the total energy of matter related to the frequency ν of the wave is E=hν

1. Chapter 3de Broglie’s postulate: wavelike properties of particles 3.1 Matter wave: de Broglie • the total energy of matter related to the frequency ν of the wave is E=hν • the momentum of matter related to the wavelength λ of the wave is p=h/λ Ex: (a) the de Broglie wavelength of a baseball moving at a speed v=10m/s, and m=1kg. (b) for a electron K=100 eV.

2. Chapter 3de Broglie’s postulate: wavelike properties of particles The experiment of Davisson and Germer • A strong scattered electron beam is detected at θ=50o for V=54 V. • The result can be explained as a constructive interference of waves scattered by the periodic arrangement of the atoms into planes of the crystal. • The phenomenon is analogous to the Bragg-reflections (Laue pattern). • 1927, G. P. Thomson showed the diffraction of electron beams passing through thin films confirmed the de Broglie relation λ=h/p. (Debye-Scherrer method)

3. Chapter 3de Broglie’s postulate: wavelike properties of particles Bragg reflection: constructive interference: consistent

4. Chapter 3de Broglie’s postulate: wavelike properties of particles Debye-Scherrer diffraction X-ray diffraction: electron diffraction : zirconium oxide crystal gold crystal Laue pattern of X-ray (top) and neutron (bottom) diffraction for sodium choride crystal

5. Chapter 3de Broglie’s postulate: wavelike properties of particles 3.2 The wave-particle duality Bohr’s principle of complementarity:The wave and particle models are complementary; if a measurement proves the wave character of matter, then it is impossible to prove the particle character in the same measurement, and conversely • Einstein’s interpretation: for radiation (photon) intensity is a probability measure of photon density • Max Born: wave function of matter is just as satisfies wave equation is a measure of the probability of finding a particle in unit volume at a given place and time. Two superposed matter waves obey a principle of superposition of radiation.

6. Chapter 3de Broglie’s postulate: wavelike properties of particles 3.3 The uncertainty principle Heisenberg uncertainty principle: Experiment cannot simultaneously determine the exact value of momentum and its corresponding coordinate. Bohr’s thought experiment: a diffraction apparatus Bohr’s thought experiment

7. Chapter 3de Broglie’s postulate: wavelike properties of particles 3.4 Properties of matter wave • wave propagation velocity: • a de Broglie wave of a particle (1) x fixed, at any time t the amplitude is one, frequency is ν. (2) t fixed, Ψ(x,t) is a sine function of x. (3) zeros of the function are at these nodes move along x axis with a velocity it is the node propagation velocity (the oscillation velocity)

8. Chapter 3de Broglie’s postulate: wavelike properties of particles modulate the amplitude of the waves

9. Chapter 3de Broglie’s postulate: wavelike properties of particles • group velocity of waves equal to moving velocity of particles • The Fourier integral can prove the following universal properties of all wave. uncertainty principle the consequence of duality uncertainty principle

10. Chapter 3de Broglie’s postulate: wavelike properties of particles Ex:An atom can radiate at any time after it is excited. It is found that in a typical case the average excited atom has a life-time of about 10-8 sec. That is, during this period it emit a photon and is deexcited. (a) What is the minimum uncertainty in the frequency of the photon? (b) Most photons from sodium atoms are in two spectral lines at about . What is the fractional width of either line, (c) Calculate the uncertainty in the energy of the excited state of the atom. (d) From the previous results determine, to within an accuracy , the energy E of the excited state of a sodium atom, relative to its lowest energy state, that emits a photon whose wavelength is centered at

11. Chapter 3de Broglie’s postulate: wavelike properties of particles uncertainty principle in a single-slit diffraction for a electron beam:

12. Chapter 3de Broglie’s postulate: wavelike properties of particles Ex: Consider a microscopic particle moving freely along the x axis. Assume that at the instant t=0 the position of the particle is measured and is uncertain by the amount . Calculate the uncertainty in the measured position of the particle at some later time t.

13. Chapter 3de Broglie’s postulate: wavelike properties of particles • Some consequences of the uncertainty principle: (1) Wave and particle is made to display either face at will but not both simultaneously. (2) We can observe either the wave or the particle behavior of radiation; but the uncertainty principle prevents us from observing both together. (3) Uncertainty principle makes predictions only of probable behavior of the particles. • Dirac’s relativistic quantum mechanics of electron: • Dirac’s assumption: a vacuum consists of a sea of electrons in negative energy levels which are normally filled at all points in space. • The philosophy of quantum theory: (1) Neil Bohr: Copenhagen interpretation of quantum mechanics. (2) Heisenberg: Principally, we cannot know the present in all details. (3) Albert Einstein: “God does not play dice with the universe” The belief in an external world independent of the perceiving subject is the basis of all natural science.