P460 - 3D S.E. 1

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.

• Solve by separating variables

ti

m

x

tzyxVtzyx

orip

),,,(),,,(22

2

2

2

2

EzyxV

tzyxtzyx

m

),,(

)(),,(),,,(2

2

2

P460 - 3D S.E. 2

• If V well-behaved can separate further: V(r) or

Vx(x)+Vy(y)+Vz(z). Looking at second one:

• LHS depends on x,y RHS depends on z

• S = separation constant. Repeat for x and y

zz

yxyx

yxyx

zyx

zz

zyx

yxyx

zyx

zyxm

VEVV

VEVV

zyxzyxassume

EzVyVxV

z

z

2

2

2

2

2

2

2

2

2

2

2

2

2

)()(

)()()(

)()()(),,(

))()()((22

SVV

SVE

yxyxyx

zdz

d

yx

z

z

)(

)(

2

2

2

2

2

2

1

P460 - 3D S.E. 3

• Example: 2D (~same as 3D) particle in a Square Box

• solve 2 differential equations and get

• symmetry as square. “broken” if rectangle

ESESSSEEE

ESEV

ESSV

ESV

zyx

zzdz

d

yydy

d

xxdx

d

z

z

y

y

x

x

)()'('

'

'

2

2

2

2

2

2

)()(),(

)()(

0

,0,,0

yxyx

yVxVVsatisfies

boxinsideV

ayyaxxV

yx

yx

)( 22

2 2

22

yxmayx nnEEE

P460 - 3D S.E. 4

• 2D gives 2 quantum numbers.

Level nx ny Energy

1-1 1 1 2E0

1-2 1 2 5E0

2-1 2 1 5E0

2-2 2 2 8E0

• for degenerate levels, wave functions can mix (unless “something” breaks degeneracy: external or internal B/E field, deformation….)

• this still satisfies S.E. with E=5E0

ionnormalizatdxdy

nnAyx

nnEEE

yxa

yn

axn

yxmayx

yx

1||

..2,1,sinsin),(

)(

2

22

2 2

22

1

sinsin

sinsin

222112

221

212

mix

ay

ax

ay

ax

A

A

P460 - 3D S.E. 5

Spherical Coordinates

• Can solve S.E. if V(r) function only of radial coordinate

• volume element is

• solve by separation of variables

• multiply each side by

ErVr

r

rErV

r

rrrrM

M

)(),,(]

)(sin)([

),,()(

2

2

22

22

2

2

sin1

sin12

2

22

)sin)(()( drrddrvold

R

R

rRr

r

rrr

rr

MRVE

2

2

22

22

2

22

sin1

sin

sin12)( )(

)()()(),,(

Rr 22 sin

P460 - 3D S.E. 6

Spherical Coordinates-Phi

• Look at phi equation first

• constant (knowing answer allows form)

• must be single valued

• the theta equation will add a constraint on the m quantum number

21 ),()(2

2

ldd mrf

lime )(

.......2,1,0

)()2()2(

limim mee ll

P460 - 3D S.E. 7

Spherical Coordinates-Theta

• Take phi equation, plug into (theta,r) and rearrange

• knowing answer gives form of constant. Gives theta equation which depends on 2 quantum numbers.

• Associated Legendre equation. Can use either analytical (calculus) or algebraic (group theory) to solve. Do analytical. Start with Legendre equation

)1()(

)]([)(

sinsin1

sin

21

2

2

2

222

ll

rVE

dd

ddm

rMdrdRr

drd

R

l

)1()( 2

2

sin

sinsin

1 lllmdd

dd

functionLegendrePz

Pllzz

l

ldzdP

dz

Pd ll

cos

0)1(2)1( 2

22

P460 - 3D S.E. 8

Spherical Coordinates-Theta

• Get associated Legendre functions by taking the derivative of the Legendre function. Prove by substitution into Legendre equation

• Note that power of P determines how many derivatives one can do.

• Solve Legendre equation by series solution

2

2

0 1

1

2

)1(

0)1(2)1(

2

2

2

2

k

kkdz

Pd

k k

kkdz

dPkkl

ldzdP

dz

Pd

zkka

kzazaP

Pllzz ll

dzPd

dzdP

dz

Pdmlm

z

z

lml

lml

l

z

z

P

z

2

21

||

||

)1(

)1(

)1(

222

122

21

220

2/||2

P460 - 3D S.E. 9

Solving Legendre Equation

• Plug series terms into Legendre equation

• let k-1=j+2 in first part and k=j in second (think of it as having two independent sums). Combine all terms with same power

• gives recursion relationship

• series ends if a value equals 0 L=j=integer

• end up with odd/even (Parity) series

0})]1()1([)1({ 2 kk

kk zallkkzakk

0})]1()1([)1)(2{( 2j

jj zalljjajj

jjjlljj

j aa )1)(2()1()1(

2

)1()1(02 lljja j

0,00,0 01 oddeven aaoraa

P460 - 3D S.E. 10

Solving Legendre Equation

• Can start making Legendre polynomials. Be in ascending power order

• can now form associated Legendre polynomials. Can only have l derivatives of each Legendre polynomial. Gives constraint on m (theta solution constrains phi solution)

2212

60

)1)(2()1()1(

210

110

010

313

,0,1,2

1,0,1

10,1,0

zP

aaal

zPaal

Paal

jjlljj

lm

Pz

l

ldzdm

lm lm

lml

l

||

)1( ||

||2/||2

P460 - 3D S.E. 11

Spherical Harmonics

• The product of the theta and phi terms are called Spherical Harmonics. Also occur in E&M.

