5.61 Fall 2007
Lecture #21
page 1
HYDROGEN ATOM
Schrodinger equation in 3D spherical polar coordinates:
−
!2 ⎡ 1 ∂ ⎛ 2 ∂ ⎞
1
∂ ⎛
∂⎞
1
∂2 ⎤
r
+
sin
θ
+
⎢
⎥ ψ r,θ ,φ + U r,θ ,φ ψ r,θ ,φ = Eψ r,θ ,φ
2 µ ⎣ r 2 ∂r ⎜⎝ ∂r ⎟⎠ r 2 sin θ ∂θ ⎜⎝
∂θ ⎟⎠ r 2 sin 2 θ ∂φ 2 ⎦
(
with Coulomb potential
U (r) =
)
(
) (
)
−Ze2
4πε 0 r
Rewrite as
⎡ 2 ∂⎛ 2 ∂⎞
⎤
r
+ 2 µr 2 ⎡⎣U r − E ⎤⎦ ⎥ ψ r,θ ,φ + L̂2ψ r,θ ,φ = 0
⎢ −!
⎜
⎟
∂r ⎝ ∂r ⎠
⎣
⎦
()
(
)
function of r only
ˆH
r is separable
⇒
(
)
function of θ,φ only
ψ is separable
Angular momentum: solutions are spherical harmonic wavefunctions
(
)
() ( )
(θ ,φ ) = ! l ( l + 1)Y (θ ,φ )
ψ r,θ ,φ = R r Yl m θ ,φ
with
L̂2Yl m
2
m
l
l = 0,1,2,...
Radial equation for the H atom:
()
(
)
2
⎤
!2 d ⎛ 2 dR r ⎞ ⎡ ! l l + 1
−
r
+
+
U
r
−
E
⎢
⎥R r = 0
⎟
⎜
dr ⎠ ⎢⎣ 2 µr 2
2 µr 2 dr ⎝
⎥⎦
()
()
()
Solutions R r are the H atom radial wavefunctions
Simplest case: l = 0 yields solution
⎛Z⎞
R r = 2⎜ ⎟
⎝ a0 ⎠
()
32
e
− Zr a0
Y
ml
exponential decay away from nucleus
(
)
5.61 Fall 2007
Lecture #21
page 2
with
E = − Z 2 e2 8πε 0 a0
lowest energy eigenvalue
Bohr radius
a0 ≡ ε 0 h2 πµe2
General case: solutions are products of (exponential) x (polynomial)
Energy eigenvalues:
−Z 2 e2
−Z 2 µe4
=
8ε 02 h2 n2
8πε 0 a0 n2
E=
n = 1,2,3,...
Radial eigenfunctions:
Rnl
12
l +3 2
⎡
⎤
n − l − 1 ! ⎥ ⎛ 2Z ⎞
⎛
⎞
− Z r na0 2l +1 2 Zr
⎢
r
= −
r le
Ln+ l ⎜
⎜
⎟
⎟
3
⎢
⎥ ⎝ na ⎠
⎝ na0 ⎠
⎡
⎤
0
2n
n
+
l
!
⎦ ⎦⎥
⎣⎢ ⎣
(
()
)
)
(
(
)
where L2ln++1l 2Zr na0 are the associated Laguerre functions, the first few of which are:
n=1
l=0
L11 = −1
n=2
l=0
⎛
⎞
L12 = −2!⎜ 2 − Zr ⎟
a0
⎠
⎝
l =1
L33 = −3!
n=3 l=0
⎛
⎞
2 2
L13 = −3!⎜ 3 − 2Zr + 2Z r
2⎟
a0
9a0 ⎠
⎝
l =1
⎛
⎞
L34 = −4!⎜ 4 − 2Zr
3a0 ⎟⎠
⎝
l=2
L55 = −5!
