Paula

Topics

Blackbody radiation

Limitation of classical mechanics for atomic systems

Wave-particle duality

Photoelectric effect

Schrdinger equation

Free particle wavefunction

Operators

Uncertainty principle

Figure 1.1

Blackbody radiation

Energy distribution

of blackbody radiation

Prediction of classical

mechanics agrees with

observation at low

frequencies (high wavelengths) but fails

at high frequencies (small wavelengths)

Blackbody radiation

Plancks distribution

Heat capacities of solids

Classical mechanics cannot accurately predict the temperature dependence of heat capacities of solids

Atomic and molecular spectra are composed of discrete lines

Photoelectric effect

Classical theory

Electrons are emitted at all frequencies, provided the light intensity is high

Kinetic energy of the electrons increases with intensity of light

Experiment

# of emitted electrons depend on the light intensity but not their kinetic energy

No electrons emitted unless the frequency of the light exceeds a threshold value

Kinetic energy of the ejected electrons depend on the frequency of the incident light

Electrons are emitted even at low intensities if the frequency exceeds a threshold value

http://hyperphysics.phy-astr.gsu.edu/hbase/imgmod2/pelec.gif

Davisson-Germer Experiment: established wave nature of electrons

The experiment (1927) involved scattering of electrons from a single crystal of Ni

Diffraction patters exhibited wave behavior

Landmark experiment that showed wave nature of matter

Confirmed the prediction of de Broglie three years ago

Source: google images

Wave-particle duality

Behavior of matter at temperatures close to absolute zero

Formation of Bose-Einstein condensates

400 nK 200 nK 50 nK

http://cua.mit.edu/ketterle_group/Animation_folder/BEC_phase_transition.htm

Wavefunctions describing systems with high kinetic energies (high momenta) are highly oscillatory

Real and imaginary part of a plane wave describing motion of a free particle

Square of the wave function denotes probability of finding the particle in a given region

Volume element for integration of the wave function in three dimensions, d=dxdydz

Wave function is an amplitude function: it may have positive an negative amplitudes but the square of the wave function, ||2 (probability) is always a positive quantity.

Volume element in spherical polar coordinates d=dxdydz=r2drsindd

Acceptable wave function must be finite, single-valued and continuous

The state of a quantum mechanical system completely specified by its wave function . The square of the wave function, (x)*(x) give the probability of finding the particle at a location x.

For every measurable property, there exists a corresponding operator. An experiment in the lab to measure the value of the property is equivalent to operating the corresponding operator on the wave function of the system.

In any single measurement of the observable corresponding to an operator, the only value that ever will be measured is the eigen value of the operator

The average value of an observable is the expectation value of its operator:

= * d

Quantum mechanical postulates