Atoms and Quanta
Quantum Mechanics
Return to PHYS 202 Outline

Outline
Supplementary Homework Problems
Answers to Problems
Return to top

1. Blackbody radiation S-42,43
1. spectral response: l _max = b/T where b = 2.9 x 10^-3 m K

power output: P_out/in = s e A T_obj/sur⁴ where s = 5.67 x 10^-8 W/m²K⁴ and
e measures whiteness (e =0) or blackness (e =1)

2. Classical theory: predicts UV catastrophe: I_ntensity/l ® ¥ as l ® 0 .
3. Planck's theory: start with: DE = hf (DE does not go to 0)
   where h = 6.63 x 10^-34 J sec (energy not continuous)
2. Photoelectric Effect S-44
1. wave theory: wrongly predicts no V_stop, time lag, no f_cutoff
2. particle theory: start with E_{one photon} = hf (same h as Planck)
3. The atom: a nucleus and electrons S-45
1. size of atoms
1. of the atom
2. of the nucleus
3. of the electrons
2. charge of electrons & protons
3. planetary model: trouble with radiation
4. Bohr theory S-46
1. assumptions: L = nh ; circular orbits (where h º h/2p )
2. get: E_n = [-mk²Z²e⁴] / [2h ²n²] ; and r = [n²h ²] / [mke²Z]
3. spectra: DE = -13.6 eV [(1/n_f²) - (1/n_i)²] = h f = h w
4. weaknesses:
1. only works for 1 electron atoms/ions
2. why assume L = nh ?

Return to top of Section 1

Supplementary Problems (S- ):

42. Assume a temperature for the skin surface of yourself and calculate the wavelength where your radiation peaks assuming you radiate as a perfect blackbody.

43. The sun's power output per wavelength vs. wavelength curve peaks at a wavelength of 480 nm. (a) Assuming the sun is a perfect blackbody [a good but not perfect assumption], calculate its surface temperature. (b) What is the total power output of the sun? (R_SUN = 7X10⁵ km.)

(c) What is the total power per area (in Watts/m²) at the position where the earth orbits the sun (radius of orbit = 93 million miles = 149 million kilometers)? [This answer is for a collector pointed directly at the sun without any atmosphere/clouds and without any day/night.]

44. The work function for Aluminum is 4.08 eV. (a) What is the cut-off frequency? (b) What is the stopping voltage for light of wavelength 250 nm (in the UV) ?

45. What is the approximate size of (a) atoms? (b) nucleons? (c) electrons? (d) How are these sizes determined?

46. (a) What is the ground state energy of the hydrogen atom? (b) How much energy does it take to ionize the hydrogen atom? (c) What would be the maximum wavelength of light that would be able to do this? (d) What kind of photon is this (i.e., ultraviolet, visible, IR, etc.) ? (e) How much energy would it take to excite a hydrogen atom in the ground state up to the n=3 state? (f) What would be the possible decay schemes for this excited state, and the energy, wavelength, and color of the photons emitted in each scheme?

Return to top of Section 1

Answers to Supplementary Problems:

42. depends on the temperature; if you assume a T of 30 °C, then wavelength of max. power is 9.6 x 10^-6 meters (I.R.).

43. a) 6000 K; b) 4.5x10²⁶ Watts; c) answer not given here - try to see if your answer is reasonable

44. a) 9.85 x 10¹⁴ Hz; b) 0.89 Volts.

45. (a) 1 x 10^-10 m; (b) 1 x 10^-14 m; (c) less than 1 x 10^-17 m; (d) by scattering high speed charged projectiles from them.

46. a) -13.6 eV; b) 13.6 eV; c) 91.4 nm; d) UV; e) 12.09 eV; f-1) 3 _® 1, E = 12.09 eV, l = 103 nm, UV; f-2) 3 _® 2 _® 1, E = 1.89 eV & 10.20 eV, l = 658 nm & 122 nm, red & UV.

Return to top of Section 1

Outline
Supplementary Homework Problems
Return to top

1. Waves and particles S-47
1. for light: DE = hf; lf = v
2. for matter: l = h/p (DeBroglie wavelength)
2. Heisenberg's Uncertainty Principle
1. position & momentum: Dx Dp_x > h/4p (here, D indicates uncertainty, not change)
1. if know l , (Dl ® 0 Þ Dp® 0), don't know x (Dx® ¥ )
2. if know x, (Dx® 0), don't know l (Dl ® ¥ Þ D p® ¥ )
2. energy & time: DE Dt > h/4p
3. Schrodinger's Equation S-48,49
1. depends on potential energy for situation
2. wave function, Y
1. Y is solution of Schrodinger's Equation
2. Y is related to probability of finding particle at x,t
3. quantum numbers
1. there are three (in 3-D) from applying boundary conditions to Schrodinger's Equation;
2. there is a fourth from relativistic considerations (spin)
4. Pauli Exclusion Principle & periodic chart
1. a consequence of Heisenburg Uncertainty Principle
2. in nature there are two types of particles:
1. fermions: Y _{2 particles} = 0 if both particles have same quantum numbers and are in same area of space
   electrons, protons, neutrons (spin 1/2 particles)
2. bosons: Y _{2 particles} > 0 if both particles have same quantum numbers and are in same area of space
   photons, alpha particles (integer spin particles)
5. emission & absorption: conservation of energy and angular momentum
4. Quantum applications S-50
1. laser
2. diode and transistor

Return to top of Section 2

Supplementary Problems (S- ):

47. List three experiments that suggest light behaves (a) as a wave; (b) as a particle; list two experiments that suggest matter behaves (c) as a wave; (d) as a particle.

48. What is Y ? How is it related to reality?

49. What is the Pauli Exclusion Principle and where does it come from?

50. Explain the operation of a laser in terms of quantum levels, absorption, spontaneous and stimulated emission, and population inversion.

Return to top of Section 2