Physical Chemistry - Hydrogen Atom_lecture21to22

5.61 Fall 2007 Lecture #21 page 1



HYDROGEN ATOM
Schrodinger equation in 3D spherical polar coordinates:

!2 ⎡ 1 ∂ ⎛ 2 ∂ ⎞ 1 ∂ ⎛ ∂⎞ 1 ∂2 ⎤
− ⎢ r +
2 µ ⎣ r 2 ∂r ⎜⎝ ∂r ⎟⎠ r 2 sin θ ∂θ ⎜⎝
sin θ + ( ) (
⎥ ψ r,θ ,φ + U r,θ ,φ ψ r,θ ,φ = Eψ r,θ ,φ
∂θ ⎟⎠ r 2 sin 2 θ ∂φ 2 ⎦
) ( ) ( )

−Ze2
with Coulomb potential U (r) =
4πε 0 r
Rewrite as

⎡ 2 ∂⎛ 2 ∂⎞ ⎤
⎢ −! ⎜ r ⎟
∂r ⎝ ∂r ⎠
()
+ 2 µr 2 ⎡⎣U r − E ⎤⎦ ⎥ ψ r,θ ,φ + L̂2ψ r,θ ,φ = 0 ( ) ( )
⎣ ⎦

function of r only function of θ,φ only
ˆH ⇒
r is separable ψ is separable

Y
ml
Angular momentum: solutions are spherical harmonic wavefunctions


( )
ψ r,θ ,φ = R r Yl m θ ,φ () ( )
with L̂2Yl m (θ ,φ ) = ! l ( l + 1)Y (θ ,φ )
2
l
m
l = 0,1,2,...


Radial equation for the H atom:


!2 d ⎛ 2 dR r ⎞ ⎡ ! l l + 1
2
() ⎤ ( )
− ⎜
2 µr 2 dr ⎝
r ⎟ + ⎢
dr ⎠ ⎢⎣ 2 µr 2
+ U r − E ⎥R r = 0 () ()
⎥⎦


()
Solutions R r are the H atom radial wavefunctions


Simplest case: l = 0 yields solution

32
⎛Z⎞
()
R r = 2⎜ ⎟
⎝ a0 ⎠
e
− Zr a0
exponential decay away from nucleus

, 5.61 Fall 2007 Lecture #21 page 2



with

E = − Z 2 e2 8πε 0 a0 lowest energy eigenvalue
a0 ≡ ε 0 h2 πµe2 Bohr radius

General case: solutions are products of (exponential) x (polynomial)

Energy eigenvalues:

−Z 2 e2 −Z 2 µe4
E= = n = 1,2,3,...
8πε 0 a0 n2 8ε 02 h2 n2


Radial eigenfunctions:

12
⎡ ⎤
( )
n − l − 1 ! ⎥ ⎛ 2Z ⎞
l +3 2
⎛
− Z r na0 2l +1 2 Zr
⎞
Rnl ()
r =− ⎢
⎢ ⎜
⎥ ⎝ na ⎠⎟ r le Ln+ l ⎜ ⎟
( )
3
2n ⎡ n + l !⎤ ⎝ na0 ⎠
⎣⎢ ⎣ ⎦ ⎦⎥ 0




( )
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!
⎛ 2 2 ⎞
n=3 l=0 L13 = −3!⎜ 3 − 2Zr + 2Z r 2⎟
⎝ a0 9a0 ⎠
⎛ ⎞
l =1 L34 = −4!⎜ 4 − 2Zr
⎝ 3a0 ⎟⎠
l=2 L55 = −5!

Normalization:
( ) ( )
2π π
Spherical harmonics ∫ 0
dφ ∫ dθ sin θ Yl m* θ ,φ Yl m θ ,φ = 1
0




() ()
∞
Radial wavefunctions ∫ 0
dr r 2 Rnl* r Rnl r = 1