Quantum Mechanics: Wave Theory of Particles

1. 1 Quantum Mechanics: Wave Theory of Particles Dr. Bill PezzagliaQM Part 2 Updated: 2010May11

2. 2 Quantum Mechanics • Bohr Model of Atom • Wave Nature of Particles • Schrodinger Wave Equation

3. 3 A. Bohr Model of Atom • Bohr’s First Postulate • Electron orbits are quantized by angular momentum • Orbits are stable, and contrary to classical physics, do not continuously radiate • Principle Quantum number “n” (an integer whose lowest value is n=1) Niels Bohr 1885-1962 1922 Nobel Prize

4. 4 1. Bohr’s First Postulate (a) Quantized Angular Momentum • 1912 first ideas by J.W. Nicholson • Postulates angular momentum of electron in atom must be a multiple of

5. 5 1. Bohr’s First Postulate (b) Stationary Orbits • Classical physics says accelerating charges (i.e. electrons in circular orbits) should radiate energy away, hence orbits decay. • Bohr says orbits are stable and do not radiate • Principle quantum number “n” has a lowest value of n=1 (lowest angular momentum of one h-bar).

6. 6 (c) The Bohr Radius • Classical equation of motion • Substitute: • Solve for radius: • Bohr Radius:

7. 7 2. Bohr’s Second Postulate (a) The sudden transition of the electron between two stationary states will produce an emission (or absorption) of radiation (photon) of frequency given by the Einstein/Planck formula

8. 8 (b) Energy of nth orbit • Viral Theorem: For inverse square law force: • Hence total energy: • Use Electrostatic energy formula, we get:

9. 9 (b) Energy of nth orbit • Substitute Bohr’s radius formula for n-th orbit gives energy of nth orbit: • Where he can calculate Rydberg’s constant from scratch!

10. 10 (c) Bohr Derives Balmer’s Formula • From Einstein-Planck Formula: • Substituting his energy formula (and divide out factor of hc), he derives Balmer’s formula!

11. 11 3. Bohr’s Correspondence Principle • 1923: Classical mechanics “corresponds” to quantum system for BIG quantum numbers. • When “n” is big, it behaves classically • When “n” is small, it behaves “quantumly” (is that a word?)

12. 12 B. Wave Nature of Particles • deBroglie Waves • Particle in a Box • Heisenberg Uncertainty

13. 13 1. deBroglie Waves (1924) • Suggest particles have wavelike properties following same rules as photon. • Proof: 1927 Electron diffraction experiment of Davisson & Germer (Nobel Prize 1937)

14. 14 (b) deBroglie’s Bohr Model • Bohr’s model had an ad-hoc assumption that orbits had quantized angular momentum (multiples of h-bar) • deBroglie postulates that only “standing waves” can yield stationary orbits, i.e. circumference must be multiple of the wavelength • Hence allowed momentums are: • Or angular momentums mustbe quantized:

15. 15 1c. Phase Velocity • Velocity of waves are FASTER than light Where “v” is the classical speed of the particle (aka “group velocity”)

16. 16 (d) Interpretation • deBroglie thought that the “wave” of a particle had two aspects. • The “group velocity” described the localized “particle” nature of the classical particle • The “phase velocity” was associated with the “pilot wave” which traveled ahead and behind the particle (faster than light), sensing the environment.

17. 17 2. Particle in a Box • Standing wave patterns • Analogous to waves on a string with fixed ends. • Momentum hence is quantized to values:

18. 18 2. Particle in a Box • Energy is hence quantized to values: • The particle can never have zero energy! The lowest is n=1 • The smaller the box, the bigger the energy. If wall is height “z”, for small enough “L”, the particle will jump and escape!

19. 19 2c. Wavepackets & Localization • A wave is infinite in extent, so the “electron” is not localized. • The superposition of waves of slightly different wavelengths will create a “localized” wavepacket, which roughly corresponds to classical particle • But now it does not have a single momentum (wavelength); it has a spread of momenta, and the packet will tend to spread out with time.

