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.