Wave properties of matter & Quantum mechanics I. Chapter 5

Transcription

1 Wave properties of matter & Quantum mechanics I Chapter 5

2 X-ray diffraction Max von Laue suggested that if x-rays were a form of electromagnetic radiation, interference effects should be observed. Crystals act as three-dimensional gratings, scattering the waves and producing observable interference effects.

3 Bragg s law William Lawrence Bragg interpreted the x-ray scattering as the reflection of the incident x-ray beam from a unique set of planes of atoms within the crystal. There are two conditions for constructive interference of the scattered x rays: 1) The angle of incidence must equal the angle of reflection of the outgoing wave. ) The difference in path lengths must be an integral number of wavelengths. Bragg s Law: d sin n (n = integer)

4 The Bragg spectrometer A Bragg spectrometer scatters x rays from several crystals. The intensity of the diffracted beam is determined as a function of scattering angle by rotating the crystal and the detector. When a beam of x rays passes through the powdered crystal, the dots become a series of rings.

5 De Broglie waves (194) Prince Louis V. de Broglie suggested that mass particles should have wave properties similar to electromagnetic radiation (or photons). Photon p E c h p h c h ( k) h Louis de Broglie (Nobel prize in 199) "for his discovery of the wave nature of electrons" Matter wave h h p m De Broglie wavelength

6 Bohr inspired de Broglie Bohr s assumptions concerning his hydrogen atom model (1913) L ( rp mr) n nh In terms of de Broglie wavelength nh mr nh r m n h p n The matter wave should form a standing wave to become a stable orbit.

7 De Broglie s view on the electron orbits The standing wave condition should be met. Artistic illustration

8 Electron diffraction (197) The 1 st demonstration of the matter wave by Davisson/Germer Diffraction patterns of the electrons by nickel Just like the x-ray diffraction by solids George P. Thomson ( ), son of J. J. Thomson, reported seeing the effects of electron diffraction in transmission experiments. The first target was celluloid, and soon after that gold, aluminum, and platinum were used. G. P. Thomson

9 Electron scattering Davisson and Germer experimentally observed that electrons were diffracted, much like x rays, in nickel crystals. Single crystal (SnO ) polycrystal

10 Davisson-Germer experiment (197)

11 Not only electrons, but also Any kind of particles can produce diffraction patterns. Atoms (H, He, etc.), neutrons, molecules,. etc. Universal nature of the matter wave Neutron diffraction pattern from a Niobium crystal

12 Wave versus particle Wave particle EM wave Photon (massless) Photoelectric effect (1905) by Einstein Compton scattering Developed to explain the mysterious experimental data Particle wave Mass Matter wave Proposed by de Broglie (194) Truly a conceptual proposition Wild, but ingenious Led to the Schroedinger equation (196) quantum mechanics

13 The displacement of a wave is The wave number k and the angular frequency ω are defined as: This is a solution to the wave equation Wave properties t kx A ft x A t x A t x A t x p p sin sin sin sin ), ( 1 x p t T f k

14 Wave properties The phase velocity is the velocity of a point on the wave that has a given phase (for example, the crest) and is given by p f T f k A more general form of wave function with a phase constant Φ that shifts the wave ( x, t) Asin kx t

15 How to represent a particle with waves Wave packet (or wave group) A probability amplitude that is spatially concentrated I am an electron Me, too ^^;

16 Beats Principle of superposition ) cos( cos 1 1 cos 1 1 cos ) cos( ) cos( t x k t x k A t x k k t x k k A y y t x k A y t x k A y av av k p g k dk d g envelop Fast modulation

17 Wave packet & Fourier integral The sum of many waves that form a wave packet is called a Fourier series: Summing an infinite number of waves yields the Fourier integral:

18 Gaussian Function A Gaussian wave packet describes the envelope of a pulse wave.

19 Dispersion The group velocity of a de Broglie wave packet g d dk d d k de dp pc E E p c m c 4 The relationship between the phase velocity and the group velocity g d dk d dk k p k p d dk p Hence the group velocity may be greater or less than the phase velocity. A medium is called non-dispersive when the phase velocity is the same for all frequencies and equal to the group velocity.

20 Wave packet in a dispersive medium The wave packet gets disperse as it propagates.

21 Group velocity

22 Matter wave for a classical particle d dk d d dk d k h m d dk h m h p k h m d d h m h E f g p / (Group velocity of the matter wave) (mass velocity) v

23 Matter wave for a relativistic particle c c k dv dk dv d dk d c h m d dk c h m d d c h m h m h p k c h mc h mc h E f p g 3/ 3/ 1/ 1/ (Group velocity of the matter wave) (mass velocity) v p > c does not violate the physics as long as v g < c. v

24 Waves or particles? Young s double-slit diffraction experiment demonstrates the wave property of light. However, dimming the light results in single flashes on the screen representative of particles.

25 Electron double-slit experiment C. Jönsson of Tübingen, Germany, succeeded in 1961 in showing double-slit interference effects for electrons by constructing very narrow slits and using relatively large distances between the slits and the observation screen. This experiment demonstrated that precisely the same behavior occurs for both light (waves) and electrons (particles).

