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.
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