• They hold whenever V is function of only r. Seen related to angular momentum

)1(

)1(

31

)1(

1

22,2

21,2

220

21,1

10

00

21

21

z

zz

z

z

z

harmonicsspherical

Y mlmlm

P460 - 3D S.E. 12

3D Schr. Eqn.-Radial Eqn.

• For V function of radius only. Look at radial equation

• can be rewritten as (usually much better...)

• 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).

Rr

llRErV

dr

dRr

dr

d

r 22

2 )1())(

21

)()(

))1(

2(

2 2

2

2

22

rrRru

Euur

llV

dr

ud

dru

drrRrP2

22

4

4)(

P460 - 3D S.E. 13

Particle in spherical box

• Good first model for nuclei

• plug into radial equation. Can guess solutions

• look first at l=0

• 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

• note plane wave solution. Supplement 8-B discusses scattering, phase shifts. General terms are

)cos()sin(

222

2

krBkrAu

kwithuk ME

drud

rarn

an

Man

n nE)/sin(

21

00

20 ....3,2,12

222

r

erR

rki

)(

arrV

arrV

)(

0)(

)()())1(

2(

2 2

2

2

22

rrRruEuur

llV

dr

ud

P460 - 3D S.E. 14

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

• 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:

• in nuclei (with j subshells)

.....222120121110 EEEEEEEnl

PSS

NeBeHeZ

221

)(1042

21

25

21

23 21111

)(1614862

sdpps

SSiOCHeZ

P460 - 3D S.E. 15

H Atom Radial Function

• For V =a/r get (use reduced mass)

• 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.

• Where fine structure constant alpha = 1/137 used. Same as Bohr model energy

• eigenfunctions depend on both n,L quantum numbers. First few:

2

2

2

222

2220

42 6.1322)4( n

eVZn

Zcm

neMZ

neE

0

0

0

0

2

200

2/21

2/20

40

/10

)2(

5.0

aZraZr

aZraZr

em

aZr

eR

eR

AaeR C

e

Rr

llRE

r

Zem

dr

dRr

dr

d

r 20

2

2

2 )1(

4

21

P460 - 3D S.E. 16

H Atom Wave Functions

P460 - 3D S.E. 17

H Atom Degeneracy

• As energy only depends on n, more than one state with same energy for n>1

• ignore spin for now

Energy n l m D

-13.6 eV 1 0(S) 0 1

-3.4 eV 2 0 0 1

1(P) -1,0,1 3

-1.5 eV 3 0 0 1

1 -1,0,1 3

2(D) -2,-1,0,1,2 5

1 Ground State

4 First excited states

9 second excited states

2nD

P460 - 3D S.E. 18

Probability Density

• P is radial probability density

• 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)

22

2

0

221

10

2

0

22

00

2

2

||)(

cos||

sin||

1||

||

nlRrrP

drddror

drddr

ionnormalizatdVolume

yprobabilit

22

0

rrrwidthrraverage

probablemost drdP

P460 - 3D S.E. 19

Most probable radius• For 1S state

• 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

0204

920

20

/2222

)1(21

023

0

0

/22/2

/2222

87.03

3

))]1(1[(

)(

)"("

20

||)(

0

20

2

0

0

20

0

aaar

adrerArr

generalin

adrrrPr

peakar

ere

eArRArrP

ar

n

llZan

ararar

drdP

ar

P460 - 3D S.E. 20

Radial Probability Density

P460 - 3D S.E. 21

Radial Probability Density

note # nodes

P460 - 3D S.E. 22

Angular Probabilities

• 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

• 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

1||

)sin(|)(||)(|),(2

22

im

m e

P

"1"

sincos

"1"

211

21,1

210

21110

00

statesPA

statesSA

P460 - 3D S.E. 23

Orthogonality

• 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

mlmnmmllnn

mlnnlm

Rwith

dddrrmlnnlm

'''

2'''

0 0

2

0

* sin

1''0)(

'''|cos|

cos

cos||

ˆtan

)1'('

10

llmmrf

mlnrnlm

polynomialLegendreis

noterEV

zintconsE

llmm

P460 - 3D S.E. 24

Wave functions• build up wavefunctions from eigenfunctions.

• example

• what are the expectation values for the energy and the total and z-components of the angular momentum?

• have wavefunction in eigenfunction components

)2(6

1),,,( /

121/

211/

100221 tiEtiEtiE eeetr

dvolt

idvolHHE **|

6

1)120(

6

1)2(

6

1

1))11(1)11(12)10(0(6

1

))1()1(2)1((6

124

7)

4

3(

6

1)2(

6

1

110

1111002

111221

zzzz LLLL

llllllL

EEEEEEE