CHAPTER NUMBER 7: Quantum Theory: Introduction and Principles

Transcription

1 CHAPTER NUMBER 7: Quantum Theory: Introduction and Principles Art PowerPoints Peter Atkins & Julio De Paula 2010

2 1 mm 1000 m 100 m 10 m 1000 nm 100 nm 10 nm 1 nm 10 Å 1 Å Quantum phenomena

3 7.1 Energy quantization -Feature of classical physics Precisely specified motion and momenta at an instant Continuous vibrational, rotational, and translation modes of motion and enenrgy classical physics doesn t hold for a small world -Failure of classical physics black body radiation & the Planck distribution Heat capacities Atomic and molecular spectra

4 Physical Chemistry Fundamentals: Figure 7.1 Feature of electromagnetic wave -speed: light -wavelength -frequency -wavenumber -energy -momentum

5 Physical Chemistry Fundamentals: Figure 7.2

6 One of blackbody: Nuclear Reactor

7 Black body radiation Energy density ( T) (, T) d 0 E( T) V ( T) d (, T) (, T) d Density of state Classical model: Rayleigh-Jeans law (, T) 8kT 4

8 -Energy distribution of black body (experiment) -Wien displacement law -Stefan-Boltzmann law

9 Physical Chemistry Fundamentals: Figure 7.5

10 Figure 7.6 -Rayleigh-Jeans law: based on classical mechanics As 0, : everything in the dark will glow (UV catastrophe) As 0, 0 (why?)

11 Planck distribution: Figure 7.7 Short wavelength limit: No UV catastrophe Planck distribution: 8hc (, T) 5 hc kt ( e 1) based on quantization of energy Long wave length limit ~Rayleigh-Jeans law ( T) 1. As 0, 0 2. For long, 8hc 5 hc kt ( e 1) for large, e hc kt 8kT Stefan-Boltzmann Law hc kt 5 4 8hc 4 8 k d at a 0 5 hc kt 3 ( e 1) 15( hc)

12 -Rayleigh-Jeans law: UV catastrophe based on classical mechanics (all oscillators share equally in the energy) -Planck distribution: based on quantization of energy E=nhv (oscillators are excited only if they can acquire an energy of at least hv.) Stefan-Boltzmann and Wien law is successfully derived from Planck distribution!!!

13 -heat capacities (close to 0K) C v =3R (from U m =3RT) However, this deviates as T 0 Einstein assumes each atom oscillates with a single freq. and the deviation can be successfully explained (E is confined to discrete values: E=nhν)

14 At high T (T>> E ), same as classical theory At low T (T<< E ), as T 0, f E 0

15 Debye formula: atoms oscillate over a range of frequencies Figure 7.8 Figure 7.9

16 -Atomic and molecular spectra -Atomic spectra: Radiation emitted by excited iron atom -Molecular spectra: Absorbing radiation at definite frequencies due to electronic, vibrational, and rotational excitation of SO 2.

17 Spectroscopic transition E hv

18 11.2 Wave particle duality -Photoelectric effect: particle character of electromagnetic radiation wave-like particle (photon) -Diffraction: wave character of particles (de Broglie wave)

19 -particle character of electromagnetic radiation Photoelectric effect 1. No e - are ejected below a threshold freq. -> phtoelectric effects occur only when hv> 2. E k of ejected e - increases linearly w/ freq of incident radiation -> E hv 3. Even at low light intensities, e - are ejected above a threshold freq. ->e- appears once the collision happens

20 Physical Chemistry Fundamentals: Figure 7.14 Energy needed to remove electron from a metal Kinetic energy of ejected electron

21 -wave character of particles: diffraction Figure 7.15

22 -De Broglie relation How short? for electron, m See Ex 7.2 Figure 7.16

23 Physical Chemistry Fundamentals: Figure 7.17

24

25 Scanning Probe Microscopy

26 7.3 Shrödinger equation For 1-d systems: 2 2m 2 d 2 dx V ( x) E For 3-d systems: In general, 2 2 V 2m H E H E 2 2m 2 V ( x) H : Hamiltonian operator Time-dep Shrödinger eqn. H i t

27 Table 7.1

28 Using the Schrödinger eqn to develop the de Broglie relation, Since E-V is equal to E k

29 7.4 The Born interpretation of the wavefunction -Normalization -Quantization

30 Born interpretation of the wave function Probability of finding a particle between x and x+dx * 2 * dx 2 dx Figure 7.18

