r R r m n m n r t c r m e iEnt n m n m The quantum central ...

1

Thequantumcentralforceproblem:threedimensions&separa8onof

variables

Reading:McIntyre7.1‐7.4,AppendixE

!n!m

r," ,#( ) = Rn!

r( )$!

m "( )%m

(#)

! r," ,#,t( ) = cn!m

Rn!

r( )$!

m "( )%m

(#)n!m

& e' iE

nt /"

2

The2‐bodyproblem

H =

!p1

2

2m1

+

!p2

2

2m2

!

"#

$

%& +V (

!r1,!r2)

H =

!PCM2

2M+

!prel2

2µ

!

"#

$

%& +V (r)

• SeparatesintoaCoMpartandarela8vepartwithsamedefini8onsasinclassicalproblemforcenterofmass,rela8vecoordinates.Again,wewilltreattheCoMpartoftheproblemas“solved”.

p(rel ) ! !i" i

""x

+ j""y

+ k""z

#$%

&'(= !i")

(rel )

Warning:alwaysaskyourself‐>isthisanoperatororanumber?H,Pandpareoperatorshere.

H(rel ) ! !

"2

2µ"2

(rel ) +V r(rel )( )

3

CMSepara8onFlowchart

4

Sphericalpolarcoordinatesr = r sin! cos"i+ r sin! sin" j+ r cos!k

r̂ = sin! cos"i + sin! sin" j+ cos!k

!̂ = cos! cos"i + cos! sin" j# sin!k

"̂ = # sin"i + cos" j

! = r̂"

"r+ #̂

1

r

"

"#+ $̂

1

r sin#

"

"$

5

Sphericalpolarcoordinates

dV = r2 sin! !d! !d" !dr

= r !d!( )! r sin! !d"( )! dr( )

= sin! !d!( )!! d"( )! r2 !dr( )= r

2!dr !d#

6

Separa8onofVariablesFlowchart

7

Sphericalcoordinates

!!

2

2µ"2

kinetic energyoperator

"#$ %$

+ V (r)potential energyoperator

&

#

$

%%%%%

&

'

(((((

) r,*,+( )eigenfunction

" #$ %$= E

eigenvalue

& ) r,*,+( )eigenfunction

" #$ %$

• Energyeigenvalueequa8onforreducedmasspartofthe2‐bodyproblemwithkine8cenergyoperatorexplicitlyinsphericalcoordinates

!!2

2µ

1

r2

""r

r2 ""r

#$%

&'(+

1

r2sin)

"")sin)

"")

#$%

&'(+

1

r2sin

2)"2

"* 2+

,-

.

/01

8

Angularmomentum

L = r ! p

Lx= yp

z" zp

y! "i" y

##z

" z##y

$%&

'()

Ly= zp

x" xp

z! "i" z

##x

" x##z

$%&

'()

Lz= ?

L2= L

x

2+ L

y

2+ L

z

2

• Thisisangularmomentuminrectangularcoordinates

Lx,L

y!" #$ = i!Lz

Ly,L

z!" #$ = i!Lx

Lz,L

x!" #$ = i!Ly

L2,L

x,y,z!" #$ = 0

L2!m!= ! ! +1( )"2 !m

!

Lz!m!= m

!" !m

!

BUTJUSTLIKESPIN!

9

Angularmomentum

L2! !"2 1

sin"##"

sin"##"

$%&

'()+

1

sin2"

#2

#* 2

+

,-

.

/0

• Thisisangularmomentuminsphericalcoordinates(homework)

Lx! i" sin!

""#

+ cos! cot#""!

$%&

'()

Ly! i" * cos!

""#

+ sin! cot#""!

$%&

'()

Lz! *i"

""!

10

Angularmomentum

• Herecomesthebigsimplifica8on:

• Assumeaseparablesolu8on:

!!

2

2µ

1

r2

""r

r2 ""r

#$%

&'(!

1

!2r

2L

2)

*+

,

-./ r,0 ,1( ) +V (r)/ r,0 ,1( ) = E/ r,0 ,1( )

HKE

= !!2

2µ

1

r2

""r

r2 ""r

#$%

&'(+

1

r2sin)

"")

sin)"")

#$%

&'(+

1

r2sin

2)"2

"* 2+

,-

.

/0

! r," ,#( ) = R r( )Y " ,#( )

11

Separa8onofvariables:

• Bluehasangulardependence,redisradial:

Algebra(followbook7.4):

!!

2

2µ

1

r2

""r

r2 ""r

#$%

&'(!

1

!2r

2L

2)

*+

,

-.R r( )Y / ,0( ) +V (r)R r( )Y / ,0( )

= ER r( )Y / ,0( )

1

R r( )d

drr 2

dR r( )dr

!

"##

$

%& '

2µ

!2(E 'V (r))r 2

function of r only

" #$$$$$$$ %$$$$$$$

=1

!2

1

Y ( ,)( )L

2Y ( ,)( )

function of ( ,) only

" #$$$ %$$$

* A

12

Separa8onofvariables:

• Oncewesolvethe(blue)angularproblem,itisthesolu8ontotheangularpartofALLcentralforceproblems!

• OncewefindA(andY),plugbackintoredequa8onandsolvetofindE(andR(r)).

d

drr

2dR r( )

dr

!

"##

$

%& '

2µ

!2

(E 'V (r))r 2R r( ) ( AR r( )

1

!2L

2Y ! ,"( ) = AY ! ,"( )

13

Separateangularequa8on

• Butweneedtoworkonblueequa8onmore,first.

Y ! ,"( ) =# !( )$ "( )

L2! !"2 1

sin"##"

sin"##"

$%&

'()+

1

sin2"

#2

#* 2

+

,-

.

/0

1

sin!d

d!sin!

d

d!"#$

%&'( B

1

sin2!

)

*+

,

-./ !( ) = (A/ !( )

d2! "( )d" 2

= #B! "( )

14

Summary

• Here’stheplan:

1

sin!d

d!sin!

d

d!"#$

%&'( B

1

sin2!

)

*+

,

-./ !( ) = (A/ !( )

d2! "( )d" 2

= #B! "( )

d

drr

2dR r( )

dr

!

"##

$

%& '

2µ

!2

(E 'V (r))r 2R r( ) ( AR r( )

!n!m

r,",#( ) = Rn!

r( )$!m "( )%m(#)

! r,",#,t( ) = cn!m

Rn!

r( )$!m "( )%m(#)

n!m

& e'iEnt/"

15

Summary• Here’stheplan:• We’llconsider3differentsystems,aring(tosolvetheφproblem),asphere(tosolvetheθandφproblem),andthefullhydrogenatom(tosolvethe(r,θ,φproblem)

• We’llfindthequantumnumbersandwavefunc8onsthatsolveeachproblem

• We’llapplyallthethingsyou’velearnedinPH424andPH425

• Pleasereadahead–themathismuchmoreintense(thoughnotharder)thanbefore