Chapter 33 Foundations of Quantum Mechanics

1. Chapter 33 Foundations of Quantum Mechanics §1 Early Quantum Theory I. Eve of Quantum Theory Classical Physics: Newton’s Mechanics; Maxwell’s Electrodynamics; Thermodynamics and Statistical Physics, Thermal Engines  First industry revolution. Dark clouds in the sky of physics: 1. Black body radiation

2. W. Wien 1894, assuming that the frequency n • radiated from molecule is only related to the • velocity v, where a, b are constants. • L. Rayleigh (1900) & J. Jeans (1905), from energy • partition according to the degree of freedom, The black body radiation field is considered as the accumulation of oscillators of electromagnetic standing waves. Figure of black-body radiation

3. 。 。 。 。 。 。 。 。 uT(l) Rayleigh-Jeans 。 。 。 。 。 。 Wien 。 。 l (mm) 2. Photoelectric Effect When ultraviolet light shone upon the surface of some metals, electrons were emitted from the surface. Maxwell’s classical wave theory could not give a satisfactory account of the observation. According to the wave theory, (a) more intense radiation should have imparted greater acceleration to the electrons and caused them to fly off with more energy; (b) the energy should not depend particularly upon the frequency of the light as long as sufficient light intensity shines on the surface.

4. In fact, (a) higher-frequency radiation causes the electrons to fly off more energetically even if the intensity is lower; (b) below some particular frequency, no electrons escape at all; (c) ordinary visible light is incapable of ejecting any electrons from most metals, whatever intensity it shines on the surface. 3. Line Spectra and Atomic Structure Pioneers of atom-theory of substances: J. Dalton, A. Avogadro, S. Cannizzaro, … In 19th century, Gustav Kirchhoff and Robert Bunson (1850s, 1860s) made spectroscopy a primary tool of chemical analysis. Since every atom emits a characteristic line spectrum and since the spectra of no two atoms are same, the spectrum emitted by any substance provides

5. unambiguous evidence about the atomic constituents of the substance. Through spectral analysis we know much about the distribution of elements in the sun and throughout the universe. The Spectrum of Hydrogen In 1885, Johann Balmer (a Switzerland high-school teacher) discovered that the frequencies of the known lines of hydrogen could be expressed mathematically by a very simple empirical law, where H = 1.0967758×107 m-1, n is an integer greater than two. Balmer speculated that other hydrogen series might exist. We now know that all the spectral lines of

6. hydrogen are described by the formula Lyman series (1906): Paschen series (1908): Brackett series (1920):

7. Pfund series (1920): Atomic Structure J. J. Thomson (1897) discovered electron (measuring the ratio of charge-to-mass), and gave rise to the model of atoms as pudding. In those early days, the mass of electron was unknown. Thus, it was not possible even to say how many negatively charged electrons a given atom contained. Atoms were electrically neutral so they must also contain some positive charge, but nobody knew what form this compensating positive charge took.

8. E. Rutherford proposed and his collaborators H. Geiger and E. Marsden, a 20-year-old student who had not yet earned his bachelor’s degree, carried out an experiment, called a-particle scattering. They fund large angle scatterings of a particles. This could not be explained by pudding model of atoms. Nuclear Model.

9. II. Naissance of Quantum Theory Early Quantum Theory Contradiction of nuclear model with Maxwell’s classical electromagnetic theory, see VCD program. 1. Planck’s Theory of Black body radiation Planck’s Hypothesis (1900): The energy of harmonic oscillators of the black body radiation can only take following values where e0=hn, h=6.6262×10-34 kg m2/s, called Planck’s constant.

10. 。 。 。 。 。 。 。 Rayleigh-Jeans 。 uT(l) 。 。 。 。 。 。 Wien 。 。 l (mm) Planck’s formula: Planck first introduced the concept of quantization of energy, which is now referred as the naissance of quantum theory. Planck was awarded the Nobel prize in physics in 1918. 2. Photon, the Photoelectric Effect In 1905, A. Einstein, a patent clerk in Berne, introduced a concept of photon to explain the phenomenon of the photoelectric effect.