20. 20 3a. Heisenberg QM • 1925 First formulation of “quantum mechanics” which correctly describes energy levels and quantum jumps. • It’s a mathematical theory, which assumes that position and momentum do not commute:

21. 21 3b. Heisenberg Uncertainty • “principle of indeterminacy” • “The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.” • 1927 Uncertainty Principle (which can be derived from [x,p]=ih …)

22. 22 C. Wave Mechanics • More Quantum Numbers • Pauli Exclusion Principle • Schrodinger Wave Mechanics

23. 23 1. Zeeman Effect (1894) (a) Zeeman effect: splitting of spectral lines due to magnetic fields, shows us sunspots have BIG magnetic fields

24. 24 1b. Bohr Sommerfeld Model 1916 use elliptical orbits to different energies (new quantum number “l”). Also, quantumnumber “m” todescribe orientation,where if l=2, mcould be{-2,-1,0,1,2}

25. 25 1c. Bohr’s Periodic Table 1921 uses quantum numbers to explain periodic table (Pauli’s contribution is that each state has 2 electrons in it, another quantum number)

26. 26 2. Pauli Spin • 1924 proposes new quantum number to explain “Anomalous Zeeman Effect” where “s” orbits split into 2 lines. • 1925 Uhlenbeck & Goudsmit identify this as description of “spin” of electron, which creates a small magnetic moment • 1927 Pauli introduces idea of “spinors” which describe spin half electrons • Famous quote: when reviewing a very badly written paper he criticized it as “It is not even wrong”

27. 27 2b. Pauli Exclusion Principle (1925) • Serious Question: Why don’t all the electrons fall down into the first (n=1) Bohr orbit? • If they did, we would not have the periodic table of elements! • Exclusion Principle: Each quantum state can only have one electron (e.g. 1s orbit can have two electrons, one with spin up, other with spin down)

28. 28 2c. Fermions & Bosons • Fermions, which have spin ½ (angular momentum of h/4) obey the Pauli exclusion principle (e.g. electrons, neutrinos, protons, neutrons, quarks) • Bosons, which have integer spin, do NOT obey the principle (e.g. photons, gravitons). • This is why we can have “laser” light (a bunch of photons with their waves all in phase).

29. 29 3. Schrodinger 1926 Bohr & Heisenberg’s quantum mechanics used abstract mathematical operations (e.g. x and p don’t commute) • Schrodinger writes a generalized equation that deBroglie waves must obey when there is Potential Energy (such that the wavelength changes from point to point in space)

30. 30 3b Electron Orbits • S orbits hold 2 electrons • P orbits hold 6 electrons • D orbits hold 10 electrons

31. 31 Electron Configurations • Bohr’s Aufbau (build up) Principle: Fill orbits of lowest energy first (e.g. the n=1 orbit before the n=2 orbit) • Madelung Rule: for states (n,l), the states with lower sum “n+l” are filled first (because they have lower energy). For example, 4s (4,0) would be filled before 3d (3,2). • Hund’s Rules (Bohr’s assistant)

32. 32 Madelung Rule

33. 33 Hund’s Rules • Rule of Maximum Multiplicity: maximize the spin (e.g. put one electron into each of the three p orbits with spins parallel, i.e. maximize unpaired electrons). • For a given multiplicity, the term with the largest value of L (orbital angular momentum), has the lowest energy • The level with lowest energy (where J=L+S) • Outer shell Less than half filled: minimum J • Outer shell more than half filled: maximum J

34. 34 3c. Max Born • 1924 coins the term “Quantum Mechanics” • 1925 helps with Heisenberg’s matrix form of quantum mechanics • 1928 The square of the quantum wave is proportional to the probability of finding the particle at that position. • Hence you can think of the quantum wave as having a “classical” probability density , and an “imaginary” quantum phase part.

35. 35 References/Notes • McEvoy & Zarate, “Introducing Quantum Theory” (Totem Books, 1996) • http://www.aip.org/history/heisenberg/p08.htm (includes audio !) • http://www.uky.edu/~holler/html/orbitals_2.html • http://www.meta-synthesis.com/webbook/30_timeline/lewis_theory.php