Particle Nature of Light h f = KE max + W o Wave-Particle Duality l =h/mv

1. Particle Nature of Light h f = KEmax + Wo Wave-Particle Duality l=h/mv

2. Complete the following statement: According to the de Broglie relation, the wavelength of a "matter" wave is inversely proportional to • Planck's constant. • (b) the frequency of the wave. • (c) the mass of the particle. • (d) the speed of the particle. • (e) the momentum of the particle. X

3. What happens to the de Broglie wavelength of an electron if its momentum is doubled? (a) The wavelength decreases by a factor of 4. (b) The wavelength increases by a factor of 2. (c) The wavelength increases by a factor of 4. (d) The wavelength decreases by a factor of 2 (e) The wavelength increases by a factor of 3. X

4. Determine the de Broglie wavelength of a neutron (m = 1.67 × 10–27 kg) that has a speed of 5.0 m/s. (a) 79 nm (b) 395 nm (c) 1975 nm (d) 162 nm (e) 529 nm X

5. Upon which one of the following parameters does the energy of a photon depend? (a) mass (c) polarization (e) phase relationships (b) amplitude (d) frequency For which one of the following problems did Max Planck make contributions that eventually led to the development of the “quantum” hypothesis? (a) photoelectric effect (d) the motion of the earth in the ether (b) uncertainty principle (e) the invariance of the speed of light through vacuum (c) blackbody radiation curves X X

6. Light is usually thought of as wave-like in nature and electrons as particle-like. In which one of the following activities does light behave as a particle or does an electron behave as a wave? (a) A Young’s double slit experiment is conducted using blue light. (b) X-rays are used to examine the crystal structure of sodium chloride. (c) Water is heated to its boiling point in a microwave oven. (d) An electron enters a parallel plate capacitor and is deflected downward. (e) A beam of electrons is diffracted as it passes through a narrow slit. X

7. Wave Function, yThe Heisenberg Uncertainty Principle

8. Description of waves Type of waves Variable physical quantity Water waves Height of the water surface Sound waves Pressure in the medium Light waves Electric and magnetic fields Matter waves Wave function,  • , the amplitude of a matter wave, is a function of time and position

9. Probability Density 2 The value of 2 for a particular object at a certain place and time is proportional to the probability of finding the object at that place at that time. For example: 2 =1: the object is definitely there 2 =0: the object is definitely not there 2 =0.4: there is 40% chance of finding the object there at that time. 2 starts from Schrodinger’s equation, a differential equation that is central to quantum mechanics

11. Why2? Why not ? • Amplitude of every wave varies from –A to +A to –A to +A and so on (A is the maximum absolute value whatever the wave variable is). • A negative probability is meaningless. • 2gives a positive quantity that can be compared with experiments.

12. The key point to the wave function is that the position of a particle is only expressed as a likelihood or probability until a measurement is made.

13. The probability that the electron will be found at the particular position is determined by the wave function illustrated to the right of the aperture. When the electron is detected at A, the wave function instantaneously collapses so that it is zero at B.

14. Question: A large value of the probability density Y2 of an atomic electron at a certain place and time signifies that the electron • is likely to be found there • is certain to be found there • has a great deal of energy there • has a great deal of charge there Answer: a

15. Question: A moving body is described by the wave function  at a certain time and place. The value of 2 is proportional to the body’s • electric field. • speed • energy • probability of being found Answer: d

16. Question: What do physicists enjoy the most at baseball games? Answer: The “wave”.

17. The Heisenberg Uncertainty Principle

18. If an object has a well-defined position at a certain time, its momentum must have a large uncertainty. • If an object has a well-defined momentum at a certain time, its position must have a large uncertainty.

19. Uncertainty Principle Momentum and position DxDp ≥ h/4p Energy and time DEDt ≥ h/4p

20. p=h/l precise x unknown Dx better defined (narrower wave packet) Dp less defined (greater spread of l) Uncertainty principleDxDp≥h/4p

21. The Heisenberg Uncertainty Principle In the microscopic world where the wave aspects of matter are very significant, these wave aspects set a fundamental limit to the accuracy of measurements of position and momentum regardless of how good instruments used are. The uncertainty principle is the physical law which follows from the wave nature of matter.

22. Example: Compare the de Broglie wavelength of 54-eV electrons with that of a 1500-kg car whose speed is 30 m/s. Solution: For the 54-eV electron: KE=(54eV)(1.6x10-19J/eV)=8.6x10-18 J KE=1/2 mv2, mv=(2mKE)1/2 l=h/mv=h/(2mKE)1/2= 1.7x10-10 m The wavelength of the electron is comparable to atomic scales (e.g., Bohr radius=5.29x10-11 m). The wave aspects of matter are very significant. For the car: l=h/mv=6.63x10-34 J•s/(1.5x103)(30m/s)= 1.5x10-38 m The wavelength is so small compared to the car’s dimension that no wave behavior is to be expected.

