Experimental Physics 4 - Hydrogen atom 1
Experimental Physics EP3 Atoms and Molecules
– Hydrogen atom –Radial density distribution, energy levels,
interaction with magnetic field
https://bloch.physgeo.uni-leipzig.de/amr/
Experimental Physics 4 - Schrödinger equation 2
Spherically symmetric potential
y
x
z
j
q( ) 0
2 22
2
2
2
22
=-+÷÷ø
öççè
涶
+¶¶
+¶¶
- yy EUzyxm
!
qjqjq
cossinsincossin
rzryrx
===
( ) 02sin1sin
sin11
2
2
2
2222
2
=-+
+¶¶
+÷øö
çèæ
¶¶
¶¶
+÷øö
çèæ
¶¶
¶¶
y
jy
qqyq
qqy
UEmrrr
rrr
!( ) )()()(,, jqjqy FQ= rRr
( ) 0sin2sinsinsin 22222
22 =-+¶¶
+÷øö
çèæ
¶¶
¶¶
+÷øö
çèæ
¶¶
¶¶ yq
jy
qyq
qqyq UErm
rr
r !
q22 sinr´
( ) 22
222
22 )(
)(1sin2)(sin
)(sin)(
)(sin
jj
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
( )Y+YÑ-=¶Y¶ rU
mti 2
2
2!
! ( )yy úû
ùêë
é+Ñ-= rU
mE 2
2
2!
Radial part of the wave function
3
( ) )1(2)()(
12
22
2 +==-+÷øö
çèæ
¶¶
¶¶ llCUErm
rrRr
rrR !( ) ( ) ( )jqjqjq F=U=Y cos)(,)(),,( mlml PrRrRr
!"($)
&&$
$!&"($)&$
+()ℏ!
$! + − - +.(. + !)ℏ!
()$!= 0 Still spherically-symmetric field!The angular part of WF remains uncahged.
- = −12!
345"$
6! ≡ −()+ℏ!
8 ≡)12!
(45"ℏ!
&&$
$!&"($)&$
+ −6!$! + 8$ − . . + ! "($) = 0
9!:9$!
− (69:9$
+8$−. . + !$!
: = 0 : $ = ;#$%
&
Experimental Physics 4 - Hydrogen atom 4
Normalized radial wave function
: $ = ;#$*+,
&
Experimental Physics 4 - Schrödinger equation 5
Spherically symmetric potential (m¹0)
( ) 01
1 22
22 =Q÷÷
ø
öççè
æ-
-+÷÷ø
öççè
æ Q-
xxx
xmC
dd
dd
( ) ( ) ( )xx
xx lmmm
ml Pd
dAP 221-==Q
lml
Experimental Physics 4 - Hydrogen atom 6
Radial density distribution
ò ò= =
=p
J
p
j
jJJjJy0
2
0
22 ddsind),,(d)( rrrrrP
Q3 $ 9$ = 34$! R $, N,O !9$r
rd
$214
$
Experimental Physics II - Magnetic field 7
Magnetic moments
Rn̂
21n̂ 22 RqR
Tq wpµ ==!AI
!!=µ
+µ!
-µ!
n̂2RmIL ww ==!!
Lmq !!
21
=µ
w
x
prwp
2
2dxrdI = 2xA p=
62
2
0
22 LQdxxr
L wrwpµ ò ==
ò=L
dmxI0
2 ò=L
dxxr0
22rp 231ML= LM
Q !!21
=µ
B!!!
´= µt
0 ≠ /+2
+10-1-2
+10-1
Experimental Physics 4 - Hydrogen atom 8
Magnetic properties of atoms
$
9$8
S = TU U =!(V $×9$
US =
!()
V $×C 98
S5S5 = −
2()
$×C XS = −2()
YZ = >YZ
giromagnetic ratio for orbital electronS6 = >Z6 = >)ℏ = S7)
the Bohr magneton2) = 3. '56…×%/%&*9/;
["
-7 = −S A [" = S7)["6 +&,*,2 = +89:*92; @ + S7)["6
l=1
l=2
Photon emission: \. = ±!
Photon spin: ^ = ! (in ℏ units)
Z3! = ^ ^ + ! ℏ! = (ℏ!
\) = ±!, 0
+ = ℏd
+Z6=ℏdℏ= d
for photons
Experimental Physics 4 - Hydrogen atom 9
The Stern-Gerlach experiment
e = −S A&[&$
. = 0 ⇒ ) = 0
-7 = S7)["6 = 0 ?
Electron has intrinsic angular momentum .and related to it magnetic moment or spin ^
S3 = >3^
^ = ^(^ + !)ℏ(^ + ! = ( ⇒ ^ = !/(−^ ≤ )3 ≤ ^
S6 = >Z6 = >)ℏ = S7) - orbiting electron
S36^6
= >3 =S7!(ℏ
= (S7ℏ= (>
S36 = S7
S3 = h5S7 ^(^ + !)For a free-standíng electron:
−2.00231930436153
S< = h
Experimental Physics 4 - Hydrogen atom 10
Nuclear magnetic resonance
-7 = −S A ["
-7 = h
Experimental Physics 4 - Hydrogen atom 11
To remember!
Ø Quantum mechanical calculations reproduce the atomic energy levels as obtained using the Bohr’s model.
Ø The calculations predict however n2-fold degeneracy of the levels.
Ø For s-orbitals the radial electron density distribution function is spherically symmetric, for others - not.
Ø The orbital angular momentum gives rise to magnetic properties of atoms.
Ø The electrons and protons have intrinsic angular momentum. Hence magnetic momentum as well.
Ø There are selection rules for spontaneousemmision of photons: ∆l = ± 1; ∆m = 0, ± 1.
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