Physics 451

Quantum mechanics I

Fall 2012

Nov 7, 2012

Karine Chesnel

Homework this week:

• HW #18 Friday Nov 9 by 7pmPb 4.10, 4.11, 4.12, 4.13

Quantum mechanics

Quantum mechanics

The hydrogen atom

What is the density of probability of the electron?

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:

12

0

2 1n

l

l n

Quantum mechanics

Quiz 24a

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

12 2

13.6n

E eVE

n n

Energies levels

Spectroscopy

221

11

fi nnE

hcE

Energy transition

E0

E1

E2

E3

E4

Lyman

Balmer

Paschen

22

111

if nnR

Rydberg constant7 11.097 10R m

Pb 4.16Pb 4.17

Quantum mechanics

Quiz 24b

A. 465 nm

B. 87.5 x 10-8 m

C. 4.65 m

D. 87.5 x10-7 m

E. 4.65 x 10-8 m

What is the wavelength of the electromagnetic radiationemitted by electrons transiting from the 7th to the 5th band

in the hydrogen atom?

7 11.097 10R m

Quantum mechanics

, , ,r R r Y

The hydrogen atom

Coulomb’s law:

2

0

1( )

4

eV r

r

0

( ) jj

j

v c

jj cljj

nljc

221

)1(21

Solution to theradial equation

)(1

)( 1 krvekrr

rR krl with mE

k2

Pb 4.10 4.11

Quantum mechanics

The hydrogen atom

0

( ) jj

j

v c

jj cljj

nljc

221

)1(21

Equivalent to associated Laguerre polynomials

)2()( 121

llnLv )(1)( xL

dx

dxL q

ppp

pq

qxq

xq xe

dx

dexL

)(

Pb 4.12

Quantum mechanics

The hydrogen atom

, , ,nlm nl mlr R r Y

Spherical harmonics

(table 4.3)

Legendre polynomials

Radial wave functions

(table 4.7)

Laguerre polynomials

0

( ) jj

j

v c

jj cljj

nljc

221

)1(21

OR

Power series expansionwith recursion formula

French mathematicians

Quantum mechanics

• Edmond Laguerre 1834 – 1886

• Adrien-Marie Legendre 1752 – 1833

Quantum mechanics

The hydrogen atom

How to find the stationary states?

),()(,, mlnlnlm YrRr

nakn

1Step1: determine the principal quantum number n

Step 2: set the azimuthal quantum number l (0, 1, …n-1)

Step 3: Calculate the coefficients cj in terms of c0 (from the recursion formula, at a given l and n)

Step 4: Build the radial function Rnl(r) and normalize it (value of c0)

Step 5: Multiply by the spherical harmonics (tables) and obtain 2l +1 functions nlm for given (n,l)

),( mlY

(Step 6): Eventually, include the time factor: /),,(),( tiEnlm

nertr

Quantum mechanics

The hydrogen atom

Representation of ,,rnlm

Quantum mechanics

The hydrogen atom

Representation of

,,rnlm

Bohr radius

2100

2

40.529 10a m

me

Quantum mechanics

The hydrogen atom

Expectation values

, , ,nlm nl mlr R r Y

2 2r r R r dr22 2 2r r R r dr

2 2sin cos sinx d d r R r dr

Pb 4.13

Most probable values

Pb 4.14 2 2

maxr

2 2

0d r

dr

Quantum mechanics

The hydrogen atom

Expectation values for potential

, , ,nlm nl mlr R r Y

22 2

04

eV R r dr

r

Pb 4.15