3D Schrodinger Equation
Download
Report
Transcript 3D Schrodinger Equation
3D Schrodinger Equation
• Simply substitute momentum operator
• do particle in box and H atom
• added dimensions give more quantum numbers. Can have
degeneracies (more than 1 state with same energy). Added
complexity.
2
p i or x 2 2
2
2m
2 ( x, y , z , t ) V ( x, y , z , t )
i
t
• Solve by separating variables
( x, y, z, t ) ( x, y, z ) (t )
2
2m
2 V ( x, y, z ) E
P460 - 3D S.E.
1
• If V well-behaved can separate further: V(r) or Vx(x)+Vy(y)+Vz(z).
Looking at second one:
2
2m
2 (Vx ( x ) V y ( y ) Vz ( z )) E
a ssu m e ( x , y , z )
2
2m
2
2m
( x 2
2
2
y 2
x
( x )
) (Vx V y )
y
( y ) z ( z )
2
2m
x y z
( x y ) x y
(Vx V y )
x y
2
2
2
z 2
2
2
• LHS depends on x,y
2
2m
( E Vz )
x y z
2 z
E Vz
z z 2
RHS depends on z
2 d 2 z
( E Vz ) S
z dz 2
2m
2
2
2
1
(
) x y Vx V y S
x y
x 2
y 2
2m
• S = separation constant. Repeat for x and y
P460 - 3D S.E.
2
2
2 d x
2 m x dx 2
Vx S ' E x
2
2 d y
2 m y dy 2
Vy S S ' E y
2
2 d z
2 m z dz 2
Vz E S E z
E x E y E z S '( S S ' ) ( E S ) E
• Example: 2D (~same as 3D) particle in a Square Box
V
V 0
x 0, x a , y 0, y a
in sid e b o x
sa tisfies
V Vx ( x ) V y ( y )
( x , y ) x ( x ) y ( y )
• solve 2 differential equations and get
E Ex E y
2 2
2 ma 2
( n x2 n y2 )
• symmetry as square. “broken” if rectangle
P460 - 3D S.E.
3
E Ex E y
( x, y ) A sin
2 2
2 ma 2
( nx2 n y2 )
n x x
a
| |2 dxdy 1
sin
n y y
a
nx , n y 1,2..
no rm alizat
io n
• 2D gives 2 quantum numbers.
Level
nx
ny
1-1
1
1
1-2
1
2
2-1
2
1
2-2
2
2
P460 - 3D S.E.
Energy
2E0
5E0
5E0
8E0
4
• for degenerate levels, wave functions can mix (unless “something”
breaks degeneracy: external or internal B/E field, deformation….)
y
12 A sin ax sin 2
a
21 A sin 2ax sin ay
mix 12 21
2 2 1
• this still satisfies S.E. with E=5E0
P460 - 3D S.E.
5
Spherical Coordinates
• Can solve S.E. if V(r) function only of radial coordinate
2
2M
2 V ( r ) E ( r , , )
2
2M
[ r 2r ( r 2
r
2
r2
1
si n
1
si n 2
r
)
(sin
2
2
)
] ( r , , ) V ( r ) E
• volume element is
d (vol) dr(rd )(r sin d )
P460 - 3D S.E.
6
Spherical Coordinates
• solve by separation of variables
( r , , ) R ( r ) ( ) ( )
( E V ) R 2 M
2
( r 2r
r2
r 2
r
1
sin 2
r
2
2
2
1
sin 2
sin
) R
R
• multiply each side by
r 2 sin 2
R
P460 - 3D S.E.
7
Spherical Coordinates-Phi
• Look at phi equation first. Have separation constant
1
d2
d 2
( ) f ( r , ) ml2
• constant (knowing answer allows form)
• must be single valued
( ) eiml
( 2 ) ( )
eiml ( 2 ) eiml ml 0,1,2.......
• the theta equation will add a constraint on the m quantum number
P460 - 3D S.E.
8
Spherical Coordinates-Theta
• Take phi equation, plug into (theta,r) and rearrange. Have second
separation constant
1
R
d
dr
ml2
(
sin 2
r 2 dR
dr
)
1
sin
2 M 2r2
2
d
d
[ E V ( r )]
( sin dd )
l (l 1)
• knowing answer gives form of constant. Gives theta equation which
depends on 2 quantum numbers.
d
1
sin d
(
sin d
d
)
ml2
sin 2
P460 - 3D S.E.
l (l 1)
9
Spherical Coordinates-Theta
d
1
sin d
(
sin d
d
)
ml2
sin 2
l (l 1)
• Associated Legendre equation. Can use either analytical (calculus) or
algebraic (group theory) to solve. Do analytical. Start with Legendre
equation
(1 z )
2
d 2 Pl
dz 2
z cos
2z
dPl
dz
l (l 1) Pl 0
Pl Legendre function
P460 - 3D S.E.
10
Spherical Coordinates-Theta
• Get associated Legendre functions by taking the derivative of the
Legendre function. Prove by substitution into Legendre equation
lml (1 z
2
)
|ml |/ 2
d |ml | Pl
dz |ml
|
20 P2
21 (1 z
2
)
22 (1 z 2 )
1
2
dPl
dz
d
2
2 1
Pl
dz 2
• Note that power of P determines how many derivatives one can do.
• Solve Legendre equation by series solution
(1 z
Pl
d 2P
dz 2
2
)
d
2
2z
Pl
dz 2
dPl
dz
a
k 0
k
z
a
k 2
k
k
dP
dz
l (l 1) Pl 0
a
k 1
k
kz k 1
k ( k 1) z k 2
P460 - 3D S.E.
11
Solving Legendre Equation
• Plug series terms into Legendre equation
k 2
k
{
k
(
k
1
)
a
z
[
k
(
k
1
)
l
(
l
1
)]
a
z
} 0
k
k
• let k=j+2 in first part and k=j in second (think of it as having two
independent sums). Combine all terms with same power
{( j 2)( j 1)a j 2 [ j ( j 1) l (l 1)]a j }z
• gives recursion relationship
a j 2
j ( j 1) l (l 1)
( j 2)( j 1)
j
aj
• series ends if a value equals 0 L=j=integer
a j 2 0 j ( j 1) l (l 1)
• end up with odd/even (Parity) series
a1 0, aeven 0 or a0 0, aodd 0
P460 - 3D S.E.
0
12
Solving Legendre Equation
• Can start making Legendre polynomials. Be in ascending power order
l 0, a0 1, a1 0 P0 1
l 1, a0 0, a1 1 P1 z
l 2, a0 1, a1 0, a2
06
21
j ( j 1) l ( l 1)
( j 2 )( j 1)
3 P2 1 3 z 2
• can now form associated Legendre polynomials. Can only have l
derivatives of each Legendre polynomial. Gives constraint on m (theta
solution constrains phi solution)
lml (1 z )
2
|ml |/ 2
d |m l |
dz |ml |
Pl
| ml | l
P460 - 3D S.E.
13
Spherical Harmonics
00 1
z cos
10 z
1, 1 (1 z
2
)
1
2
20 1 3 z 2
2 , 1 (1 z
2
)
1
2
z
2 , 2 (1 z 2 )
• The product of the theta and phi terms are called Spherical
Harmonics. Also occur in E&M. See Table on page 127 in book
• They hold whenever V is function of only r. Saw related to angular
momentum
Ylm lm m
spherical harm onics
P460 - 3D S.E.
14
3D Schr. Eqn.-Radial Eqn.
• For V function of radius only. Look at radial equation. L comes in
from theta equation (separation constant)
1 d r 2d R
2
V ( r ) E ) R
2
r dr
d
r
l (l 1)
R
r2
• can be rewritten as (usually much, much better...)