26 Scanning Electron Microscope (SEM) The principle of microscopes Interaction between an object and a probing wave Optical microscope employs visible light as the probing wave. SEM Utilize the wave nature of moving electrons Small wavelength high resolution E/B applied to electrons easy beam control

27 When Hitachi (e-gun) met Hamamatsu (detector)

28 Which slit? To determine which slit the electron went through: We set up a light shining on the double slit and use a powerful microscope to look at the region. After the electron passes through one of the slits, light bounces off the electron; we observe the reflected light, so we know which slit the electron came through. For the photon, For the electron, The difficulty is that the momentum of the photons used to determine which slit the electron went through is sufficiently great to strongly modify the momentum of the electron itself, thus changing the direction of the electron! The attempt to identify which slit the electron is passing through will in itself change the interference pattern.

29 Wave-particle duality solution The solution to the wave particle duality of an event is given by the Bohr s principle of complementarity. It is not possible to describe physical observables simultaneously in terms of both particles and waves. NO Physical observables are those quantities such as position, velocity, momentum, and energy that can be experimentally measured. In any given instance we must use either the particle description or the wave description. YES

30 Photon s own words

31 Uncertainty principle (197) It is impossible to know both the exact position and exact momentum of an object at the same time. In classical physics, it is believed to be possible.

32 Werner Heisenberg Werner Heisenberg (Nobel prize in 193) "for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen"

33 Uncertainty principle (Intuitive approach) x: small Narrow wavepacket λ? λ Wide wavepacket x: large

34 Uncertainty principle (Mathematical Approach)

35 Uncertainty principle The localization of the wave packet over a small region to describe a particle requires a large range of wave numbers. Conversely, a small range of wave numbers cannot produce a wave packet localized within a small distance. ( x) g( k) cos kx dk pulse wave packet wave train Gaussian

36 Wave packet envelope The range of wave numbers and angular frequencies that produce the wave packet have the following relations: x k t In case of adding two sinusoidal waves A Gaussian wave packet has similar relations: x k t 1 1 * Minimum value of the product

37 Uncertainty principle It is impossible to measure simultaneously, with no uncertainty, the precise values of k and x for the same particle. Since the wave number k may be rewritten as for a single particle we have Heisenberg s uncertainty principle: p x k p p h p p h k 1 k x For position and momentum

38 Energy-time uncertainty t: small Narrow in time T? T Wide in time f t 1 t: large hf t E t h E t

39 Heisenberg's microscope As soon as we induce a scattering between a photon and an electron in an attempt to measure the electron state, we inevitably alter the initial state of the electron.

40 Uncertainty principle: A particle approach To measure the exact position We need a high resolution. We need a photon with small wavelength. The photon has a large momentum. The electron is strongly scattered off. We cannot know the exact momentum of the electron. To measure the exact momentum We should use a photon with a small momentum. The photon has a long wavelength. We have a bad resolution. We cannot measure the position of the electron exactly.

41 Waves of what? What should the amplitude Ψ of the matter wave be? What is the physical meaning of Ψ? WAVE water sound light AMPLITUDE height pressure E or B matter? Max Born s interpretation (196) - Ψ = the probability (density) of finding the mass at (x,y,z,t) - Probability density, to be exact - The mass itself does not swell, but Ψ may. Ψ is the wave of the probability.

42 Probability, wave Functions, and the Copenhagen interpretation The wave function determines the likelihood (or probability) of finding a particle at a particular position in space at a given time. The total probability of finding the electron is 1. Forcing this condition on the wave function is called normalization.

43 The Copenhagen interpretation Bohr s interpretation of the wave function consisted of 3 principles: 1) The uncertainty principle (of Heisenberg) ) The complementarity principle (of Bohr) 3) The statistical interpretation (of Born), based on probabilities determined by the wave function Together these three concepts form a logical interpretation of the physical meaning of quantum theory. According to the Copenhagen interpretation, physics depends on the outcomes of measurement.

44 The Copenhagen interpretation Supporters Niels Bohr Werner Heisenberg Max Born Wolfgang Pauli Opponents Albert Einstein Max Planck Louis de Broglie Erwin Schrödinger Today the great majority of physicists accept it.

45 Schrödinger s Cat A thought experiment proposed by Schrödinger in 1935 To illustrate the bizarreness of quantum mechanics Transposed the superposition of an atom to a large-scale system: a living-and-dead cat

46 Schrödinger s Cat Q: Is it dead or alive? A: The cat remains both dead and alive (to the universe outside the box) until the box is opened (or until measured)

47 Bohr versus Einstein

48 Bohr versus Einstein

49 Well God seems to play dice after all.

50 And more, maybe. Not only does God play dice, but He sometimes throws them where they cannot be seen. Stephen Hawking

51 Particle in a box A particle of mass m is trapped in a onedimensional box of width L. The particle is treated as a (sinusoidal) wave. The box puts boundary conditions on the wave. The wave function must be zero at the walls of the box and on the outside. In order for the probability to vanish at the walls, we must have an integral number of half wavelengths in the box. n L L n n ( n 1,,3,...)

52 Particle in a box The energy of the particle is given by E KE p h m m matter wave The possible wavelengths are quantized, which yields the particle energy: E n h m n L n h 8mL n ml 1,,3,...) The possible energies of the particle are quantized. The lowest energy level is not E 0 = 0 but E 1 =h /8mL. ( n

53 Probability of finding the particle The probability of observing the particle between x and x + dx in each state is The most probable position is neither at the center nor uniform, but depends on the energy level n!!!