STSF2223 Quantum Mechanics I

Transcription

1 STSF2223 Quantum Mechanics I What is quantum mechanics? Why study quantum mechanics? How does quantum mechanics get started? What is the relation between quantum physics with classical physics? Where is quantum mechanics applied? 1

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3 The Nobel Prize Physics for discovery or invention Chemistry for discovery or improvement Physiology or Medicine for discovery Literature and Peace

4 Classical Physics Old Quantum Physics New Quantum Physics Quantum Field Theory String Theory /80 s 10-6 m m m m 4

5 Physics Classical Physics Quantum Physics Applications Statistical Physics Electromagnetism

6 Blackbody Radiation (Beginning of Quantum Physics) Quantum theory began when scientists study the spectrum from the energy radiated by bodies that were heated. The distribution spectrum is the relative amount of energy that is associated with each wavelength Blackbody is object that absorbs all the radiation and when equilibrium state is achieved, it will reflect all the radiation. 6

7 Example; black carbon cavity with a small opening where radiation can enter and escape. Figure 1 - Experimental black body curve for 5000 K 7

8 The black body radiates energy at every wavelength. The curve gets infinitely close to the horizontal (x)-axis but never touches it. Black body shows a peak wavelength, at which most of the radiant energy is emitted. At 5000 K - peak wavelength ~ 5 x 10-7 m (500 nm) which is in the visible light region, in the yellow-green section. At each temperature the black body emits a standard amount of energy. This is represented by the area under the curve. 8

9 Fig 2: Black body radiation curves showing peak wavelengths at various temperatures As the temperature increases, the peak wavelength emitted by the black body decreases. It moves from the infra-red towards the visible part of the spectrum. 9

10 Visible radiation is emitted even at these lower temperatures and at any temperature above absolute zero a black body will emit some visible light. As temperature increases, the total energy emitted increases - the total area under the curve increases.. Wien's Displacement Law The displacement of the peak is given by an empirical relation λpt = constant = x 10-3 m K λp: wavelength at the peak T: temperature in K Wein s Law E(λ,T) = ae b / λt λ 5 a and b are constant. 10

11 Stefan Boltzmann s Law The total power radiated per unit area at temperature T is E(T) = σt 4 σ = x 10-8 watt/m 2 -K 4 These laws are not derived from any physical model. 11

12 Rayleigh-Jeans Theory Assumption - blackbody consists of oscillators with energy 1 2 k B T where kb = 1.38 x J/K is the Boltzmann constant. The total energy is 2 H = T + V = p x 2m kx 2 where k is the spring constant. 12

13 Rayleigh-Jeans Law Energy per unit volume is given as: Derivation: I(λ,T) = 8πk B T λ 4 Oscillating charged particles will emit electromagnetic waves. Each degree of freedom will contribute 1 2 k B T. The spectral energy density is given by I(λ,T) = n(v)kbt n(v): numbr of oscillators per unit volume at frequency v, also known as Jeans number. n(v) = 8πv2 c 3 n(v)dv = n(λ) dλ dv = n(λ) c v 2 13

14 n(λ) = 8π λ 4 I(λ,T) = 8π λ 4 kt Rayleigh-Jeans Experimental Rayleigh-Jeans theory is only applicable for large λ. At lower wavelength, it predicted ultraviolet catastrophe which was not observed in reality A fundamental mistake in this model. 14

15 Planck Radiation Law Postulate 1 The amount of energy (ε) emitted or absorbed by an oscillator is directly proportional to its frequency (v) Δε = hv where h = x J s is called the Planck s constant. 15

16 Postulate II An oscillator cannot have an arbitrary energy but must occupy one of the discrete energy states given by εn = nhv, where n = 0, 1, 2, 3,.. n = 4 n = 3 n = 2 n = 1 n = 0 The amount energy emitted or absorbed are quantized Each energy quanta of electromagnetic waves are called photons. At T > 0 K, the oscillators are in the higher energy states. 16

17 The Maxwell-Boltzmann distribution gives the distribution of the oscillators at temperature T N(n) = N o e ε n / k B T N(n): number of oscillator with energy εn No: constant at all T. The average energy of the oscillator can be written as ε = n= 0 n= 0 N(n)ε n N(n) = n= 0 N 0 e nhv kbt nhv n= 0 N 0 e nhv kbt = 0 + hve hv kbt 1+ e hv kbt + 2hve 2hv kbt e 2hv kbt

18 " = hvx 1+ 2x + 3x % $ ' where x = e hv/ k BT # 1+ x + x 2 + x & ' hvx% & 2 ( 1 x) $ hvx hv 1 " = = / k T ( 1 ) 1 e hv B x x 1 = I = n(v)ε = 8πhv3 1 c 3 e hv/ k B T 1 = n(λ)ε = 8πhc 1 λ 5 e hc/ λkbt 1 # 18

19 Wave-Particle Duality Classical physics states that energy can be transported by either waves or particles. Example; i) a disturbance on the surface of a pond is a wave phenomenon ii) a ball thrown into the air shows the transport of energy by a particle. However, the Davisson-Germer experiment shows that electron which is known to be a particle exhibits wave properties In the e/m experiment, electron shows the particle property (denoted by mass m). 19

20 The de Broglie Wave The de Broglie wave - relates the particle property with the wave property h: Planck constant p: momentum λ: wavelength p = mv = h λ Example: An electron with kinetic energy of 100 ev. Find the de Broglie wavelength of the electron. Using the de Broglie relation p = h λ. Need to determine p. K = 1 2 mv2 = p2 2m hence p = 2mK m is the mass of electron. 20

21 λ = h p = 2( Js kg)( J ) = 1.2Å Example 2: A ball of mass m = 1.0 kg, travel at 10 m/s. Find the de Broglie wavelength. λ = 34 h h Js = = = m = Å p mv (1.0 kg)(10 m/s) 21

22 Davisson and Germer Experiment (Wave nature of electrons) Using electron beams off nickel crystals and analyzed how the electrons were more likely to appear at certain angles than others. Intensity maximum at θ = 50 o and V = 54 V This result can be explained as constructive interference (a wave phenomenon). 22

23 Similar to Bragg s relation in X-ray diffraction. Wave nature of electron nλ =2d sinθ θ = = 65 Lattice spacing for nickel d = 0.91 Å 23

24 λ = 2dsinθ = 2 (0.91) sin 65 = 1.65 Å Particle nature of electron h h λ = = = 1. 67Å p 2mK de Broglie wavelength 24

25 The wave-like properties of light were demonstrated by the famous experiment first performed by Thomas Young 25

26 Electron beam also shows the same pattern 26

27 Important experiments - development of quantum physics 1. Black body radiation Electromagnetic waves are made up of particles known as photon with quantized energy 2. Compton effect Shows the particle property of electromagnetic waves 3. Photoelectric effect Shows the particle property of electromagnetic waves 4. Frank-Hertz Experiment Shows that the energy level of atoms are quantized 5. Davisson-Germer Experiment Shows that electron has wave nature 27

28 6. Stern-Gerlach Experiment Shows that the z-component of angular momentum is quantized and the existence of electron spin 28

29 Relation between de Broglie and Planck λ = h p de Broglie λ = 2π k k: wavevector p = h λ = hk 2π = k = h 2π (h-bar) E = hv Planck = h ω 2π = h 2π ω = ω 29