4e : the quantum universe5 the hydrogen atom in its full quantum mechanical glory r 222 2 222 2 2 2...
TRANSCRIPT
2
The Coulomb Attractive Potential That Binds the electron and Nucleus (charge +Ze) into a
Hydrogenic atom
F V
me
+e
r
-e
2
( ) kZeU rr
=
3
The Hydrogen Atom In Its Full Quantum Mechanical Glory
2 2 2
As in case of particle in 3D box, we should use seperation of variables (x,y,z ??) to derive 3 independent differential. eq
1 1( ) x,y,z all mixe
This approach
d up
willns.
get very
!
ugly
U rr x y z
∝ = ⇒+ +
To simplify the situation, choose more appropriate variablesCartesian coordinates (x,y,z)
since we have a "co
Spherical Polar (r
njoined triplet"
, , ) coordinatesθ φ→
r
2
( ) kZeU rr
=
4
Spherical Polar Coordinate System
dV
2
( sin )Vol
( )( ) = r si
ume Element dV
ndV r d rd dr
drd dθ φ θ
θ θ φ
=
5
The Hydrogen Atom In Its Full Quantum Mechanical Glory
r 2 2 22
2 2 2
2
2 22 2
2
222
for spherical polar coordinates:
1=
Instead of writing Laplacian ,
write
1sin
Thus the T.I.S.Eq. for (x,y,z) = (r, , ) be
sin
come
i1r s n
rr r r
x y z
r θ φ
ψ ψ
θ θ
φ
θθ
θ
∂ ∂⎛ ⎞⎜ ⎟∂ ∂⎝ ⎠
∂ ∂ ∂∇ = + +
∂ ∂ ∂
∂
∇
∂ ∂⎛ ⎞∇ + +⎜ ⎟∂ ∂⎝ ∂⎠
2
2 2
2
2 2 2
2
22
2
1 (r, , ) (r, , )r
(r, , ) (r, , ) =
s
1
1
0
2m+ (E-U(r))si
sinsin
1 1with )
n
(
r
r
U r
r r
r x y z
r
θ
θ
φ
ψ θ φ ψ θ φ
ψ θ φ ψ
θ
θ
θ
φ
θ∂ ∂⎛ ⎞⎜ ⎟∂ ∂⎝ ⎠
∂ ∂⎛ ⎞+ +⎜ ⎟∂ ∂⎝ ⎠
∝ =+
∂
+
∂
6
The Schrodinger Equation in Spherical Polar Coordinates (is bit of a mess!)
22
2 2 2
2
22
2 2
2
2
The TISE is :1 2m+ (E-U(r))
sin
Try to free up second last
1 (r, , ) =0 r
all except
T
term fro
1 sinsin
his requires multiplying thr
m
uout by sin
sin
rr r
r
rr
r rψ ψ ψ ψ θ φ
θ φθ
θ θ
θ
θ
θ
φ
∂ ∂⎛ ⎞+ +⎜ ⎟∂ ∂⎛ ⎞⎜ ⎟∂ ∂⎝ ∂ ∂⎠
⇒
∂
⎠∂
∂
⎝ ∂
2
2
2 2
2
22m ke+ (E+ )r
(r, , ) = R(r)
sin sin
. ( ) . ( ) Plug it into the TISE above & divide thruout by (r, , )=R(r). ( ). ( )
sin =0
For Seperation of Variables, Write
rr
θ θφ
ψ θ φ θ φψ θ φ θ
θ θψ θ
φ
ψ ψ ψ∂ ∂⎛ ⎞+ +⎜∂⎛ ⎞
⎜ ⎟∂⎝ ⎠∂∂⎟∂
Θ
∂ ⎠
ΦΦ
⎝
Θ
R(r) r
( ) when substituted in TISE
( )
( , ,
) ( ). ( )r
( , , )Note that : ( ) ( )
( , , ) ( ) ( )
r
r R r
r R r
θθ
θ φ θ φ
θ φ φθθ φ φθ φ
θ
∂Ψ= Θ Φ
∂∂Ψ
= Φ∂
∂Ψ= Θ
∂
∂∂
∂Θ⇒
∂∂Φ∂
7
Solving For the Hydrogen Atom: Separation of VariablesD
on’t
Pani
c: It
s sim
pler
than
you
thin
k !