31 Figure 7.19

32 Figure 7.20

33 Normalization N N dx 1 2 * 1 * dx 12 For a normalized wavefunction, * dx 1 * dxdydz 1 * (or d 1)

34 Figure 7.21 Figure 7.22

35

36 Quantization [1] Figure 7.23 Since it is 2 nd derivative. But, delta ftn is valid!

37 Quantization [2] A particle may possess only certain energies, for otherwise its wave function would be physically unacceptable

38 7.5 The information in a wavefunction -the probability density distribution -Eigenvalues and eigenfunctions -Construction of operators -Hermittian operators -Superpositions and expectation values

39 The information in a wave function When V=0, 2 2 d 2 2m dx E Solution: k 2 2 ikx ikx Ae Be, E 2m

40 (a) The probability density if B=0, ikx Ae Then, where is the particle? 2 * 2 ikx ikx * ikx ikx Ae Ae A e Ae A Equal probability of finding the particles Figure 7.24 a

41 if A=B, ikx ikx 2 cos A e e A kx Then, where is the particle? 2 * 2 2 2Acos kx 2Acos kx 4 A cos kx Node: particles will never be found Figure 7.24 b

42 The probability density wavefunction 2 2 waveamplitude 2

43 (b) Operators, Eigenvalues and eigenfunctions operator H 2 2 d V ( x) 2 2m dx H E H : hamiltonian (operator) -total E of a system Eigenvalue

44 Eigenvalue equation (Operator)(same function)=(constant factor)x(same function) Ex. (Energy Operator) Xψ=(energy)Xψ (Operator corresponding to an observable) ψ =(constant factor) ψ

45 Illustration for orthogonality of eigen functions f ( x) (sin x)(sin 2x) Area = 0

46 (c) Construction of Operators Observables are represented by operators, Position operators xˆ x Momentum operators ˆ p x i d dx

47 V E k kx px 2m Vˆ E k kx d d d 2m idx idx 2m dx 2 2 ˆ ˆ ˆ d H E ( ) ˆ k V x V ( x) 2 2m dx 2 2 2

48 KE of a particle is an average contribution from the entire space. Figure 7.26

49 Wave ftn of a particle in a potential decreasing towards right Figure 7.27

50 Physical Chemistry Fundamentals: Figure 7.28

51 (d) Hermitian operators Hermiticity d * i j * d * j i Orthogonality * i j d 0

52 (e) Superpositions and expectation values Ex. if ψ A cos kx not an eigenfunction any more!!! When the wavefunction of a particle is not an eigen function of an operator, the property to which the operator corresponds does not have a definite value. Linear combination!

53 Weighted mean of a series of observations Weighted mean of a series of observations : the expectation value is the sum of the two eigencalues weighted by the probabilities that each one will be found in a series of measurements

54 Expectation value Ex. Mean kinetic energy 2 2 * * d Ek Ek d d 2 2m dx

55 1. When the momentum is measured, one of the eigenvalue will be measured 2. Measuring an eigenvalue ~ Proportional to the square of the modulus in the linear combinations 3. Average value = Expectation value

56 Summary of Operators position momentum Kinetic E Potential E Note) Expectation value

57 7.6 The uncertainty principle

58 Heisenberg uncertainty principle In Quantum Mechanics, Position and momentum cannot be predicted simultaneously But, in Classical Mechanics, Position and momentum can be predicted simultaneously Ex. Particle travelling to the right -> position is unpredictable But, the momentum is definite ikx Ae px k

59 Wave function for a particle at a welldefined location An infinite number of waves is needed to construct the wavefunction of perfectly localized partile.

60 Quantitative version of uncertainty principle pq , p p p q q q Note) p and q are the same direction Standard deviation

61 More general version of uncertainty principle Complementary observables~ they do not commute commutator ˆ ( ˆ ) ˆ ( ˆ ) ˆ, ˆ ˆ ˆ ˆ ˆ Ex. xˆ, pˆ x i pq 1 2 Uncertainty principle 1 ˆ 1 ˆ ˆ ˆ 2 1, 2 2

62 If there are a pair of complementary observables, (non-commuting) The uncertainty principle should be applied!!!

63 Physical Chemistry Fundamentals: Table 7.2

64 7.7 Postulates of quantum mechanics -wave function -Born interpretation -Acceptable wave function -Observables -Uncertainty relation

65 Physical Chemistry Fundamentals: Mathematic Background 7.1

66 Physical Chemistry Fundamentals: Mathematic Background 7.2

67 Physical Chemistry Fundamentals: Mathematic Background 7.3

68 Physical Chemistry Fundamentals: Mathematic Background 7.4