Experimental Physics 4 - Schrödinger equation 1

Experimental Physics EP4 Atoms and Molecules

– Schrödinger equation II –Potential box, harmonic oscillator,

spherically symmetric potential

https://bloch.physgeo.uni-leipzig.de/amr/

Experimental Physics 4 - Schrödinger equation 2

Potential box

ikxikx BeAex -+=)(y

0=+ BA 0=+ -ikbikb BeAe

22 2!mEk =

x

U(x)

0 b

022

2

=+ yy kdxd

( ) )sin(2)( kxiAeeAx ikxikx =-= -y( ) )sin(20 kbiAeeA ikbikb =-= -

)sin(2)( bxniAx py =

bnk p

=

22

22

2n

mbE p!=

Y, |Y

|2

x0 b

hpx ³DD2

22

22 mbh

mp

=Þ

Zero-point energy

Experimental Physics 4 - Schrödinger equation 3

Two-dimensional potential box

x

U

0

y

b

a

022

2

2

2

=++ yyy kdyd

dxd

Ansatz: )()(),( yYxXyx =y

0),()0,0( ====== aybxyx yy

)sin()( 1 bx

xnCxX p= )sin()( 2 ay

ynCxY p=

)sin()sin(2),( ay

ybx

x nnab

yx ppy =

mkE2

22!= ÷÷

ø

öççè

æ+= 2

2

2

222

2 an

bn

myxp!

There are degenerate energy levels!

Experimental Physics 4 - Schrödinger equation 4

Harmonic oscillator

kxF -=mk

=wyywy Exm

dxd

m=+- 22

2

22

21

2!

022

2

2

=÷÷ø

öççè

æ-- y

wwy

w !!! Exmdxd

m

w!2

( ) 022

2

=-+ yxxy Cdd

( ) ( ) 2/2xxxy -= eH

w!EC 2

º

!wx mxº

0)1('2'' =-+- HCHH x

( ) ( ) ( )22

1 xx

xx --= e

ddeH n

nn

n

Hermitian polynomials, n = 0, 1, 2, 3.. xx

x

x

128,324,2

2,11,0

3

2

-

-

! = #$%&

!

( ) 2/0

2xxy -= e

Experimental Physics 4 - Schrödinger equation 5

Harmonic oscillator( ) ( ) 2/2xxxy -= eH

0)1('2'' =-+- HCHH x

( ) ( ) ( )22

1 xx

xx --= e

ddeH n

nn

n

xx

x

x

128,324,2

2,11,0

3

2

-

-( ) å

=

=n

k

kkn aH

0xx

!wx mxº

w!EC 2

º

12 +=Þ nC

n=3

n=2

n=1

n=0

0)1(2 =-+- nn aCna ( )nE += 21w!Þ

- normalization factor

!' " = $'%'&'ℏ exp −&'2ℏ "

(

#$")!

! +,-ℏ

Experimental Physics 4 - Schrödinger equation 6

Experimental Physics 4- Schrödinger equation 7

Spherically symmetric potential

y

x

z

j

q

( ) 02 2

2

2

2

2

22

=-+÷÷ø

öççè

涶

+¶¶

+¶¶

- yy EUzyxm

!q

jqjq

cossinsincossin

rzryrx

===

( ) 02sin1sin

sin11

2

2

2

2222

2

=-+

+¶¶

+÷øö

çèæ

¶¶

¶¶

+÷øö

çèæ

¶¶

¶¶

y

jy

qqyq

qqy

UEmrrr

rrr

!( ) )()()(,, jqjqy FQ= rRr

( ) 0sin2sinsinsin 2222

222 =-+

¶¶

+÷øö

çèæ

¶¶

¶¶

+÷øö

çèæ

¶¶

¶¶ yq

jy

qyq

qqyq UErm

rr

r !

q22 sinr´

( ) 2

222

22

2 )()(

1sin2)(sin)(

sin)()(

sinjj

jq

qqq

qqqq

¶F¶

F-=-+÷

øö

çèæ

¶Q¶

¶¶

Q+÷øö

çèæ

¶¶

¶¶ UErm

rrRr

rrR !

( ) ÷øö

çèæ

¶Q¶

¶¶

Q-=-+÷

øö

çèæ

¶¶

¶¶

qqq

qqqq)(sin

sin)(1

sin2)(

)(1

212

22 CUErm

rrRr

rrR !

= C1

= C2

Experimental Physics 4 - Schrödinger equation 8

y

x

z

j

q

Spherically symmetric potential

12

2 )()(

1 C=¶F¶

F-

jj

jjj 11 '' CiCi eBeA -+=F

j1CiAe±=F

111 2 CijCiCi eAeAe pjj ±±± =

)2()( jpjj +F=F - wave function is unambiguously defined …

mC =1any integer number

(magnetic quantum number)

221 )(sin

sin)(1

sinCC

=÷øö

çèæ

¶Q¶

¶¶

Q-

qqq

qqqq 22

2 )(sinsin)(1

sinCm

=÷øö

çèæ

¶Q¶

¶¶

Q-

qqq

qqqq

qx cos=( ) 0

11 2

2

22 =Q÷÷

ø

öççè

æ-

-+÷÷ø

öççè

æ Q-

xxx

xmC

dd

dd

qqx dd sin-=

Experimental Physics 4 - Schrödinger equation 9

( ) 01 22 =Q+÷÷

ø

öççè

æ Q- C

dd

dd

xx

x

Spherically symmetric potential (m=0)

( ) å=

=Ql

k

kkl a

0xx

0)1(2 2 =+--- Caallla lll )1(2 += llCÞ

( ) ( ) ( )qxx coslll APAP ==Q - Legendre’s polynomialsy

x

z

j

q

2)(cosqlPNot normalized!

0

1

2

3

0,0 0,5 1,0 1,5 2,0 2,5 3,0

cos(q)

angular momentum quantum number

Experimental Physics 4 - Schrödinger equation 10

Spherically symmetric potential (m¹0)

( ) 01

1 2

2

22 =Q÷÷

ø

öççè

æ-

-+÷÷ø

öççè

æ Q-

xxx

xmC

dd

dd

( ) ( ) ( )xx

xx lm

mmml P

ddAP 221-==Q

lml ££-

Otherwise you cannot define the differential

m = -3 -2 -1 0 1 2 3

l = 0

l = 1

l = 2

positive

negative

( ) ( ) ( )jqjq F=U cos, ml

ml P

Experimental Physics III - Schrödinger equation 11

To remember!

Ø A particle contained in a potential box quite naturally exhibits energy quantization (analogue of a string fixed at both ends).

Ø For multidimensional boxes the number of allowed energies increases and there might be degenerate energy levels.

Ø The shape of the potential well dictates the energy levels (compare them for square and harmonic wells).

Ø The Heisenberg uncertainty principle requires the existence of zero-point energy.

Ø For any spherically symmetric potential the angular parts of the wave functions will be identical.

Ø Their forms are determined by the magnetic (m) and angular momentum (l) quantum numbers.