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.