PHYS1220 – Quantum Mechanics

1. PHYS1220 – Quantum Mechanics Lecture 1 August 20, 2002 Dr J. Quinton Office: PG 9 ph 49-21-7025 phjsq@alinga.newcastle.edu.au

2. Early Quantum Theory Physics circa 1900 The Revolution in Physics Blackbody Radiation Photoelectric Effect Compton Effect Pair Production Wave-Particle Duality de Broglie’s Hypothesis Early Atomic Models Thompson Rutherford Bohr Correspondence Principle Quantum Mechanics Wave functions Quantum Mechanics Schrödinger Equation Heisenberg Uncertainty Principle Particle in a box Infinite Potential Well Finite Potential Well Barrier potential Electron Tunnelling Applications of Quantum Mechanics PHYS 1220 – Quantum Mechanics

3. The Success of Classical Physics • At the turn of the 20th Century, it was thought that physics had just about explained all natural phenomena. • The known fundamental gravitational, electric and magnetic forces were quite well understood and (successful!) theories existed to describe them. • During the preceding 3 centuries (~1600-1900) • Newtonian Mechanics • Forces and motion of Particles, fluids, waves, sound • Universal theory of gravity • Maxwell’s Theory of Electromagnetism (EM) • Unified electric and magnetic phenomena • Thoroughly explained electric and magnetic behaviour • Predicted existence of electromagnetic waves • Thermodynamics • Thermal processes • Kinetic theory of gases and other materials

4. IR 700nm 600nm 500nm 400nm Wave Theory of Light - Classical Physics • Light is an electromagnetic wave, produced by accelerating charges (Maxwell) • Electromagnetic Spectrum UV

5. Nature of Light - Classical Physics • Light propagates by mutual induction of orthogonal electric and magnetic fields (without the need for a medium, ie aether) • We know velocity (in free space) from wave theory 299792458 m/s (exact)

6. The Birth of Modern Physics • ~ 1900 – only a few phenomena were not fully understood, and were not explainable using then-known principles. • The spectrum of light emitted by hot objects • “Light electricity” (Hertz, 1887; Hallwarchs 1888) • Hydrogen emission spectrum (Balmer, 1885) • X-rays (Roentgen, 1896) • Cathode Rays, discovery of electron (J.J. Thomson 1895-97) • Radioactivity (Becquerel 1896, Marie and Pierré Curie 1898) • a, b and g radiation • The big question - “What is the structure of the atom?” • However, attempts to explain these led to a revolution in physics during the early part of the 20th century, primarily due to the emergence of two new theories. • Quantum Theory & Relativity • We will be discussing Quantum Mechanics from its beginnings

7. Blackbody Radiation • Recall Stefan-Boltzmann law (1879, 1884) • Describes energy dissipated through radiation • Stefan-Boltzmann constant s=5.67x10-8 W/m2.K4 • The emissivity (0<e<1) is a measure of the materials’ ability to emit (and absorb) radiation • For very black surfaces, e is close to 1 • For bright, shiny surfaces, e is closer to zero • A blackbody is the theoretical name for the ‘ideal’ case • All radiation that falls upon it is absorbed • Emissivity e=1 • A cavity is the closest real approximation • Perfect absorbers are perfect emitters • All thermal energy is converted to radiation • A reasonable approximation for crystalline solids, most liquids, many gases

8. Blackbody Emission Spectrum • Total Intensity increases with T • Peak wavelength moves to shorterwavelengths with increasing T • illustrates that the apparent colourof an object depends on its temperature. • Question: What is the colourprogression (with increasing T)for incandescent materials? Wien’s Law • where lp is the wavelength at the peak of the spectrum • Wien was awarded the 1911 Nobel Prize in Physics for this work.

9. Example - The Solar Spectrum • Question: The solar radiation spectrum possesses a maximum intensity at a wavelength of ~ 502nm (visible, green!). Assuming that ‘Sol’ is a blackbody, calculate its approximate ‘surface’ temperature in degrees Celcius. • Answer: Using Wien’s law • And so converting to degrees Celsius

10. Rayleigh-Jeans Theory Rayleigh-Jeans Law Experimental Data Theory Development • I(l,T)dl is the radiated power/area in wavelength interval dl • Radiation results from oscillating charges (due to molecular vibrations) within the material • Full classical treatment led Lord Rayleigh and J. Jeans to where kB = 1.381x10-23 J/K is the Boltzmann constant • fits data well for long wavelengths • major disagreement at short wavelengths • Limit as l  0, I(l,T)   • Energy density should become infinite for short wavelengths • Known in scientific folklore as the “The Ultraviolet Catastrophe”

