The Heisenberg Uncertainty Principle

1. The Heisenberg Uncertainty Principle Inderjit Singh

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

3. 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

4. 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 ! AFTER ELECTRON-PHOTON COLLISION scatteredphoton recoiling electron

5. 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

6. 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

7. Example of Baseball (cont’d) • The uncertainty in position is then • No wonder one does not observe the effects of the uncertainty principle in everyday life!

8. Example of Electron • Same situation, but baseball replaced by an electron which has mass 9.11 x 10-31 kg traveling at 40 m/s • So momentum = 3.6 x 10-29 kg m/s and its uncertainty = 3.6 x 10-31 kg m/s • The uncertainty in position is then

9. Classical World • The observer is objective and passive • Physical events happen independently of whether there is an observer or not • This is known as objective reality

10. Role of an Observer in Quantum Mechanics • The observer is not objective and passive • The act of observation changes the physical system irrevocably • This is known as subjective reality

11. The Heisenberg Uncertainty Principle • Whenever a measurement is made there is always some uncertainty • Quantum mechanics limits the accuracy of certain measurements because of wave –particle duality and the resulting interaction between the target and the detecting instrument

12. 4. The Heisenberg’s uncertainty principle • In the example of a free particle, we see that if its momentum is completely specified, then its position is completely unspecified • When the momentum p is completely specified we write: (because: and when the position x is completely unspecified we write: • In general, we always have: This constant is known as: (called h-bar) h is the Planck’s constant

13. So we can write: That is the Heisenberg’s uncertainty principle “ it is impossible to know simultaneously and with exactness both the position and the momentum of the fundamental particles” N.B.: •We also have for the particle moving in three dimensions • With the definition of the constant

14. Energy Uncertainty The energy uncertainty of a Gaussian wave packet is combined with the angular frequency relation • Energy-Time Uncertainty Principle: .

15. Derivation of Uncertainty Principle .

16. Derivation - Continued w

17. Derivation - Continued The condition for the formation of the node is that amplitude should be zero or

18. Derivation - Continued If x1 and x2 be the position of two consecutive nodes, then So that and

19. Derivation - Continued So uncertainity in measurement of position of the particle (x1-x2)

20. Physical Origin of the Uncertainty PrincipleHeisenberg (Bohr) Microscope • The measurement itself introduces the uncertainty • When we “look” at an object we see it via the photons that are detected by the microscope • These are the photons that are scattered within an angle 2θ and collected by a lens of diameter D • Momentum of electron is changed • Consider single photon, this will introduce the minimum uncertainty

21. Bohr Microscope-continued

22. (a)Limitation in determing the position of electron =half angle subtended by the objective at the object i.e. electron

23. (b)Limitation in determining the momentum of the electron If photon is scattered along OQ,Then

24. If photon is scattered along OP,Then

25. .

26. Diffraction of Electron by a slit

27. Electron Diffraction -Continued

28. Electron Diffraction -Continued

29. Applications of Heisenberg uncertainty Principle-Non existence of electron in the nucleus Size of Nucleus =10-14 m If electron is present in the nucleus uncertainty in the position of electron is =10-14 m

30. The minimum momentum of the electron must be at least equal to uncertainty in momentum

31. Zero point Energy or minimum energy of a particle in the box The minimum energy of a system at 0K is called zero point energy. Let a particle of mass m0 is moving in a one dimensional box of length L So uncertainty of the position of the particle in the box Δx=L Uncertainty in the momentum Δp h/Δx = h/L

32. Minimum energy of a particle in the box Minimum momentum of the particle is at least equal to uncertainty in momentum p=Δp=ℏ/L K.E. of the particle is =p2 /2m0 =(Δp)2 /2m0 K.E.=ℏ/ 2m0 L2 This is the energy of the particle. Because the energy of the system is minimum at 0K .

33. Minimum energy of a particle in the box Since the Δp  0 at 0K, So the particle will have some energy even at 0K. This minimum energy is called end point energy. So a particle confined to a region of space cannot have zero energy.

34. Binding energy of an electron Electron-revolving around the nucleous in an orbit of radius r So uncrtainity in the position of the electron is equal to the radius of the atom Δx=r Uncertainty in the momentum Δp h/Δx = h/r

35. Binding energy of an electron Minimum momentum of the particle is at least equal to uncertainty in momentum p=Δp=ℏ/r K.E. of the particle is =p2 /2m0 =(Δp)2 /2m0 K.E.=ℏ/ 2m0 r2

36. Binding energy of an electron The potetial energy of the electron in the field of the nucleus of atomic no Z is

37. Binding energy of an electron .

38. Binding energy of an electron .

39. Binding energy of an electron .

40. Minimum Energy of the Hermonic Oscillator Consider a mechanical oscillator –spring attached at one end and mass attached at the other end. Classical mechanics-when at rest-energy of the oscillator is zero Quantum mechanics—Energy of the oscillator can never be zero. Let Δx is the displacement of the mass. Uncertainity in position= Δx

41. Harmonic Oscillator-Continued Uncertainty in the momentum Δp h/Δx Minimum momentum of the particle is at least equal to uncertainty in momentum p=Δp=ℏ/ Δx Energy of the Hermonic oscillator= PE+KE

42. Harmonic Oscillator-Continued .

43. Harmonic Oscillator-Continued .

44. Harmonic Oscillator-Continued .

45. Harmonic Oscillator-Continued

46. THANKS

47. Heisenberg (Bohr) Microscope As a consequence of momentum conservation Trying to locate electron we introduce the uncertainty of the momentum

48. Heisenberg (Bohr) Microscope • θ~(D/2)/L, L ~ D/2 θ is distance to lens • Uncertainty in electron position for small θ is • To reduce uncertainty in the momentum, we can either increase the wavelength or reduce the angle • But this leads to increased uncertainty in the position, since

49. Heisenberg (Bohr) Microscope

50. PROBLEM 3 An electron is moving along x axis with the speed of 2×106 m/s (known with a precision of 0.50%). What is the minimum uncertainty with which we can simultaneously measure the position of the electron along the x axis? Given the mass of an electron 9.1×10-31 kg SOLUTION From the uncertainty principle: if we want to have the minimum uncertainty: We evaluate the momentum: The uncertainty of the momentum is: