Lecture 19: The Hydrogen Atom

• Reading: Zuhdahl 12.7-12.9

• Outline– The wavefunction for the H atom– Quantum numbers and nomenclature– Orbital shapes and energies

H-atom wavefunctions• Recall from the previous lecture 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

8ε02h2

⎛

⎝ ⎜

⎞

⎠ ⎟= −2.178x10−18J

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).

ml, the “z component” of orbital angular momentum. Ranges in value from -l to 0 to l.

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)

€

ψ1s =1

π

Z

ao

⎛

⎝ ⎜

⎞

⎠ ⎟

32

e−Z

a0

r

=1

π

Z

ao

⎛

⎝ ⎜

⎞

⎠ ⎟

32

e−σ

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

€

En = −2.178x10−18JZ 2

n2

⎛

⎝ ⎜

⎞

⎠ ⎟= −2.178x10−18J

1

1

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

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

€

ψ1s =1

π

Z

ao

⎛

⎝ ⎜

⎞

⎠ ⎟

32

e−Z

a0

r

=1

π

Z

ao

⎛

⎝ ⎜

⎞

⎠ ⎟

32

e−σ

€

ψ*ψProbability =

• Probability is spherical

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.

QuickTime™ and aCinepak Codec by Radius decompressorare needed to see this picture.

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

• not spherical, but lobed.

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

€

ψ2pz=

1

4 2π

Z

ao

⎛

⎝ ⎜

⎞

⎠ ⎟

32

σe−σ

2 cosθ

Orbital Shapes (cont.)3p orbitals

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

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

€

ψ3pz=

2

81 π

Z

ao

⎛

⎝ ⎜

⎞

⎠ ⎟

32

6σ −σ 2( )e

−σ3 cosθ

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

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

Orbital Shapes (cont.)

3d (l = 2) orbitals

• dxy • dx2-y2

Orbital Shapes (cont.)

3d (l = 2) orbitals

• 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 (“degenerage”).

• the “ground” or lowest energy orbital is the 1s.