Quantum Mechanics. Reilly V. Bautista. September 5, Reilly V. Bautista Quantum Mechanics September 5, / 78

Transcription

1 Quantum Mechanics Reilly V. Bautista September 5, 2016 Reilly V. Bautista Quantum Mechanics September 5, / 78

2 Contents Properties of electromagnetic radiation Issues with classical model of physics Atoms for physics Wave nature of matter Probabilistic nature of matter and Schr odinger's Equation Quantum Systems Reilly V. Bautista Quantum Mechanics September 5, / 78

3 Properties of electromagnetic radiation Reilly V. Bautista Quantum Mechanics September 5, / 78

4 Properties of electromagnetic radiation ) B d dφ l E = µ 0 (i c + ɛ 0 dt encl E d l = dφ B dt E da = Q encl ɛ 0 (1) B da = 0 (2) (3) (4) Reilly V. Bautista Quantum Mechanics September 5, / 78

5 Properties of electromagnetic radiation The magnitudes of E and B are related by this equation: The speed of electromagnetic radiation in vacuum is given by: E = cb (5) c = 1 ɛ0 µ m/s (6) Reilly V. Bautista Quantum Mechanics September 5, / 78

6 Properties of electromagnetic radiation Reilly V. Bautista Quantum Mechanics September 5, / 78

7 Properties of electromagnetic radiation The intensity I of an electromagnetic radiation emitted at rate P follows inverse-square law: I = The radiation pressure p rad is related to intensity I: P 4πr 2 (7) p rad = I c p rad = 2I c (radiation absorbed) (8) (radiation reected) (9) Reilly V. Bautista Quantum Mechanics September 5, / 78

8 Properties of electromagnetic radiation Frequency ν and wavelength λ are related: c = νλ (10) The electromagnetic spectrum is divided to following parts: Region Wavelength Gamma ray λ 1 nm X-ray 1 nm λ 10 nm Ultraviolet 10 nm λ 400 nm Visible 400 nm λ 700 nm Infrared 700 nm λ 1 mm Microwave 1 mm λ 10 cm Radio 10 cm λ Reilly V. Bautista Quantum Mechanics September 5, / 78

9 Properties of electromagnetic radiation A blackbody absorbs all of the incoming radiation (ideal absorber) and emits all of the radiation (ideal emitter) The wavelength λ of the radiation emitted by an object with temperature T is related by Wien's displacement law: λt = m K (11) Reilly V. Bautista Quantum Mechanics September 5, / 78

10 Properties of electromagnetic radiation The luminosity L is related to the temperature T of the object (Stefan-Boltzmann law) L = AσT 4, σ = W /m 2 K 4 (12) Reilly V. Bautista Quantum Mechanics September 5, / 78

11 Properties of electromagnetic radiation Since electromagnetic radiation has wave nature, the following phenomena exist: Law of Reection: the angle of reection is equal to angle of incidence θ r = θ i (13) Snell's Law n 1 θ 1 = n 2 θ 2 (14) Reilly V. Bautista Quantum Mechanics September 5, / 78

12 Properties of electromagnetic radiation Since electromagnetic radiation has wave nature, the following phenomena exist: Diraction: light spreads upon encountering a slit or an edge Reilly V. Bautista Quantum Mechanics September 5, / 78

13 Properties of electromagnetic radiation Since electromagnetic radiation has wave nature, the following phenomena exist: Interference: two or more waves interact with each other, overlapping, resulting to changes in their amplitudes Reilly V. Bautista Quantum Mechanics September 5, / 78

14 Issues with classical model of physics Classical model of blackbody radiation does not match observations at frequencies approaching ultraviolet range, thus ultraviolet catastrophe Reilly V. Bautista Quantum Mechanics September 5, / 78

15 Issues with classical model of physics Solution: Max Planck came up with an equation that ts with observation: B λ (λ, T ) = 2hc 2 λ 5 1 e hν λk B T 1 (15) where h = J s and k B = J K Planck arrived at the equation by assuming that the radiation comes from atomic oscillator system Reilly V. Bautista Quantum Mechanics September 5, / 78

16 Issues with classical model of physics Planck's Law requires that the energy of an oscillator can have only certain discrete values, implying quantization of energy E = nhν (16) Reilly V. Bautista Quantum Mechanics September 5, / 78

17 Issues with classical model of physics Certain materials emit electrons when illuminated by light (photoelectric eect) The resulting photocurrent depends on frequency, not on intensity There is no measurable time delay between absorption of light and emission of electrons The resulting potential dierence depends on frequency, not on intensity Reilly V. Bautista Quantum Mechanics September 5, / 78