2 d 2u
2 l (l 1)
(V
)u Eu
2
2
2 dr
2
r
u ( r ) rR ( r )
• and then have probability
P ( r ) 4R 2 r 2 dr
4u 2 dr
P460 - 3D S.E.
15
3D Schr. Eqn.-Radial Eqn.
2 d 2u
2 l (l 1)
(V
)u Eu
2
2
2 dr
2
r
u ( r ) rR ( r )
• note L(L+1) term. Angular momentum. Acts like repulsive potential
and goes to infinity at r=0 (ala classical mechanics)
• energy eigenvalues typically depend on 2 quantum numbers (n and L).
Only 1/r potentials depend only on n (and true for hydrogen atom only
in first order. After adding perturbations due to spin and relativity,
depends on n and j=L+s.
P460 - 3D S.E.
16
Particle in spherical box
u ( r ) rR ( r )
• Good first model for nuclei
V (r) 0
V (r)
r a
r a
• plug into radial equation. Can guess solutions
2 d 2u
2 l ( l 1)
(V
)u Eu
2 d r2
2
r2
2 d 2u
2 l ( l 1)
u El u
2
2
2 d r
2
r
• look first at l=0
d 2u
dr 2
k 2u
with
k
2 ME
u A sin(kr ) B cos(kr )
P460 - 3D S.E.
17
Particle in spherical box
•
l=0
d 2u
dr 2
k 2u
with
k
2 ME
u A sin (kr ) B co s(kr )
• boundary conditions. R=u/r and must be finite at r=0. Gives B=0. For
continuity, must have R=u=0 at r=a. gives sin(ka)=0 and
Enlm En 00
n 00
1
2a
n 2 2 2
2 Ma 2
n 1,2....
sin( nr / a )
r
• note “plane” wave solution. Supplement 8-B discusses scattering,
phase shifts. General terms are
ik r
R( r )
P460 - 3D S.E.
e
r
18
Particle in spherical box
• ForLl>0 solutions are Bessel functions. Often arises in scattering off
spherically symmetric potentials (like nuclei…..). Can guess shape
(also can guess finite well)
• energy will depend on both quantum numbers
Enl E10 E11 E12 E20 E21 E22 .....
• and so 1s 1p 1d 2s 2p 2d 3s 3d …………….and ordering (except
higher E for higher n,l) depending on details
• gives what nuclei (what Z or N) have filled (sub)shells being different
than what atoms have filled electronic shells. In atoms:
Z 2
1S
• in nuclei (with j subshells)
Z 2
6
8
14
2
( He C O Si S )
16
1s 1 p 3 1 p 1 1d 5
2
4 10 ( He Be Ne)
2S 2 P
2
2s 1
2
P460 - 3D S.E.
19
H Atom Radial Function
• For V =a/r get (use reduced mass)
1 d r 2 dR
2m
Ze2
l (l 1)
E
R
R
2
2
r dr dr
4 0 r
r
• Laguerre equation. Solutions are Laguerre polynomials. Solve using
series solution (after pulling out an exponential factor), get recursion
relation, get eigenvalues by having the series end……n is any integer
> 0 and L<n. Energy doesn’t depend on L quantum number.
En
MZ 2 e 4
( 4 0 ) 2 2 2 n 2
me c 2 2 Z 2
2n2
13.6 eVZ 2
n2
• Where fine structure constant alpha = 1/137 used. Same as Bohr
model energy
P460 - 3D S.E.
20
H Atom Radial Function
• Energy doesn’t depend on L quantum number but range of L restricted
by n quantum number.
l<n n=1 only l=0 1S
n=2
l=0,1
2S 2P
n=3
l=0,1,2
3S 3P 3D
2 2 2
2
En
me c Z
2 n2
13.6 eVZ
n2
• eigenfunctions depend on both n,L quantum numbers. First few:
R10 e
Zr / a0
R20 ( 2
R21
Zr
a0
Zr
a0
a0
4 0 2
me e 2
C
0.5 A
)e Zr / 2 a0
e Zr / 2 a0
P460 - 3D S.E.
