1. Lecture 17: The Hydrogen Atom
Reading: Zuhdahl
Outline
The wavefunction for the H atom
Quantum numbers and nomenclature
Orbital shapes and energies

2. 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

3. 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

4. Potentials and Quantization (cont.)
• What does the energy look like?
n = 1, 2, …
Energy is quantized E y
y*y

5. 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 n=1 to n=2?
(should be 680 nm)

6. H-atom wavefunctions
Recall that the Hamiltonian is composite of kinetic (KE) and potential (PE) energy.
• The hydrogen atom potential energy is given by:

7. H-atom wavefunctions (cont.)
• The Coulombic potential can be generalized: Z
• Z = atomic number (= 1 for hydrogen)

8. 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)

9. 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.

10. 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).

11. Orbital Shapes
Let’s 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 = Å 1 1

12. Orbital Shapes (cont.)
Note that the “1s” wavefunction has no angular dependence (i.e., Q and F do not appear).
Probability =
• Probability is spherical

13. Orbital Shapes (cont.)
Radial probability (likelihood of finding the electron in each spherical shell

14. 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 (-l…0…l) is usually “dropped”

15. Orbital Shapes (cont.) • Table 12.3: Quantum Numbers and Orbitals
n l Orbital ml # of Orb. s s
p , 0, s
p , 0,
d -2, -1, 0, 1,

16. 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)

17. Orbital Shapes (cont.) s (l = 0) orbitals • r dependence only
• as n increases, orbitals
demonstrate n-1 nodes.

18. Orbital Shapes (cont.) 2p (l = 1) orbitals • not spherical, but lobed.
• labeled with respect to orientation along x, y, and z.

19. Orbital Shapes (cont.) 3p orbitals
• more nodes as compared to 2p (expected.).
• still can be represented by a “dumbbell” contour.

20. Orbital Shapes (cont.) 3d (l = 2) orbitals
• labeled as dxz, dyz, dxy, dx2-y2 and dz2.

21. Orbital Shapes (cont.) 4f (l = 3) orbitals
• exceedingly complex probability distributions.

22. 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.