Hydrogen Atom

1
Hydrogen Atom

• Coulomb force confines electron to region near
  proton gt standing waves of certain energy

2
Electron in n2 level makes a transition to
lower level by emitting a photon of frequency
fE/h (E2-E1)/h c/?
3
Transitions

• Electron in ground state E1 can move to the
  excited state E3 if it absorbs energy equal to
  E3-E1
• absorb a photon hf E3-E1
• electron will not stay in the excited state for
  long gt emits a photon or a sequence of photons

hf E3 - E1
hf E3 - E2 hc/?
hf E2 - E1 hc/?
photon
Emission spectrum
4
Continuous visible spectrum
Line spectra from H, He, Ba, Hg
5
Hydrogen Atom

• Coulomb force confines electron to region near
  proton gt standing waves of certain energy

6
Atoms

• In 1913 Neils Bohr proposed a model of hydrogen
  based on a particle in an orbit
• electron with charge -e in a circular orbit about
  a nucleus of charge Ze
• Coulomb attraction provides centripetal force
  mv2/r kZe2/r2
• potential energy is U kq1q2/r -kZe2/r
• kinetic energy K(1/2)mv2(1/2)kZe2/r
• hence U -2K (same for gravity!)
• total E K U -K -(1/2)kZe2/r
• e/m theory states that an accelerating
  charge radiates energy!
• Should spiral into the nucleus!
• Why are atoms stable?

7
Bohrs postulates

• Bohr postulated that only certain orbits were
  stable and that an atom only radiated energy when
  it made a transition from one level to another
• the energy of a photon emitted was hf Ei -
  Ef
• since the energies of the orbits are related to
  their radii, f (1/2)(kZe2/h)(1/r2 - 1/r1)
• experimentally observed photon frequencies
  satisfied fc/? cR(1/n22 - 1/n12) where n1
  and n2 are integers Rydberg-Ritz formula
• do the allowed values of r ? n2 ?
• If we think of the allowed orbits as standing
  waves then we need 2?r n? for constructive
  interference

8
Stable orbits

• 2?r n? for constructive interference
• but de Broglie says ?h/p
• hence pr nh/2? but Lrp for circular
  orbits
• hence L mvr n h n1,2,3,
• angular momentum is quantized!

Bohr Atom
9
Bohr Theory

• How do we find the allowed radii?
• Coulomb force kZe2/r2 mv2/r gt v2kZe2/mr
• but Bohr says mvr n h gt v2
  n2 h2/m2r2
• solve for r r n2 (h2/mkZe2) n2
  a0/Z where a0 is a radius corresponding to n1
  and Z1(Hydrogen)
• a0 h2/mke2 0.0529 nm (called the
  Bohr radius)
• hence only certain orbits are allowed gt only
  certain energies
• energy differences (1/2)kZe2(1/r2 -
  1/r1) (1/2)(kZ2e2/a0)(1/n22 - 1/n12)

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Bohr Atom

• compare with Rydberg-Ritz formula for observed
  wavelengths in Hydrogen 1/? R(1/n22 -
  1/n12) where R is Rydberg constant
• frequency of photons fc/? c R(1/n22 - 1/n12)
  (E2 - E1)/h
• using Z1, Rmk2e4/4?ch3 1.096776 x 107 m-1
  in agreement with experiment!
• Energy levels can be determined from allowed
  radii
• E-(1/2)kZe2/r -(mk2e4/2h2)(Z2/n2) -E0 Z2/n2
• E0 is the lowest energy for hydrogen 13.6 eV
• hence hydrogen atom(Z1) has energies En
  -13.6eV/n2 n1,2,...

11
E1 -13.6eV
En -13.6eV/n2 n1,2,3,...
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Hydrogen Atom

• En -13.6eV/n2 n1,2,3,
• ground state has E1 -13.6eV
• ionization energy is 0- E1 13.6eV gt energy
  needed to remove electron
• excited state n2 E2 -(13.6/4)eV
• electron must absorb a photon of energy hf E2
  - E1 hc/? (3/4)(13.6eV)

13
Electron in n2 level makes a transition to
lower level by emitting a photon of frequency
fE/h (E2-E1)/h c/?
14
En-E2 13.6eV ( 1/4 -1/n2) hf hc/?
?maxhc/(13.6eV)(5/36) 658 nm gt 4
lines visible
En-E1 13.6eV ( 1 -1/n2) hf hc/?
?maxhc/(13.6eV)(.75) 122 nm
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Balmer Series