Quantum Theory of Atoms

1
Quantum Theory of Atoms

• The Bohr theory of Hydrogen(1913) cannot be
  extended to other atoms with more than one
  electron
• we have to solve the Schrödinger equation(1925)
• since the Coulomb force only depends on r, we
  should use spherical coordinates (r,?,?) instead
  of Cartesian coordinates (x,y,z)

2
Quantum Theory of Atoms

• For a particle in a cubic or rectangular box,
  Cartesian coordinates are more appropriate and
  there are three quantum numbers (n1,n2,n3) needed
  to label a quantum state
• similarly in spherical coordinates, there are
  also three quantum numbers needed
• n 1, 2, 3, ..
• l 0, 1, 2, ,n-1 gt n values of l
  for a given value of n
• m -l,(-l1),0,1,2,,l gt 2l1 values of m
  for a given l
• n is the principal quantum number and is
  associated with the distance r of an electron
  from the nucleus
• l is the orbital quantum number and the angular
  momentum of the electron is given by
  Ll(l1)1/2 h
• m is the magnetic quantum number and the
  component of the angular momentum along the
  z-axis is Lz m h

3
Quantum Theory of Atoms

• The fact that both l and m are restricted
  to certain values is due to boundary
  conditions
• in the figure, l2 is shown
• hence L(2(21))1/2 h h(6)1/2
• and m -2, -1, 0, 1, 2
• the Schrödinger equation can be
  solved exactly for hydrogen
• the energies are the same as in the Bohr
  theory En -Z2 (13.6 eV)/n2
• they do not depend on the value of l and m
• this is a special property of an inverse-square
  law force

4

• The lowest energy has n1 gt l0 and m0
• the second lowest energy has n2 gt l0,
  m0 l1, m-1,0,1
• hence 4 states!
• Notation l0 S-state
• l1 P-state
• l2 D-state
• l3 F-state
• l4 G-state

5

• When a photon is emitted or absorbed we must
  have ?m 0 or 1 ?l 1
• conservation of angular momentum and conservation
  of energy

6
Wave Functions

• We denote the solutions of the SE as ?nlm
  (r,?,?)
• the probability of finding the electron at some
  position (r,?,?) is P (r,?,?) (?nlm)2 dV where
  dV is the volume element
• in spherical coordinates dVr2drsin?d?d?

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Wave Functions

• the ground state wave function is Cexp(-Zr/a0)
  where C is determined from normalization

• Hence for the ground state, the probability
  density ?1002 is independent of ? and ? and is
  maximum at the origin

8
Wave Functions

• What is the probability of finding the electron
  between r and rdr?
• in other words, in a spherical shell of thickness
  dr
• volume of shell is 4?r2 dr (area x
  thickness)
• P(r)dr (4?r2 ?2)dr P(r) is the radial
  probability density

9
Wave Functions

• Maximum is at r a0/Z a0 for Hydrogen (first
  Bohr orbit)
• not a well defined orbit but rather a cloud

10
Probability density