11. The stopping potential Vstop , at which the reading of meter A has just dropped to zero. When V=Vstop , the most energetic ejected electrons are turned back just before reaching the collector C. In this case we have Kmax does not depend on the intensity of the light source. (b) The cutoff frequency

12. If the frequency of light is below the cutoff frequency n0 , or the wavelength is greater than the corresponding cutoff wavelength l0=c/n0 , the photoelectric effect does not occur.

13. Einstein’s Explanation of Photoelectric Effect Einstein extended the concept of quantization of energy of Planck , introduced the concept of photon, whose energy has a form E=hn. Photons collide with electrons in metal, and part (or all) energy may be absorbed by electrons, so that the electrons can eject from the metal. His photoelectric equation is where F is the work function of the target, which depends on the properties of the target metal. From this equation, the kinetic energy of photoelectrons depends on the frequency of light, but not on the intensity of light. However, according to the wave theory of electromagnetic field, the energy of light is proportional to the intensity of light. This does not agree with the experimental observation.

14. Due to the contribution of explanation to the photoelectric effect, Einstein was awarded the Nobel prize in physics in 1921. Einstein’s theory of the photoelectric effect was further confirmed by accurate experiments of R. Milikan in 1916. Due to the contributions to the measurement to the charge of electron and to the photoelectric effect, Milikan was awarded Nobel prize in physics in 1923. 3. Bohr’s Atomic Theory Niels Henrik David Bohr was born in Copenhagen in 1885, the year Balmer published his formula on the hydrogen spectrum. His father was a professor of physiology at the University of Copenhagen, his brother Harrald became a distinguished mathematician, and his son Aghe Bohr won a Nobel prize in physics.

15. In 1911, a few months with J. J. Thomson at Cambridge; then he went to Manchester to join Rutherford’s group. This was about the time that Rutherford had proposed his planetary model of the atom, and Bohr was aware of the difficulties inherent in that model. Bohr’s Postulations: 1. Electrons in atoms can exist in certain discrete orbits with values of radii, r1 , r2 , r3 , …, rn with corresponding energies in those orbits E1 , E2 , E3 , …, En . Such a state is called a stationary state. These means the energy of states, the orbits , as well as the angular momentum Ln in an atom are quantized. • Atoms do not radiate when they are on the stationary states. The radiation happens only when an atom undergoes sudden transition (quantum jump), from one stationary state to another.

16. 4 a c 3 2 d b 1 3. Submicroscopic Energy Conservation The law of energy conservation remained rigorously valid. For the jumps in the figure, the energy conservation requires 4. The Correspondence Principle The classical mechanics should approximately correctly describe an atomic transition when the transition takes place between adjacent stationary states that differ very little energy, vibration frequency, and other properties. In other words, quantum mechanics must have a “classical limit”.

17. From the Coulomb’s law and the quantum postulations Bohr obtained

18. §2 Foundations of Quantum Mechanics De Broglie Matter Waves In 1924 Louis de Broglie postulated that any particles of momentum p has associated with it a wave of wavelength l and that the two are related by De Broglie’s prediction of the existence of matter waves was first verified experimentally in 1927, by C. J. Davisson and L. H. Germer of the Bell Telephone Laboratories and by George P. Thomson of the University of Aberdeen in Scotland.

19. The wave function y expresses the amplitude of the probability of particles. The probability density is given by Interference pattern by a beam of electrons in a two-slit interference experiment. Matter waves, like light waves, are probability waves. The approximate numbers of electrons are 7, 100, 3000, 20 000, and 70 000.

20. An experimental • arrangement used to • demonstrate, by • diffraction technique, • the wavelike character • of the incident beam. • (b) x-ray beam • (c) an electron beam • (matter wave). Waves and Particles Duality of microscopic world Superposition Principle of Wave Function

21. If you substitute the quantities by their operator forms, you have What does it mean? The quantities as operators should act on a wave function, that is This is known as the famous Schrödinger equation. For one dimensional case,

22. satisfies equation Solution of the Schrödinger equation can be given by or where which is called the Hamiltonian (operator).

