04/19/23 Lecture XVII 1

Today

• Monday, March 21, 2005• Event: The Elliott W. Montroll Lecture • Speaker: Prof. David Gross, University of California,

Santa Barbara• Title: The future of physics (25 questions that might

guide physics in the next 25 years)• Talk:3:45 pm, Hoyt Hall Tea:3:15 pm, B&L Foyer 

Recipient of the 2004 Nobel Prize In Physics

04/19/23 Lecture XVII 2

PHY123

Hydrogen Atom

04/19/23 Lecture XVII 3

Concepts

• Electron distributions

• Quantum numbers, quantum state

• Zeeman effect

04/19/23 Lecture XVII 4

Schrödinger equation for Hydrogen atom

• Potential energy – electron is in Coulomb’s potential of the nucleus:

• Spherically symmetric potential• 3-D Schrödinger's equation

r

eU

2

04

1

E

r

e

zyxm

h

2

02

2

2

2

2

22

4

1

2

04/19/23 Lecture XVII 5

Hydrogen atom• Energy levels in H

eVn

En 2

6.13

• n- principle quantum number – determines the energy level

• What about electron distribution in atom?

04/19/23 Lecture XVII 6

Electron distributions• Same n, different shapes. Other quantum numbers play a role

• Quantum state of electron in atom is defined by a set of 4 numbers

• Principle quantum number n

• Orbital quantum number l

• Magnetic quantum number ml (same as lz)

• Spin projection ms

• Wave functions depend on 3 quantum numbers lnlm

04/19/23 Lecture XVII 7

Electron quantum state• Principle quantum number n=1,2,3,4,…

– determines energy level, higher E for higher n

• Orbital quantum number l – For each n l can be 0,1,2,3, …(n-1)

– l states are leveled by letters

– s: l=0; p: l=1; d: l=2; f: l=3; g:l=4

• E.g. n=5, then l can be 0, 1, 2, 3, 4– Possible l states are s,p,d,f,g

• n=1, only l=0 s-state is possible

04/19/23 Lecture XVII 8

Electron quantum state• Orbital quantum number is a

vector length l• Orbital angular momentum:

• Its projection on z axis is Lz =mlh another q.n. – magnetic quantum number ml

• ml can be only integer

z

0,1,2),...2();1(; lllml

2;2 lml

1;2 lml

0;2 lml

1;2 lml

2;2 lml

hllL )1(

04/19/23 Lecture XVII 9

Zeeman effect• Normally energy does not depend on

ml, but under magnetic field energy levels split- fine structure

• Magnetic moment (dipole) associated with orbital angular momentum

• Potential energy of the magnetic dipole in magnetic field

Lm

e 2

1

BU

04/19/23 Lecture XVII 10

Electron quantum state

• All electrons have spin=1/2• It is a vector • Its projection on z axis is

another q.n. – spin ms

• ms can be only

z

2

1sm

2

1;

2

1 sms

2

1;

2

1 sms

04/19/23 Lecture XVII 11

Ground state wave function• n=1l=0ml=0100 spin could be up or down

• No “preferred direction” system is spherically symmetric expect wave function to depend only on r– Not true if l≠0

• n=10 knots

• Wave function• Bohr’s radius:

0 r

nmme

hr

er

r

r

0529.0

1

20

2

0

30

1000

04/19/23 Lecture XVII 12

Probability to find e at r• dV=4r2dr

• dP=||2dV= ||24r2dr=Prdr

• Most probable radius – where Pr has maximum

0

2

30

2

4 r

r

r er

rP

0max

0

2

30

2

40

22

30

2

30

2

188840 0000

rr

r

re

r

re

r

re

r

re

r

r

rPr

r

r

r

r

r

r

r

r

r

0)(;0)0( rr PP

04/19/23 Lecture XVII 13

n=2 wave function• n=2l=0,1

• First consider200

• Still no “preferred direction” system is spherically symmetric expect wave function to depend only on r

• n=21 knots

• Wave function

0 r

0)2(

232

1

0200

2

03

0

2000

r

er

r

r

r

r

04/19/23 Lecture XVII 14

Probability to find e at r for n=2

• Most probable radius – where Pr

has global maximum

0

2

03

0

2

28

r

r

r er

r

r

rP

0max

2

03

0

2

5~

)2(8

0 0

rr

er

r

r

r

rPr

r

r

r

04/19/23 Lecture XVII 15

n=2 wave function• n=2l=0,1

• Now consider21m

• There is a “preferred direction” – direction of vector L system is NOT spherically symmetric, but axial symmetry is there expect wave function to depend not only on r

0

0

2

50

121

2

50

210

64

32

r

r

r

r

er

iyx

er

z