2 2 22
2 2
2 2
2 2
2 2
2 2 22
1 2m ke+ (sin sin =0
Rearrange b
E+ )r
2m ke 1+ (E+ )r
LHS
y taking
i
the term
s f
sin sin
sin
on RHS
sin sin =-s
n.
in
R rrR r r
R rrR r r
θ θ
φ
θ θ
θ θ
θ θ
θφθ θ
φ
θ
∂ ∂Θ⎛ ⎞+ +⎜ ⎟Θ ∂ ∂∂ ∂⎛ ⎞⎜ ⎟∂ ∂⎝ ⎠
∂ ∂⎛ ⎞⎜ ⎟∂ ∂
∂ ΦΦ ∂
∂ ΦΦ ∂
⎝ ⎠
∂ ∂Θ⎛ ⎞+ ⎜ ⎟Θ ∂ ∂⎠ ⎝ ⎠⎝
2
of r, & RHS is fn of only , for equality to be true for all r, ,
LHS= constant = RHS = m l
θ φ θ φ
⇒
8
Deconstructing The Schrodinger Equation for Hydrogen
2 2 22 2
2
2
2
2
sin sin =m
Divide Thruout by sin and arrange all terms with r aw
Now go break up LHS to seperate the terms...r .. 2m keLHS: + (E+ )
a
& sin si
y from
r
1
n lR r
r
rr
r
R r
R
θ θ
θ
θ θθ θ
θ
θ
∂ ∂Θ⎛ ⎞+ ⎜ ⎟Θ ∂ ∂⎝ ⎠∂ ∂⎛
∂
⎞
∂∂
⎜ ⎟∂ ∂⎝ ⎠
⇒2
2
2 2
2
m 1 sinsin sin
Same argument : LHS is fn of r, RHS is fn of ;
For them to be equal for a LHS = const = RHS
What is the mysterious ( 1) ?
2m ke(E+ )=
ll r, =
r
( 1)
l
l l
l
r
l
Rr
θθ θ θ θ
θ
θ
∂ ∂Θ⎛ ⎞− ⎜ ⎟Θ ∂ ∂⎝ ⎠⎛ ⎞ +
+
⎟⎠
+
⎜ ∂
⇒
⎝
Just a number like 2(2+1)
9
2
22
2
22
2
do we have after all the shuffling!
m1 sin ( 1) ( ) 0.....(2)si
So What
d ..... ............(1)
1
n sin
m 0..
l
l
d d l l
d Rrr d
d
d
r r
dθ θ
θ θ θ
φ
θ⎡ ⎤Θ⎛ ⎞ + + − Θ =⎜ ⎟ ⎢ ⎥
⎝
Φ
⎠
∂⎛ ⎞ +⎜ ⎟∂⎝
Φ
⎣ ⎦
⎠
+ =
2 2
2 2
2m ke ( 1)(E+ )- ( ) 0....(3)r
These 3 "simple" diff. eqn describe the physics of the Hydrogen atom.
All we need to do now is guess the solutions of the diff. equations
Each of them, clearly,
r l l R rr
⎡ ⎤+=⎢ ⎥
⎣ ⎦
has a different functional form
10
And Now the Solutions of The S. Eqns for Hydrogen Atom
22
2
dThe Azimuthal Diff. Equation : m 0
Solution : ( ) = A e but need to check "Good Wavefunction Condition"Wave Function must be Single Valued for all ( )= ( 2 )
( ) = A e
l
l
l
im
im
dφ
φ
φ
φφ φ φ π
φ
Φ+ Φ =
Φ⇒ Φ Φ +
⇒Φ ( 2 )
2
2
A e 0, 1, 2, 3....( )
m1The Polar Diff. Eq: sin ( 1) ( ) 0 sin sin
Solutions : go by the name of "Associated Legendr
Q
e Functions"
uantum
#
liml
l
m
d d l
Magneti
d d
c
l
φ π
θ θθ θ θ θ
+= ⇒ = ± ± ±
⎡ ⎤Θ⎛ ⎞ + + − Θ =⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦
only exist when the integers and are related as follows 0, 1, 2, 3.... ; positive number
: Orbital Quantum Number
l
l
l mm l l
l
= ± ± ± ± =
Φ