Lecture 17: The Hydrogen Atom

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Lecture 17 The Hydrogen Atom

• Reading Zuhdahl 12.7-12.9
• Outline
• The wavefunction for the H atom
• Quantum numbers and nomenclature
• Orbital shapes and energies

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Schrodinger Equation

• Erwin Schrodinger develops a mathematical
  formalism that incorporates the wave nature of
  matter

Kinetic Energy
The Hamiltonian
d2/dx2
x
The Wavefunction
E energy
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Potentials and Quantization (cont.)

• What if the position of the particle is
  constrained by a potential

Particle in a Box
Potential E
0 for 0 x L
? all other x
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Potentials and Quantization (cont.)

• What does the energy look like?

n 1, 2,
Energy is quantized
E
y
yy
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Potentials and Quantization (cont.)

• Consider the following dye molecule, the
  length of which can be considered the length of
  the box an electron is limited to

L 8 Å
What wavelength of light corresponds to DE from
n1 to n2?
(should be 680 nm)
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H-atom wavefunctions

• Recall that the Hamiltonian is composite of
  kinetic (KE) and potential (PE) energy.

The hydrogen atom potential energy is given by
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H-atom wavefunctions (cont.)
The Coulombic potential can be generalized
Z
Z atomic number ( 1 for hydrogen)
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H-atom wavefunctions (cont.)
The radial dependence of the potential
suggests that we should from Cartesian
coordinates to spherical polar coordinates.
r interparticle distance (0 r ?)
e-

• angle from xy plane
• (?/2 ? - ?/2)

p
? rotation in xy plane (0 ? 2?)
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H-atom wavefunctions (cont.)
If we solve the Schrodinger equation using
this potential, we find that the energy
levels are quantized
n is the principle quantum number, and ranges
from 1 to infinity.
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H-atom wavefunctions (cont.)
In solving the Schrodinger Equation, two
other quantum numbers become evident
l, the orbital angular momentum quantum
number. Ranges in value from 0 to (n-1).
m, the z component of orbital angular
momentum. Ranges in value from -l to 0
to l.
We can then characterize the wavefunctions
based on the quantum numbers (n, l, m).
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Orbital Shapes

• Lets take a look at the lowest energy orbital,
  the 1s orbital (n 1, l 0, m 0)

• a0 is referred to as the Bohr radius, and
  0.529 Å

1
1
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Orbital Shapes (cont.)

• Note that the 1s wavefunction has no angular
  dependence (i.e., Q and F do not appear).

Probability
Probability is spherical
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Orbital Shapes (cont.)

• Radial probability (likelihood of finding the
  electron in each spherical shell

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Orbital Shapes (cont.)

• Naming orbitals is done as follows
• n is simply referred to by the quantum number
• l (0 to (n-1)) is given a letter value as
  follows
• 0 s
• 1 p
• 2 d
• 3 f

- ml (-l0l) is usually dropped
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Orbital Shapes (cont.)
Table 12.3 Quantum Numbers and Orbitals

• n l Orbital ml of Orb.
• 0 1s 0 1
• 0 2s 0 1
• 1 2p -1, 0, 1 3
• 0 3s 0 1
• 1 3p -1, 0, 1 3
• 2 3d -2, -1, 0, 1, 2 5

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Orbital Shapes (cont.)
Example Write down the orbitals associated
with n 4.
Ans n 4
l 0 to (n-1) 0, 1, 2, and 3 4s, 4p, 4d,
and 4f
4s (1 ml sublevel) 4p (3 ml sublevels) 4d (5 ml
sublevels 4f (7 ml sublevels)
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Orbital Shapes (cont.)
s (l 0) orbitals
r dependence only
as n increases, orbitals demonstrate n-1
nodes.
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Orbital Shapes (cont.)
2p (l 1) orbitals
not spherical, but lobed.
labeled with respect to orientation along x,
y, and z.
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Orbital Shapes (cont.)
3p orbitals
more nodes as compared to 2p (expected.).
still can be represented by a dumbbell
contour.
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Orbital Shapes (cont.)
3d (l 2) orbitals
labeled as dxz, dyz, dxy, dx2-y2 and dz2.
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Orbital Shapes (cont.)
4f (l 3) orbitals
exceedingly complex probability distributions.
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Orbital Energies
energy increases as 1/n2
orbitals of same n, but different l are
considered to be of equal energy
(degenerate).
the ground or lowest energy orbital is
the 1s.