Lecture physics a2 angular momentum, atomic states, the pauli principle huynh quang linh

Summary of s-states of H-atomof the Coulomb potential, one needs to solve the radial SEQ  The zeros in the subscripts below are a reminder that these are states with zero angular moment

Lecture 11: Angular Momentum, Atomic

States, &

the Pauli Principle

2

0

2

Parametric Curve

2

2

x20( ) t

2

|Y 20 |

1 0

1

Parametric Curve

1.2

1.2 x21( ) t

1.2

|Y 21 |

1 0

1

Parametric Curve

1.2

1.2 x( ) t

1.2

|Y 22 |

z

1 0

1

Parametric Curve

1.2

1.2 x11( ) t

1.2

1

0

1

Parametric Curve

1.2

1.2

x10( ) t

1.2

z

1

0

1

Parametric Curve

1.2

1.2

y( ) t

1.2

|Y 00 |

z

 Radial wave functions

 Angular wave functions

 Quantization of Lz and L 2

 Spin and the Pauli exclusion principle

 Stern-Gerlach experiment

 Nuclear spin and MRI

Summary of s-states of H-atom

of the Coulomb potential, one needs to solve the radial SEQ

 The zeros in the subscripts below are a reminder that these are states with zero angular momentum.

0

0.5

1

1

0

f( ) x

4

R 10

0

/ 0

,

1 (r) e r a

r

R 20

0 0.5

1 1

.2 h( ) x

10

0

0 2 / 0

0 , 2

2

1 )

a

r r











 



R 30

0

2 3

.5 d4( ) x

15

0

3 / 2

0 0

0 , 3

3 2

2 3 )

a

r a

r r































) r ( ER )

r (

R r

e r

r r

1 m 2

2 2

2 2





















(spherically symmetric)

The Ylm(q,f) are known as “Spherical harmonics”.

They are related to the angular momentum of the electron

Wavefunction of the H-atom

(SEQ) in spherical coordinates is a product

wave function of the form:

x

y z

r

q

f

) , ( )

( )

, ,



with quantum numbers: n l and m

principal orbital magnetic

(angular momentum)

Last lecture we studied the properties of the radial part

Today we will examine the angular part

Quantized Angular Momentum

 The  dependence of our

wavefunction must have the form

Re( y )

f

r

But, l = h/pt, where pt is the tangential momentum of ‘orbiting’ electron:

The angular momentum along a given axis can have only quantized values:

Lz = 0,   ,  2,  3,

pt r = Lz = m

m = 0, 1,  2,  3,

Reminder: e im f = cos(m f ) + i sin(m f )

2 p r = m l m = 0,  1,  2,  3, m = ‘orbital’ magnetic quantum number

 Semi-classical ‘argument’ for angular momentum quantization: An integer

number, m, of wavelengths must fit around the circle (radius r ) above!

Quantization arises because only certain wavelengths fit around path.

l m

Y q f  e 

Summary of quantum numbers for the H-atom:

The “l” Quantum Number

 The quantum number l in the angular wave function Ylm(q,f) tells the total angular momentum L

L 2 = Lx2 + Ly2 + Lz2 is quantized The possible values of L 2 are:

momentum about a given axis

, 2 , 1 , 0 l

where )

1 l

( l

, ,

0 m

m

Principal quantum number:

n = 1, 2, 3, ….

Orbital quantum number:

l = 0, 1, 2, …, n-1

Orbital magnetic quantum number:

m = -l, -(l-1), … 0, … (l-1), l ( = m l )

Classical Picture of L-Quantization

e.g., l = 2 L  l ( l  1 ) 2  2 ( 2  1 ) 2  6 

Classically, the angular momentum vector precesses around the z axis.

Why can’t the orbital angular momentum vector simply point in a

specific direction, e.g., along z ?

If it did, then that would mean that r and p must both be in the x-y

plane But that means that there would be no position uncertainty

D z in the z-direction, nor any momentum uncertainty D pz, i.e., D z =

D pz =0 , in violation of the Uncertainty Principle.

L =  6 

Lz

+2

+

0

-

-2

L   r p

Probability Density of Electrons

Probability density = Probability per unit volume = ynlm 2  Rnl2 Ylm2

The density of dots plotted below is proportional to ynlm 2

The Angular Wavefunction, Ylm( q ,  )

 The angular wavefunction may be written: Ylm(q,) = P(q)eim 

where P(q) are polynomial functions of q

1 0

1

Parametric Curve

1.2

1.2 y( ) t

1.2

|Y 00 |

z

0

q

l =0

p

4

1

0

,

Y

polar plots of the q-dependent part of |Ylm( q ,  )| (i.e., P(q)):

Y 00

z

x

y

1 0

1

Parametric Curve

1.2

1.2 x10( ) t

1.2

|Y 10 |

z

l =1

0

q

q cos

0 ,

1 

Y

Y 10

y z

x

Re{Y 11 }

z

x

y

1 0

1

Parametric Curve

1.2

1.2 x11( ) t

1.2

q

q

sin

1 ,

1 

Y

(Length of the dashed arrow is the magnitude of Ylm as a function of q )

The Angular Wavefunction, Ylm( q ,  )

q

l = 2

2 0

2

Parametric Curve

2

2 x20( ) t

2

1 0

1

Parametric Curve

1.2

1.2 x21( ) t

1.2

1 0

1

Parametric Curve

1.2

1.2 x( ) t

1.2

|Y 20 |

|Y 21 |

|Y 22 |

z

3 cos2 1

0

,

2  q 

Y

q

q cos sin

1 ,

2 

Y

q

2 2

,

2  sin

Y

q

q

z

z

z

x