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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) - PowerPoint PPT Presentation
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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 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