23. Operational Expresses of Mechanical Quantities and Schrödinger Equation In quantum mechanics, mechanical quantities are described by operators, such as energy, momentum, and angular momentum, and so on. The classical energy:

24. substituting into the equation Stationary Solutions Separating variables:

25. Mean Value of a Mechanical Quantity and Heisenberg’s Uncertainty Principle In quantum mechanics, an instantaneous measurement has no any significant meaning. An significant measurement is the mean value of this quantity. For mechanical quantity Â, its mean value is defined by For example, the average energy of a system is given by For any given values of position x0, and momentum p0 ,

26. the mean square deviations of x and p are The Heisenberg’s uncertainty is

27. Commutation Relations (对易关系) In quantum mechanics, because the mechanical quantities are expressed by operators, they have different algebraic relations. Let’s see the relations of position and momentum. or

28. If we define a commutator (对易子) for any two operators , So the commutator for the position x and momentum px we have By the similar way, one can show that Commutators of positions and momenta in different directions are also commutative.

29. §3 One Dimensional Infinite Potential Well or This is an oscillation equation. Its solution is

30. Continuity of wave function and Therefore,

31. The wave function becomes Normalization condition: CONNECTION

32. Here, , ( n=1, 2, …)} are a set of eigenfunctions of the system. Any state of the system can be expressed or expanded by these eigenfunctions, From previous slide we have The expansion coefficients cnhas a property: |cn|2 expresses the probability of measuring the particle in state yn(x). It is easy to see that, for the Hamiltonian Ĥ, following equation holds:

33. The equation is known as eigen equation of energy, and En are called the eigenvalues of energy. The mean value of measuring the energy is given by The denominator in the equation expresses the normalization.

34. Quantum jump in the well A quantum jump (or transition) happens when the electron in the well absorbs a photon hn when it is from a lower level to an excited level, or emits a photon when it from a higher level to a lower level.

35. Probability of Detection or To find the probability that the electron can be detected in any finite section of the well—say, between point x1 and point x2—we must integrate p(x) between those points, such as

36. Example: A particle with mass m is confined in a one dimensional potential well 0≤x≤ a, the initial wave function at t = 0 is Then, • What is the wave-function in the time t0? • What is the average energy from t = 0 to t = t0? • At t = t0 , what is the probability finding the particle • in the left half potential box (0≤x≤ a/2)?

37. Solution: (a) According to the equation and So we have

38. The wavefunction at any time t is given by It can be seen that the probabilities in states y1 and y2 are |c1|2= 4/5 and |c2|2= 1/5, respectively. (b) • According to the statistical explanation of wave- • function, the probability between 0a/2 is given by

39. By substituting the expressions of c’s and y’s, we have

40. I II III §4 An Electron in a Finite Well Outside the well the potential energy of an electron is U0 which is called the well depth. In this case the Schrödinger equation is in the form,

41. where Region I: Region II: Region III: The solutions are in the forms

42. Without the detailed discussion, the probability densities can be shown by figures, L=100 pm, U0 = 450eV

43. Nanocrystallites Quantum Dots and Quantum Wells size~ Quantum Corrals (using STM to manipulate atoms) connection

44. Using , we have §5 One Dimensional Harmonic Oscillation 1. Hamiltonian The potential of a harmonic oscillator is given by The Hamiltonian is given by

45. â If we introduce a pair of operators âandâ: Then the Hamiltonian can be written as It can be shown that 2. Schrödinger Equation and Its Solution

46. Making variables substituting: The Schrödinger equation can be written as Without detailed discussion, we just give the solution of one dimensional harmonic oscillator

47. where Hn is known as Hermte polynomial (厄米多项式). the energy eigen values

48. A simple representation for this can be introduced known asparticle representation. Due to introducing operator Its eigen states defined by |n> (n=0,1,2,3,…), so that Following commutation relations can be shown easily

49. â and â are called annihillation operator and creation operator, respectively. They act on the states |n> as follows |0> is defined as vacuum state, â|0>=0, â |0> =1|1> , so that The Hamiltonian acts on the states |n> leading to

50. Assignments (Friday, June 14): 39 — 49, 68 Assignments (Monday, June 17): 40 — 16P, 22P