Normalization:
Spherical harmonics
∫
2π
Radial wavefunctions
∫
∞
0
0
( ) ( )
π
dφ ∫ dθ sin θ Yl m* θ ,φ Yl m θ ,φ = 1
0
() ()
dr r 2 Rnl* r Rnl r = 1
5.61 Fall 2007
Lecture #21
page 3
TOTAL HYDROGEN ATOM WAVEFUNCTIONS
(
)
() ( )
ψ nlm r,θ ,φ = Rnl r Yl m θ ,φ
principle quantum number
angular momentum quantum number
magnetic quantum number
ENERGY depends on n:
n = 1,2,3,...
l = 0,1,2,..., n − 1
m = 0, ±1, ±2,..., ±l
E = −Z 2 e2 8πε 0 a0 n2
(
ORBITAL ANGULAR MOMENTUM depends on l:
)
L = ! l l +1
ANGULAR MOMENTUM Z-COMPONENT depends on m:
Lz = m!
Total H atom wavefunctions are normalized and orthogonal:
∫
2π
0
π
∞
0
0
(
)
(
)
*
dφ ∫ sin θ dθ ∫ r 2 dr ψ nlm
r,θ ,φ ψ n ' l ' m ' r,θ ,φ = δ nn 'δ ll 'δ mm '
() ( )
since components Rnl r Yl m θ ,φ are normalized and orthogonal.
Lowest few total H atom wavefunctions, for n = 1 and 2 (with σ = Zr a0 ):
5.61 Fall 2007
n=1
n=2
Lecture #21
l=0
l=0
m=0
m=0
page 4
3/ 2
ψ 100
1 ⎛Z⎞
=
⎜ ⎟
π ⎝ a0 ⎠
3/ 2
ψ 200
⎛Z⎞
=
⎜ ⎟
32π ⎝ a0 ⎠
3/ 2
ψ 210
⎛Z⎞
=
⎜ ⎟
32π ⎝ a0 ⎠
e− σ
= ψ 1s
1
1
(2 − σ ) e
−σ / 2
= ψ 2s
σ e− σ / 2 cosθ = ψ 2 p
l =1
m=0
l =1
⎛Z⎞
−σ / 2
m = ±1 ψ 21±1 =
sin θ e± iφ
⎜ a ⎟
σ e
64π ⎝ 0 ⎠
z
3/ 2
1
or the alternate linear combinations
ψ2p
ψ2p
⎛Z⎞
=
⎜ ⎟
32π ⎝ a0 ⎠
1
x
⎛Z⎞
=
⎜ ⎟
32π ⎝ a0 ⎠
1
y
3/ 2
σ e− σ / 2 sin θ cos φ =
3/ 2
σ e− σ / 2 sin θ sin φ =
1
2
1
2i
(ψ
21+1
+ ψ 21−1
)
(ψ
21+1
− ψ 21−1
)
The value of l is denoted by a letter: l = 0,1,2,3...
s,p,d,f orbitals
The value of m is denoted by a letter for l = 1: m = 0, ± 1 linear combinations
pz , px ,p y orbitals
HYDROGEN ATOM ENERGIES
Potential energy of two electrons separated by the Bohr radius:
U = e2 4πε 0 a0 _ one “atomic unit” (a.u.) of energy.
H atom energies:
E = −Z 2 e2 8πε 0 a0 n2 = −Z 2 2 n2 a.u.
n
1
2
3
4
5
!
En (a.u.)
-1/2
-1/8
-1/18
-1/32
-1/50
!
∞
0
5.61 Fall 2007
Lecture #21
page 5
H atom energies &transitions
n
E
E/hc
(a.u.) (cm-1)
4
3
0
0
-1/32 -6,855
-1/18-12,187
2
-1/8
1
-1/2 -109,680
H atom emission spectra
Lyman series
Balmer
Paschen
λ(A)
ν(1012 Hz)
DEGENERACIES OF H ATOM ENERGY LEVELS
As n increases, the degeneracy of the level increases.
What is the degeneracy gn of each level as a function of n?
Does this help understand the periodic table?