18 Issues with classical model of physics Einstein adapted Max Planck's postulate but generalized it: E = hν (17) He argued that energy itself is quantized, that is, divided into small bundles, particles called photons Reilly V. Bautista Quantum Mechanics September 5, / 78

19 Issues with classical model of physics Photoelectric eect is described by the following equation: ev 0 = hν φ (18) where V 0 is stopping potential, e is charge of electron, h is Planck's constant, ν is frequency of light, and φ is work function, minimum energy needed to remove electron Reilly V. Bautista Quantum Mechanics September 5, / 78

20 Issues with classical model of physics Compton scattering occurs when a light wave interacts (collides) with a particle, leading to scattering. If light is a wave, the scattered light will have same frequency and wavelength as incident light the scattered light will be scattered in a variety of directions the electron would oscillate in response to oscillating electric eld (which is one component of electromagnetic radiation) Reilly V. Bautista Quantum Mechanics September 5, / 78

21 Issues with classical model of physics Reality: in Compton scattering, the light wave suddenly decided to follow the law of conservation of momentum! Light wave loses some of its energy, transferred to the particle Light wave scatters at a direction which depends on its energy The electron moves, not oscillates Reilly V. Bautista Quantum Mechanics September 5, / 78

22 Issue with classical model of physics Compton scattering is described by the following equation: λ λ = h (1 cos φ) (19) mc Reilly V. Bautista Quantum Mechanics September 5, / 78

23 Issue with classical model of physics So, basically, light consists of a photon, with a denite energy: E = hν (20) and momentum: p = h λ (21) Reilly V. Bautista Quantum Mechanics September 5, / 78

24 A laser pointer with a power output of 5.00 mw emits red light (λ = 650 nm). (a) What is the magnitude of the momentum of each photon? (b) How many photons does the laser pointer emit each second? Reilly V. Bautista Quantum Mechanics September 5, / 78

25 While conducting a photoelectric-eect experiment with light of a certain frequency, you nd that a reverse potential dierence of 1.25 V is required to reduce the current to zero. Find (a) the maximum kinetic energy and (b) the maximum speed of the emitted photoelectrons. Reilly V. Bautista Quantum Mechanics September 5, / 78

26 Issues with classical model of physics X-rays are produced by thermionic emission of electrons from a cathode and then sending them to hit anode. As it hits the anode atoms, bremsstrahlung occurs. ev ac = hf max = hc λ min (22) Reilly V. Bautista Quantum Mechanics September 5, / 78

27 Electrons in an x-ray tube accelerate through a potential dierence of 10.0 kv before striking a target. If an electron produces one photon on impact with the target, what is the minimum wavelength of the resulting x rays? Find the answer by expressing energies in both SI units and electron volts. Reilly V. Bautista Quantum Mechanics September 5, / 78

28 Wave nature of matter Electron diraction experiment Electrons were made to pass through a double-slit setup The electrons acted like waves! Reilly V. Bautista Quantum Mechanics September 5, / 78

29 Wave nature of matter The de Broglie wavelength is dened as wavelength of matter: λ = h p = h mv = h 2mK (23) (philosophy: if light has both particle and wave nature, then matter would also have both particle and wave nature) Reilly V. Bautista Quantum Mechanics September 5, / 78

30 Find the de Broglie wavelength of a baseball of mass 0.17 kg moving at 10 km/h. Find the wavelength of an electron whose kinetic energy is 10.0 ev. Reilly V. Bautista Quantum Mechanics September 5, / 78

31 Wave nature of matter Reilly V. Bautista Quantum Mechanics September 5, / 78

32 Atoms for physics History of atomic physics: Greek age Atoms are objects of (Greek) argumentation Dalton age Atoms are real, for chemists Boltzmann age Atoms are only mathematical constructs and are not real, so kill yourself, Boltzmann 1 1 he did, in 1905 Reilly V. Bautista Quantum Mechanics September 5, / 78

33 Atoms for physics When the solid is heated, a continuous spectrum of radiation is produced When the gas is heated, a discrete line spectrum is produced Reilly V. Bautista Quantum Mechanics September 5, / 78

34 Atoms for physics Line spectrum is unique for each element and compound He was discovered this way Reilly V. Bautista Quantum Mechanics September 5, / 78

35 Atoms for physics Balmer series The wavelengths of spectral lines of Balmer series can be calculated by: ( 1 1 λ = R 1 ) nl 2 nu 2 (24) where R = m 1 Reilly V. Bautista Quantum Mechanics September 5, / 78

36 Atoms for physics Reilly V. Bautista Quantum Mechanics September 5, / 78

37 Atoms for physics Atoms should collapse! Reilly V. Bautista Quantum Mechanics September 5, / 78

38 Atoms for physics The energy of atoms are also quantized. The emitted radiation due to change in energy level is given by hf = hc λ = E i E f (25) Reilly V. Bautista Quantum Mechanics September 5, / 78

