Wave-Particle Duality and de Broglie Hypothesis in Matter Waves

1. Wykład 2

3. We have seen experimental examples wherelight behaves both as a particle and as a wave. • This is referred to as “wave-particle” duality. • We noticed that one needs something more than classical physics to describe EMR properly ...quantisation of energy

4. We have seen experimental examples wherelight behaves both as a particle and as a wave. • This is referred to as “wave-particle” duality. • We noticed that one needs something more than classical physics to describe EMR properly ...quantisation of energy BUT

5. We have seen experimental examples wherelight behaves both as a particle and as a wave. • This is referred to as “wave-particle” duality. • We noticed that one needs something more than classical physics to describe EMR properly ...quantisation of energy BUT • It turns out that wave-particle duality is not limited to light. In fact, also matter demonstrates this behavior.

6. Matter waves: de Broglie hypothesis • The path to the wavelength expression for a particle is by analogy to the particle velcity (as suggested by De Broglie in 1923)

7. Matter waves: de Broglie hypothesis • The path to the wavelength expression for a particle is by analogy to the particle velcity (as suggested by De Broglie in 1923) E = mc2 hν = E mc2 = hν mV2 = hν ν = V/λ mV2 = hν/λ λ= hν/mν2 = h/mV

8. Matter waves: de Broglie hypothesis „All carriers of energy and momentum, such as light and electrons, propagate like a wave and exchange energy like a particle”

9. Matter waves: de Broglie hypothesis „All carriers of energy and momentum, such as light and electrons, propagate like a wave and exchange energy like a particle”

10. Matter waves: de Broglie hypothesis „All carriers of energy and momentum, such as light and electrons, propagate like a wave and exchange energy like a particle”

11. Matter waves: de Broglie hypothesis

12. Matter waves: de Broglie hypothesis NOTE: most X-rays have a wavelength ranging from 0.01 to 10 nanometers, corresponding to frequencies in the range 30petahertz.

13. Matter waves: de Broglie hypothesis

15. Matter waves: Davisson-Germer exp. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be only exhibited by waves. [Think again about the LIGHT] Therefore, the presence of any diffractioneffects should demonstrate the wave-like nature of matter.

17. Matter waves: Davisson-Germer exp. In 1927, Clinton J. Davisson and Lester H. Germer shot electron particles onto a nickel crystal. 

18. Matter waves: Davisson-Germer exp.

19. Matter waves: Davisson-Germer exp.

20. Matter waves: Davisson-Germer exp. When the de Broglie wavelength was inserted into theBragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.

24. Matter waves: de Broglie hypothesis

25. Matter waves: Davisson-Germer exp. Just as the photoelectric effect demonstrated the particle nature of light, the Davisson–Germer experiment showed the wave-nature of matter, and completed the theory of wave-particle duality. For physicists this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use wave equations to describe phenomena in matter if one uses the de Broglie wavelength.

26. Indeed in 1926, Erwin Schrödinger published an equation describing how a matter wave should evolve and used it to derive the energy spectrum of H. In other words: the Schrödinger equation can be considered an analogue of Maxwell’s equations for matter waves.

27. Towards quantum mechanics As a fundamental constraint, higher level descriptions of the universe must obey all quantum mechanical descriptions which includes Heisenberg's uncertainty relationship.

28. Heisenberg realized that ... • In the world of very small particles, one cannot measure any property of a particle without inter-acting with it in some way • This introduces an unavoidable uncertainty into the result • One can never measure all the properties exactly

32. Measuring Position and Momentum of an Electron • Shine light on electron and detectreflected light using a microscope • Minimum uncertainty in position is given by the wavelength of the light • So to determine the position accurately, it is necessary to use light with a short wavelength BEFORE ELECTRON-PHOTON COLLISION incidentphoton electron

33. Measuring Position and Momentum of an Electron • By Planck’s law E = hc/λ, a photon with a short wavelength has a large energy • Thus, it would impart a large ‘kick’ to the electron • But to determine its momentum accurately, electron must only be given a small kick. This means using light of long wavelength ! AFTERELECTRON-PHOTON COLLISION scatteredphoton recoiling electron

34. Implications • It is impossible to know both the position and momentum exactly, i.e., Δx=0 and Δp=0 • These uncertainties are inherent in the physical world and have nothing to do with the skill of the observer • Because h is so small, these uncertainties are not observable in normal everyday situations

35. Example of Baseball • A pitcher throws a 0.1-kg baseball at 40 m/s • So momentum is 0.1 x 40 = 4 kg m/s • Suppose the momentum is measured to an accuracy of 1 percent , i.e., Δp = 0.01p = 4 x 10-2 kg m/s

36. Example of Baseball • The uncertainty in position is then • No wonder one does not observe the effects of the uncertainty principle in everyday life ;)

37. Uncertainty principle So, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.

38. Uncertainty principle In general, pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can’t be known simultaneously.

39. Summary • Light is made up of photons, but in macroscopic situa-tions it is often fine to treat it as a wave • Photons carry both energy & momentum • Matter also exhibits wave properties. For an object of mass m, and velocity, v, the object has a wavelength, λ = h / mV • One can probe ‘see’ the fine details of matter by usinghigh energy particles (they have a small wavelength !) but only with limited accuracy