21
H Atom Wave Functions
P460 - 3D S.E.
22
H Atom Degeneracy
• As energy only depends on n, more than one state with same energy
for n>1 (only first order)
n
l
m
D
• ignore spin for now Energy
-13.6 eV
1
0(S)
0
1
-3.4 eV
2
0
0
1
1 Ground State
4 First excited states
9 second excited states
-1.5 eV
1(P)
-1,0,1
3
0
0
1
3
1
D n2
2(D)
P460 - 3D S.E.
-1,0,1
3
-2,-1,0,1,2
5
23
Probability Density
| |2 probability
2
|
|
dVolum e 1 norm alization
2
| |
0
0
0
1
or
0
2
r 2 sin d d dr
2
| |
1
2
r 2 d d cos dr
0
• P is radial probability density P(r ) r | Rnl |
• small r naturally suppressed by phase space (no volume)
• can get average, most probable radius, and width (in r) from P(r).
(Supplement 8-A)
dP
2
2
m o st p ro b a b le
dr
0
a vera g e r r
wid th r
P460 - 3D S.E.
r2 r 2
24
Most probable radius
• For 1S state
P ( r ) A r 2 | R |2 A r 2 e 2 r / a0
dP
dr
0 2 re 2 r / a0
r a0
2r2
a0
e 2 r / a0
(" p ea k" )
r
rP ( r )d r
3
2
a0
0
(
n 2 a0
Z
r2
r
[1
r
2
1
2
(1
l ( l 1)
n2
)]in g en era l)
Ar 2 e 2 r / a0 d r 3a02
3a02
9
4
a02 0.8 7a0
•
Bohr radius (scaled for different levels) is a good approximation of
the average or most probable value---depends on n and L
• but electron probability “spread out” with width about the same size
P460 - 3D S.E.
25
Radial Probability Density
P460 - 3D S.E.
26
Radial Probability Density
note #
nodes
P460 - 3D S.E.
27
Angular Probabilities
P ( , ) | ( ) |2 | ( ) |2 sin ( )
m eim | |2 1
•
no phi dependence. If (arbitrarily) have phi be angle around z-axis,
this means no x,y dependence to wave function. We’ll see in angular
momentum quantization
00 "1"
S sta tes
10 A co s
11
A
2
sin
P sta tes
2
2
10
12, 1 11
"1"
• L=0 states are spherically symmetric. For L>0, individual states are
“squished” but in arbitrary direction (unless broken by an external
field)
• Add up probabilities for all m subshells for a given L get a spherically
symmetric probability distribution
P460 - 3D S.E.
28
Orthogonality
n lm nl m
nn ' ll ' mm '
2
*
2
r
sin d rd d
nlm
n
'
l
'
m
'
0 0
0
with Rn lm m
• each individual eigenfunction is also orthogonal.
• Many relationships between spherical harmonics. Important in, e.g.,
matrix element calculations. Or use raising and lowering operators
• example
E const ant in zˆ
V | E | r cos
note
cos is Legendre polynom ial10
nlm | r cos | n' l ' m'
mm ' l ( l ' 1) f ( r ) 0
m m'
P460 - 3D S.E.
l l '1
29
Wave functions
• build up wavefunctions from eigenfunctions.
• example
( r, , , t )
1
( 100 e iE1t / 2 211e iE2t / 211e iE2t / )
6
• what are the expectation values for the energy and the total and zcomponents of the angular momentum?
E
H |
*
H dvol
*
i
dvol
t
• have wavefunction in eigenfunction components
E
L2
Lz
1
1
5
9
( E1 4 E2 E2 )
( E1
E1 )
E1
6
6
4
24
1
(l0 ( l0 1) 4 l1 (l1 1) l1 ( l1 1))
6
1
10
( 0( 0 1) 4 1(1 1) 1(1 1))
6
6
1
1
3
( Lz 0 4 Lz1 Lz 1 )
( 0 4 1)
6
6
6
P460 - 3D S.E.
30