11. Planck’s Law Planck’s Approach • Planck proposed an empirical formula (Dec 1900) which nicely fit the data. • The constant, h, introduced by Planck, was measured from fitting the equation to data (currently accepted as 6.626x10-34 J.s) • Example: Calculate the value of I(l,T) using the (a) Rayleigh-Jeans and (b) Planck’s theories for l=100nm (UV) and T=300K (a) I(l,T) = 2pckT/l4 = (2 x 3.14159 x 2.997x108 x 1.38x10-23J/K x 300K)/(100x10-9m)4= 7.8x1016 W.m-3 (b) I(l,T) = 2phc2/[l5(ehc/lkT-1)]= 1.6x10-189 W.m-3 • Difference is 205 orders of magnitude!

12. Planck’s Law - Implications • To produce a theory that resulted in his equation, Planck had to make a radical assumption, called Planck’s Quantum Hypothesis • Oscillating charges possess quantised (or discrete) energies, related to the oscillation frequency (cf. acoustic modes of strings and pipes) is referred to as the quantum of energy. • Planck (and everyone else) didn’t believe this to be the ‘real’ story • Merely a mathematical tool to “get the right answer” • Continued looking for a theory based on classical approaches • Won the 1918 Nobel Prize in Physics for this work • Question: Is Planck consistent with Wien and Stefan-Boltzmann? • Tutorial Exercise: Giancoli Chapter 38, problem 7.

13. Photoelectric Effect • Hertz (1887) observed that light can produce electricity • After receiving energy from the incoming light, electrons are ejected from the surface of a metal • Light strikes the photocathode (P) and ejects electrons, which get accelerated to the collector (C). • The applied potential V creates an accelerating electric field between the collector and the Photocathode • If the metal is continually illuminated, a steady state current is produced and can be read at the ammeter. • The photoelectron current increases with light intensity

14. Photoelectric Effect • If the polarity of the voltage source is reversed and the potential varied, the maximum KE of the electrons may be measured. • When the current goes to zero, i.e. no electrons make it to the collector, the maximum KE of all emitted electrons is given by: • V0 is called the stopping potential • Experiments by Lenard (1902) showed that KEmax is linearly dependent on light frequency! • constant of proportionality = h !! • A ‘cut-off’ frequency, f0 exists. Below this, no current will be produced, regardless of the incident light intensity

15. Predictions of Classical Wave Theory • The electric field of an EM wave can exert a force on electrons in the metal and eject some of them • Light has two important properties • Intensity • Wavelength (or frequency) • If the light intensity is increased, • Electric field amplitude is greater • number of electrons ejected (and measured current) increases • kinetic energy (and KEmax) of ejected electrons increases • If the frequency of the light is increased, • Nothing should happen. The kinetic energy of photoelectrons should be independent of the incident light frequency • A time delay should exist before electrons are emitted • The energy required to remove electrons will need to build up   

16. Einstein’s Corpuscular Theory of Light • Albert Einstein proposed the following (1905) • Light ‘quanta’ possess a corpuscular nature • Energy is related to frequency and wavelength bywhere h is Planck’s constant • The KE of an emitted electron is given by where W is the energy required to remove that electron from (the surface of) the material • If the light frequency is below f0 , then no electrons will be emitted (no matter how great the intensity) • The minimum energy required to eject electrons from the material is called the work function, W0 and is related to the cut-off frequency (and KEmax) by • More intensity → more quanta → more electrons • Ejection of the first electron should be instantaneous • Einstein won the 1921 Nobel Prize in Physics principally for this work ü ü ü

17. Work functions of Materials • The work function of a metal is typically ~ a few eV Source: V. S. Fomenko, Handbook of Thermal Properties, G. V. Samsanov, ed., Plenum Press Data Division, New York, 1966. (Values given are the author’s distillation of many different experimental determinations)

18. Example • What is the energy of near infrared light of wavelength 1mm? • A photocell made from Tungsten has a work function of 4.50 eV. Calculate the cut-off frequency. • If light of wavelength 10nm (UV) is used to illuminate the surface, what is the maximum kinetic energy of emitted electrons? • What is the stopping potential?

19. Applications of the Photoelectric Effect • Photonic switches, burglar and smoke alarms • The phototube acts much like a switch in an electric circuit. • Photodiodes and light dependent resistors (LDRs) and are modern equivalent to phototube • IR detectors, such as remote controls, etc • Light meters • Photosynthesis • Optical sound track on movie film • The first lasers (optically pumped) • X-ray Photoelectron Spectroscopy (XPS) is used for chemical analysis by obtaining elemental fingerprints of material surfaces • And many others • The photoelectric effect dominates interactions between light (near IR-soft X-rays) and matter