Matter Waves and the Uncertainty Principle _________________________

1. Matter Waves and the Uncertainty Principle_________________________ Anu Venugopalan Centre for Philosophy and Foundations of Science New Delhi

2. The Birth of Quantum Mechanics___________________________ At the turn of the last century, there were several experimental observations which could not be explained by the established laws of classical physics and called for a radically different way of thinking This led to the development of Quantum Mechanics which is today regarded as the fundamental theory of Nature.

3. Some key events/observations that led to the development of quantum mechanics�_________________________________ Black body radiation spectrum (Planck, 1901) Photoelectric effect (Einstein, 1905) Model of the atom (Rutherford, 1911) Quantum Theory of Spectra (Bohr, 1913)

4. Some key events/observations that led to the development of quantum mechanics�_________________________________ Scattering of photons off electrons (Compton, 1922) Exclusion Principle (Pauli, 1922) Matter Waves (de Broglie 1925) Experimental test of matter waves (Davisson and Germer, 1927)

7. The nature of light��.____________________________ The birth of quantum mechanics is intimately linked with the theories and discoveries relating to the nature of light Is the nature of light that of a wave or a particle???

8. The story of light��.._______________________________ Corpuscular theory (Newton) Wave nature (Huygens) Double-slit interference experiment (Young)

9. The story of light (contd.)��.._______________________________ Light is an electromagnetic wave (Maxwell) Photoelectric effect � existence of light quanta � photons (Einstein) Photons have momentum (Compton)�..

11. Light has a dual nature___________________________ Wave (electromagnetic) - Interference - Diffraction Particle (photons) - Photoelectric effect - Compton effect Wave - Particle Duality for light

12. What about Matter?_______________________________ If light, which was traditionally understood as a wave also turns out to have a particle nature, might matter, which is traditionally understood as particles, also have a wave nature? Yes!

13. Louis de Broglie�s hypothesis____________________________ The dual nature of matter A particle with momentum p has a matter wave associated with it, whose wavelength is given by

14. The connecting link � Planck�s constant_______________________________ Dual Nature Radiation Matter

15. Why isn�t the wave nature of matter more apparent to us�?___________________________________ Planck�s constant is so small that we don�t observe the wave behaviour of ordinary objects � their de Broglie wavelengths could be many orders of magnitude smaller than the size of a nucleus!

16. Particle______________________________ Our traditional understanding of a particle� �Localized� - definite position, momentum, confined in space

17. Wave____________________________ Our traditional understanding of a wave�. �de-localized� � spread out in space and time

18. How do we associate a wave nature to a particle?___________________________________ What could represent both wave and particle? Find a description of a particle which is consistent with our notion of both particles and waves�� Fits the �wave� description �Localized� in space

19. ____________________________________ A �Wave Packet�

20. What happens when you add up waves?________________________________

21. Adding up waves of different frequencies�..____________________________________

22. Constructing a wave packet by adding up several waves ����___________________________________

23. A wave packet describes a particle____________________________ A wave packet is a group of waves with slightly different wavelengths interfering with one another in a way that the amplitude of the group (envelope) is non-zero only in the neighbourhood of the particle A wave packet is localized � a good representation for a particle!

24. Wave packet, phase velocity and group velocity ____________________________ The spread of wave packet in wavelength depends on the required degree of localization in space � the central wavelength is given by What is the velocity of the wave packet?

25. Wave packet, phase velocity and group velocity ________________________________ The velocities of the individual waves which superpose to produce the wave packet representing the particle are different - the wave packet as a whole has a different velocity from the waves that comprise it Phase velocity: The rate at which the phase of the wave propagates in space Group velocity: The rate at which the envelope of the wave packet propagates

26. Wave packet, phase velocity and group velocity ________________________________ Phase velocity Group velocity Here is the velocity of light and is the velocity of the particle

27. Heisenberg's Uncertainty Principle___________________________________ The Uncertainty Principle is an important consequence of the wave-particle duality of matter and radiation and is inherent to the quantum description of nature Simply stated, it is impossible to know both the exact position and the exact momentum of an object simultaneously A fact of Nature!

28. Heisenberg's Uncertainty Principle__________________________________ Uncertainty in Position : Uncertainty in Momentum:

29. Heisenberg's Uncertainty Principle - applies to all �conjugate variables�___________________________________ Position & momentum Energy & time

30. Uncertainty Principle and the Wave Packet___________________________________

31.  Some consequences of the Uncertainty Principle ___________________________________ The path of a particle (trajectory) is not well-defined in quantum mechanics Electrons cannot exist inside a nucleus Atomic oscillators possess a certain amount of energy known as the zero-point energy, even at absolute zero.

32. Why isn�t the uncertainty principle apparent to us in our ordinary experience�? Planck�s constant, again!! ___________________________________ Planck�s constant is so small that the uncertainties implied by the principle are also too small to be observed. They are only significant in the domain of microscopic systems

33. Summary___________________________________ Matter and radiation have a dual nature � of both wave and particle The matter wave associated with a particle has a de Broglie wavelength given by

34. Summary (contd.)___________________________________ A (localized) particle can be represented by a group of waves called a wave packet The group velocity of the wave packet is The phase velocity of the wave packet is

35. Summary (contd.)___________________________________ Heisenberg�s uncertainty principle is an important consequence of the wave-particle duality of matter and radiation and is inherent to the quantum description of nature It applies to all conjugate variables and also to the notion of the wave packet

36. Are matter waves for real?! __________________________________ In 1927 Davisson and Germer showed that electrons can diffract � they act like waves Big application � Electron Microscopes

38. Are matter waves for real?! __________________________________ Double �slit experiment with electrons (1989) (www.hqrd.hitachi.co.jp/em/doubleslit.cfm)

39. Are matter waves for real?! __________________________________ Today, advances in technology have led to matter wave interference experiments being demonstrated successfully not only with electrons but neutrons, atoms, and big and small molecules! Infact, the largest molecule showing interference has almost a 100 atoms!

40. Are matter waves for real?! __________________________________ C60 molecules (Fullerenes or Bucky Balls) have a wave nature! (A. Zeilinger et al, Vienna, 1999) Biomolecules have it too! Porphyrin (2003)

41. Are matter waves for real?! __________________________________ What next? � Matter wave interference for a Virus?!

42. Matter waves are real�. _______________________________ If there is wave, there must be a wave equation��� The Schr�dinger Equation