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

Schrodinger Equation• Erwin Schrodinger develops a mathematical

formalism that incorporates the wave nature of matter:

ˆ H E

ˆ H The Hamiltonian:

ˆ p 2

2m (PE)

Kinetic Energy

The Wavefunction:x

E = energy

d2/dx2

Potentials and Quantization (cont.)

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

“Particle in a Box”

Potential E

x0

inf.

0 L = 0 for 0 ≤ x ≤ L

= all other x

Potentials and Quantization (cont.)

• What does the energy look like?

Energy is quantized

E

E n2h2

8mL2

n = 1, 2, …

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:

E h2

8mL2 n final2 ninitial

2 h2

8m(8Å)2 22 1 2.8x10 19 J

What wavelength of light corresponds to E from n=1 to n=2?

L = 8 Å

700nm(should be 680 nm)

N

N

+

H-atom wavefunctions

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

• The hydrogen atom potential energy is given by:

e-

P+r

r0

V (r) e2

r

H-atom wavefunctions (cont.)• The Coulombic potential can be generalized:

e-

P+r

V (r) Ze2

r Z

• Z = atomic number (= 1 for hydrogen)

H-atom wavefunctions (cont.)• The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates.

p+

e-

r = interparticle distance (0 ≤ r ≤ )

= angle from “xy plane” (/2 ≤ ≤ - /2)

= rotation in “xy plane” (0 ≤ ≤ 2)

H-atom wavefunctions (cont.)• If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized:

En Z 2

n2

me4

802h2

2.178x10 18 J

Z 2

n2

• n is the principle quantum number, and ranges from 1 to infinity.

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).

Orbital Shapes• Let’s take a look at the lowest energy orbital, the

“1s” orbital (n = 1, l = 0, m = 0)

1s 1

Z

ao

32

e

Z

a0

r

1

Z

ao

32

e

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

En 2.178x10 18 JZ 2

n2

2.178x10 18 J

1

1

Orbital Shapes (cont.)• Note that the “1s” wavefunction has no angular

dependence (i.e., and do not appear).

1s 1

Z

ao

32

e

Z

a0

r

1

Z

ao

32

e

*Probability =

• Probability is spherical

Orbital Shapes (cont.)• Radial probability (likelihood of finding the

electron in each spherical shell

1s 1

Z

ao

32

e

Z

a0

r

1

Z

ao

32

e

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”

Orbital Shapes (cont.)

• Table 12.3: Quantum Numbers and Orbitals

n l Orbital ml # of Orb.

1 0 1s 0 12 0 2s 0 1

1 2p -1, 0, 1 33 0 3s 0 1 1 3p -1, 0, 1 3

2 3d -2, -1, 0, 1, 2 5

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 sublevels4f (7 ml sublevels)

Orbital Shapes (cont.)s (l = 0) orbitals

• r dependence only

• as n increases, orbitals demonstrate n-1 nodes.

Orbital Shapes (cont.)2p (l = 1) orbitals

• not spherical, but lobed.

• labeled with respect to orientation along x, y, and z.

2 pz

1

4 2Z

ao

32

e

2 cos

Orbital Shapes (cont.)3p orbitals

• more nodes as compared to 2p (expected.).

• still can be represented by a “dumbbell” contour.

3 pz

2

81 Z

ao

32

6 2 e 3 cos

Orbital Shapes (cont.)3d (l = 2) orbitals

• labeled as dxz, dyz, dxy, dx2-y2 and dz2.

Orbital Shapes (cont.)4f (l = 3) orbitals

• exceedingly complex probability distributions.

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.