39 A hypothetical atom has energy levels at 0.00 ev (the ground level), 1.00 ev, and 3.00 ev. (a) What are the frequencies and wavelengths of the spectral lines this atom can emit when excited? (b) What wavelengths can this atom absorb if it is in its ground level? Reilly V. Bautista Quantum Mechanics September 5, / 78

40 Atoms for physics Bohr Model of Atom Postulate: The electrons in the hydrogen atom can move only in certain nonradiating, circular orbits called stationary states. Reilly V. Bautista Quantum Mechanics September 5, / 78

41 Atoms for physics Reilly V. Bautista Quantum Mechanics September 5, / 78

42 Atoms for physics If we set principal quantum number n as indicating the orbit of electron, then angular momentum L n is L n = mv n r n = n h 2π The wave nature of electron leads naturally to quantization of electron's angular momentum (26) Reilly V. Bautista Quantum Mechanics September 5, / 78

43 Atoms for physics Bohr Model of Atom Bohr radius a 0 : a 0 = ɛ 0 h 2 πme 2 = m (27) Orbital radii: Orbital speed: r n = n 2 n 2 h 2 a 0 = ɛ 0 πme 2 (28) v n = 1 e 2 ɛ 0 2nh (29) Reilly V. Bautista Quantum Mechanics September 5, / 78

44 Atoms for physics Bohr Model of Atom Energy of electron: Rydberg constant: R = E n = hcr n 2, (30) me4 8ɛ 2 0 h3 c = m 1 (31) Reilly V. Bautista Quantum Mechanics September 5, / 78

45 Find the kinetic, potential, and total energies of the hydrogen atom in the rst excited level, and nd the wavelength of the photon emitted in a transition from that level to the ground level. Reilly V. Bautista Quantum Mechanics September 5, / 78

46 Atoms for physics Bohr Model of Atom Bohr radius a 0 : a 0 = ɛ 0 h 2 πme 2 = m (32) Orbital radii: Orbital speed: r n = n 2 n 2 h 2 a 0 = ɛ 0 πme 2 (33) v n = 1 e 2 ɛ 0 2nh (34) Reilly V. Bautista Quantum Mechanics September 5, / 78

47 Atoms for physics Quantum numbers are used to describe the properties of electrons in an atom Principal quantum number n Orbital angular momentum l E n = 1 4πɛ 0 m r e 4 2n 2 2 = 13.60eV n 2, n = 1, 2, 3,... (35) L = l(l + 1), l = 0, 1, 2,..., n 1 (36) Reilly V. Bautista Quantum Mechanics September 5, / 78

48 Atoms for physics Zeeman eect Zeeman eect is the splitting of spectral lines into three when the emitting gas is placed in a magnetic eld Interaction energy Bohr magneton U = µ z B = m l e 2m, m l = 0, ±1, ±2,..., ±l (37) µ B = e 2m = ev /T = J/T (38) Reilly V. Bautista Quantum Mechanics September 5, / 78

49 Atoms for physics Zeeman eect Magnetic quantum number m l is dened from the interaction energy of Zeeman eect L z = m l, m l = 0, ±1, ±2,..., ±l (39) Reilly V. Bautista Quantum Mechanics September 5, / 78

50 Atoms for physics Selection rules for transitions to dierent levels Allowed transitions must have changes in quantun numbers in... l must change by 1 m l must change by 0 or ±1 Reilly V. Bautista Quantum Mechanics September 5, / 78

51 Atoms for physics Stern-Gerlach Experiment Reilly V. Bautista Quantum Mechanics September 5, / 78

52 Atoms for physics The Stern-Gerlach experiment conrmed quantization of angular momentum pointed to existence of half-integer angular momentum Reilly V. Bautista Quantum Mechanics September 5, / 78

53 Atoms for physics Spin Angular Momentum S = 1 2 ( ) = (40) 4 Component of spin angular momentum S z = ± 1 2 = m s, m s = ± 1 2 (41) Reilly V. Bautista Quantum Mechanics September 5, / 78

54 Atoms for physics Spin Angular Momentum Spin magnetic moment Energy µ sz = ( ) e 2m S z (42) U = µ sz B (43) Reilly V. Bautista Quantum Mechanics September 5, / 78

55 How many distinct (n, l, m l ) states of the hydrogen atom with n = 3 are there? What are their energies? Reilly V. Bautista Quantum Mechanics September 5, / 78

56 An atom in a state with emits a photon with wavelength nm as it decays to a state with If the atom is placed in a magnetic eld with magnitude what are the shifts in the energy levels and in the wavelength that result from the interaction between the atom's orbital magnetic moment and the magnetic eld? Reilly V. Bautista Quantum Mechanics September 5, / 78