5.61 Fall 2007
Lecture #21
page 6
SHAPES AND SYMMETRIES OF THE ORBITALS
S ORBITALS
( )
ψ 1s = π a03
−1 2
e
−r / a0
(
)(
)
ψ 2s = 32π a03 2 − r a0 e
l=0
−r / 2a0
spherically symmetric
n - l -1 = 0
radial nodes
l=0
angular nodes
n -1 = 0
n - l - 1 = 1
l = 0
n - 1 = 1
total nodes
(
Electron probability density given by ψ r,θ ,φ
)
2
Probability that a 1s electron lies between r and r + dr of the nucleus:
∫
2π
0
P ORBITALS:
(
π
) (
( )
)
dφ ∫ dθ sin θ ψ 1s* r,θ ,φ ψ 1s r,θ ,φ r 2 dr = 4π π a03
0
1l=
−1
e
−2r / a0 2
r dr
wavefunctions
Not spherically symmetric: depend on θ ,φ
m = 0 case:
ψ 2 p independent of φ
⇒
z
(
ψ 210 = ψ 2 p = 32π a03
z
) (r a ) e
−1/ 2
0
−r / 2a0
cosθ
symmetric about z axis
radial nodes
n - l -1 = 0
angular nodes
l = 1
total nodes
n - 1 = 1
()
(note difference from 2s: Rnl r
depends on l as well as n)
xy nodal plane – zero amplitude at nucleus
m = ±1 case: Linear combinations give
Equivalent probability distributions
( ) (r a ) e
= ( 32π a ) ( r a ) e
ψ 2 p = 32π a03
x
ψ2p
y
3
0
−1/ 2
0
−1/ 2
0
−r / 2a0
sin θ cos φ
−r / 2a0
sin θ sin φ
5.61 Fall 2007
Lecture #21
page 7
H atom radial probability densities
0.6
0.5
()
0.4
r 2 Rnl2 r a0
220nlrRra
()
0.3
0.6
0.6
0.2
0.5
0.5
2s
0.1
0.4
0.4
1s
0.3
0
0.6
0.3
0.5
0.2
5
10
15
20
25
30
10
30
15
20
25
30
0.2
0.4
0.1
0
2p
0.1
0.6
0
0.3
0
5
10
15
0
20
0
25
5
0.5
0.2
0.4
0.1
0.6
3s
0.3
0
0.5 0
5
10
15
20
25
30
20
25
30
20
25
30
0.2
0.4
0.1
3p
0.3
0
0
5
10
15
0.2
3d
0.1
0
0
5
10
15
s = r/a0
5.61 Fall 2007
Lecture #21
page 8
MAGNETIC FIELD EFFECTS
Electron orbital angular momentum (circulating charge)
μ=−
magnetic moment
e
L
2me
Magnetic field B applied along z axis interacts with μ:
Potential energy
U = −μ•Β = −μ zΒz = eBz Lz
2me
Include in potential part of Hamiltonian operator:
Ĥ = Ĥ 0 +
eBz
2me
L̂z
H atom wavefunctions are eigenfunctions of both Ĥ 0 and L̂z operators
eigenfunctions of new Ĥ operator.
Energy eigenvalues are the sums
eB
Z 2 e2
E=
+ z m
2
2me
8 0 a0 n
Energy depends on magnetic quantum number m when a magnetic field is applied.
2p orbitals: m = -1,0,+1 states have different energies
Splitting proportional to applied field B z.
5.61 Fall 2007
Lecture #21
page 9
Applied
magnetic field
No magnetic
field
2p
m = +1
m=0
m = -1
Energy
m=0
1s
2p →
1s Emission spectra
One line
Three lines
Complex functions ψ 21−1 and ψ 21+1 are eigenfunctions of L̂z with eigenvalues ±m! .
ψ 2 p and ψ 2 p are eigenfunctions of Ĥ 0 but not of L̂z ⇒ no longer energy
x
y
eigenfunctions once magnetic field is applied.