Physics 451

Quantum mechanics I

Fall 2012

Nov 5, 2012

Karine Chesnel

Homework this week:

• HW #17 Tuesday Nov 6 by 7pm

• HW #18 Thursday Nov 8 by 7pm

Announcements

Quantum mechanics

Quantum mechanics

The hydrogen atom

What is the density of probability of the electron?

The hydrogen atom

2

2 2

1 1 1sin ( 1)

sin sin

Y Yl l

Y

The angular equation (same)

, , ,r R r Y

Angular function (same) , cosm im m

l lY Ae P

Azimutal quantum number

Magnetic quantum number

l

m0l m l

Phys 451

, , ,r R r Y

The radial equation

( )u rR r

2 2 2

2 2

( 1)( )

2 2

d u l lV r u Eu

m dr m r

The hydrogen atom

Coulomb’s law:

2

0

1( )

4

eV r

r

2 2 2 2

2 20

( 1)

2 4 2

d u e l lu Eu

m dr r m r

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kr

Asymptotic behaviors

The hydrogen atom

The radial equation2

02 2

( 1)1

d u l lu

d

2mEk

2

2

d uu

d 0

2

2 2

( 1)d u l lu

d

u Ae 1lu B

Phys 451

2

0 202

me

k

The hydrogen atom

1 ( )lu e v

Peeling off the asymptotic behaviors

2

022( 1 ) 2( 1) 0

d v dvl l v

d d

Power expansion0

( ) jj

j

v c

Recursion formula:0

1

2( 1) 2

( 1)( 2 2) 1j j j

j lc c c

j j l j

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The hydrogen atom

1 ( )lu e v

The series must terminate

0

( ) jj

j

v c

02( 1)0

( 1)( 2 2)

j l

j j l

max 02( 1)j l

Principal quantum number0

max 12

n j l

Phys 451

Quantum mechanics

Quiz 23a

A. one

B. two

C. n

D. n -1

E. An infinity

For a given quantum number n,how many values of l can exist?

Quantum mechanics

The hydrogen atom

Ground state: “binding energy”

22

1 20

13.62 4

m eE eV

Quantization of the energy22

2 20

1

2 4n

m eE

n

Bohr 1913

Principal quantum number

0 2n

Quantum mechanics

The hydrogen atom

2

20

1

4

mek

n

1~

tank

dis ce

Bohr radius2

1002

40.529 10a m

me

nak

1

Quantum mechanics

The hydrogen atom

21

n

EEn na

kn1

Energies levels

Stationary states ),()(,, mlnlnlm YrRr

n: principal quantum number

l: azimuthal quantum number 1nl

m: magnetic quantum number lm

Degeneracy of nth energy level:

1

0

12n

l

l

Quantum mechanics

Quiz 23b

A. 5

B. 9

C. 11

D. 25

E. 50

What is the degeneracy of the 5th energy bandof the hydrogen atom?

Quantum mechanics

The hydrogen atom

21

n

EEn Energies levels

Spectroscopy

221

11

fi nnE

hcE

Energy transition

22

111

if nnR

Rydberg constant

E0

E1

E2

E3

E4

Lyman

Balmer

Paschen