57 Calculate the interaction energy for an electron in an l = 0 state in a magnetic eld with magnitude 2.00 T. Reilly V. Bautista Quantum Mechanics September 5, / 78

58 Atoms for physics Exclusion Principle No two electrons can occupy the same quantum-mechanical state in a given system. No two electrons can have the same values of all four quantum numbers. Reilly V. Bautista Quantum Mechanics September 5, / 78

59 Atoms for physics Screening eect E n = Z eff 2 (13.6eV ) (44) n2 Reilly V. Bautista Quantum Mechanics September 5, / 78

60 Probabilistic nature of matter Reilly V. Bautista Quantum Mechanics September 5, / 78

61 Probabilistic nature of matter The interference patterns produced is actually a probability distribution of position of photons. Reilly V. Bautista Quantum Mechanics September 5, / 78

62 Probabilistic nature of matter Schrodinger equation Ψ(r, t) i = 2 2 Ψ(r, t) + V Ψ(r, t) (45) t 2m x 2 Any quantum system must be mathematically described by a solution to Schrodinger equation Reilly V. Bautista Quantum Mechanics September 5, / 78

63 Probabilistic nature of matter How to measure something in a system? 1 Measure it 2 Measure it again 3 Measure it again 4 Determine the probability distribution Reilly V. Bautista Quantum Mechanics September 5, / 78

64 Probabilistic nature of matter The psi-function (wave-function) is a probability distribution of position of a particle, giving a description of the system Each wave function must be normalized; that is, its total area under the curve is equal to unity (1) Reilly V. Bautista Quantum Mechanics September 5, / 78

65 Probabilistic nature of matter The expectation value z of physical quantity z is the "most probable" value of a variable that is being described by the wave-function. Ehrenfest's theorem: expectation values obey classical laws The position and momentum wave-functions are complementary; they are mathematically connected through Fourier transform. Reilly V. Bautista Quantum Mechanics September 5, / 78

66 Probabilistic nature of matter Uncertainty principle σ x σ p 2 (46) Caveat: The uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology σ is therefore a description of precision of probability distributions Reilly V. Bautista Quantum Mechanics September 5, / 78

67 Probabilistic nature of matter Correspondence principle: quantum mechanics reproduces classical physics in limit of large quantum numbers Reilly V. Bautista Quantum Mechanics September 5, / 78

68 Probabilistic nature of matter Schrodinger's cat (meow) A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small ask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The rst atomic decay would have poisoned it. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. Reilly V. Bautista Quantum Mechanics September 5, / 78

69 Probabilistic nature of matter Reilly V. Bautista Quantum Mechanics September 5, / 78

70 Probabilistic nature of matter Copenhagen Interpretation The system is in superposition of states The act of observation forces the system to "collapse" into a single state, which is what an instrument "observes" Reilly V. Bautista Quantum Mechanics September 5, / 78

71 Probabilistic nature of matter Concerning measurement and uncertainty: Classical model: Uncertainty is due to instrument and human error. The act of measurement does not change the quantity being measured. Reality: Uncertainty is inherent and rooted to nature. The act of measurement changes the quantity being measured. Reilly V. Bautista Quantum Mechanics September 5, / 78

72 Quantum systems Particle in a box Describe the position of particle inside a box given its potential barriers Reilly V. Bautista Quantum Mechanics September 5, / 78

73 Quantum systems Particle in a box Momentum p n = h λ n = nh 2L (47) Energy E n = p2 n 2m = n2 h 2 8mL = n2 π 2 2, n = 1, 2, 3, mL2 (48) Reilly V. Bautista Quantum Mechanics September 5, / 78

74 Quantum systems Potential well Describe the position of particle inside a box given a potential well Reilly V. Bautista Quantum Mechanics September 5, / 78

75 Quantum systems Particle in a box Energy E 1 IDW = π2 2 2mL 2 (49) Reilly V. Bautista Quantum Mechanics September 5, / 78

76 Quantum systems Tunneling probability T = Ge 2κL, G = 16 E U 0 ( 1 E U 0 ), κ = 2m(U0 E) (50) Reilly V. Bautista Quantum Mechanics September 5, / 78

77 A 2.0-eV electron encounters a barrier 5.0 ev high. What is the probability that it will tunnel through the barrier if the barrier width is (a) 1.00 nm and (b) 0.50 nm? Reilly V. Bautista Quantum Mechanics September 5, / 78

78 Quantum systems Harmonic oscillator E n = ( n + 1 ) ( k 2 m = n + 1 ) ω, n = 0, 1, 2,... 2 (51) Reilly V. Bautista Quantum Mechanics September 5, / 78