23. Question: The narrower the wave packet of a particle is • the shorter its wavelength • the more precisely its position can be established • the more precisely its momentum can be established • the more precisely its energy can be established Answer: b

24. Question: The wave packet that corresponds to a moving particle • has the same size as the particle • has the same speed as the particle • has the speed of light • consists of x-ray Answer: b

25. Question: If Planck’s constant were larger than it is, a. moving bodies would have shorter wavelength b. moving bodies would have higher energies c. moving bodies would have higher momenta d. The uncertainty principle would be significant on a larger scale of size Answer: d

26. If Planck’s constant were changed to 660 J•s, what would be the minimum uncertainty in the position of a 120-kg football player running at a speed of 3.5 m/s? (a) 0.032 m (b) 0.13 m (c) 0.50 m (d) 0.065 m (e) 0.25 m X

27. ChapterAtomic Physics The Hydrogen Atom The Bohr Model Electron Waves in the Atom

28. Line Spectrum

29. Rutherford Model Predicts: • A continuous range of frequencies of light emitted • Unstable atoms • These are inconsistent with experimental observations Why ?

30. Niels Henrik David Bohr 1885-1962

31. Quantized orbits Each orbit has a different energy

32. Excited Electron

33. Photon emitted: hf=Eu-El

35. 1/l=R(1/22-1/n2), n=3,4,… for Balmer series where Rydberg constant R=1.097x107 m-1

36. Line Spectrum: hf=Eu-El

37. Equations Associated with The Bohr Model Electron’s angular momentum L=Iw=mvrn=nh/2p, n=1,2,3 n is called quantum number of the orbit Radius of a circular orbit rn=n2h2/4p2mkZe2=(n2/Z)r1 where r1=h2/4p2mke2=5.29x10-11 m (n=1) r1 is called Bohr radius, the smallest orbit in H Total energy for an electron in the nth orbit: En=(-2p2Z2e4mk2/h2)(1/n2)=(Z2/n2)E1 where E1=-2p2Z2e4mk2/h2 =-13.6 eV (n=1) E1 is called Ground State of the hydrogen Both orbits and energies depend on n, the quantum number

38. To break a hydrogen atom apart requires 13.6 eV + 13.6 eV = + electron Proton Hydrogen atom e V=2.2x106 m/s p Electron orbit r=0.053 nm

39. Question: The classical model of the hydrogen atom fails because • a moving electron has more mass than an electron at rest • a moving electron has more charge than an electron at rest • the attractive force of the nucleus is not enough to keep an electron in orbit around it • an accelerated electron radiates electromagnetic waves Answer: d

40. Question: In the Bohr model of the hydrogen atom, the electron revolves around the nucleus in order to • emit spectral lines • produce X rays • form energy levels that depend on its speed • keep from falling into the nuclues Answer: d

41. Question: A hydrogen atom is in its ground state when its orbital electron • is within the nucleus • has escaped from the atom • is in its lowest energy level • is stationary Answer: c

42. Example: Find the orbital radius and energy of an electron in a hydrogen atom characterized by principal quantum number n=2. Solution: For n=2, r2=r1n2=0.0529nm(2)2=0.212 nm and E2=E1/n2=-13.6/22 eV=-3.40 eV

43. Line and Absorption Spectra hf=Eu- El hc/l=Eu - El 1/l=(1/hc)Eu- El 1/l=(2p2Z2e4mk2/ch3)(1/nl2-1/nu2)

47. Orbits and energies are quantized! Why?

48. Wave-Particle Duality de Broglie proposed that electrons, too, have a wave nature and a wavelength and that all material objects have a wave nature. In particular, deBroglie proposed that the wavelength of a body could be found from l = h/p= h/mv This wave is often called its de Broglie wave. Planck's constant h has such an extremely small value that the wavelength associated with any ordinary object is far too small to be experimentally detected. Louis de Broglie (1892-1987)

49. The quantized orbits and energy states in the Bohr model are due to the wave nature of the electron, and the electron wave functions can only occur in the form of standing waves. Implication: The wave-particle duality is at the root of atomic structure

50. Condition for orbit An electron can circle an atomic nucleus only if its orbit is a whole number of electron wavelengths in circumference Condition for orbit stability nl=2prn, n=1,2,3…