phys 342 - lecture 19 notes - f12

8/11/2019 PHYS 342 - Lecture 19 Notes - F12

1/13

Lecture 19

Hydrogen-Like Atoms

Atoms with all but one electrons stripped

Energy level:

En

13.6eVn

2 Z

2

Bohr radius:

rn

n2a0

Z

Bohrs theory can be adapted to other hydrogen-like systems.


2/13

Lecture 19

Many-Electron Systems

Bohrs shell hypothesis:

Electrons in an atom are arranged in shells.

Each shell is associated with each of the principle quantum

numbers n, with its radius increasing as n2 and its energy

decreasing as n-2 .

There can be no more than 2n2 electrons in each shell.

Electrons fill the innermost shell first and then outer shells.


3/13

Lecture 19

Shells have letter names:

K shellfor n= 1

L shellfor n= 2

Energy Diagram and X-rays

The atom is most stable in its ground state.

When it occurs in a heavy atom, the

radiation emitted is an x-ray. It has the

energyE (x-ray) =Eu!E".

If there is a vacancy in an inner shell, an

electron from higher shells will fill thevacancy at lower energy.


4/13

Lecture 19

X-ray Lines

K

K

Vacancies may be created by collisional

or photo ionization.

The x-rays have names:

L shell to K shell: K!x-ray

M shell to K shell: K"x-ray

Moseley found this relation

holds for the K#x-ray: f1/2 An(Z bn )

Kseries: due to transitions from n>1 to n=1, b1=1

Lseries: due to transitions from n>2 to n=2, b2=7.4


5/13

Lecture 19

Moseley Plot


6/13

Lecture 19

Wave and Particle Duality

de Broglies hypothesis:

A photon, quantum of light, has all the properties of a particle

of normal matter. This is the wave-particle duality of light.

The wave properties and particle properties of light are relatedin the following way:

p

h

h

Ef == !,

The relation above is a perfectly general one, applying

to radiation and matter alike.


7/13

Lecture 19

Plane Waves

For a wave that propagates alongx-axis, we have

Wave equation:2

2

22

21

tvx !

!=

!

! ""

!"#

$%&

+')

*,

= -.

/0T

txAtx 2sin),(Solutions:

Definitions:

period:T

initial phase:!

wavelength:"=vT

frequency:f=1/T

wave number: k=2#/"

angular frequency: $=2#/T


8/13

Lecture 19

Phase Velocity

Alternative expression for the plane wave:

[ ]!"# ++= tkxAtx sin),(in the negativexdirection:

[ ]!"# +$= tkxAtx sin),(in the positivexdirection:

more definitions:

phase of the wave: !" +=# tkx

phase velocity: the velocity of a point that moves with awave at constant phase.

kTvvphase

!"==#


9/13

Lecture 19

Wave Description of a Particle

Consider a particle of rest mass m0that moves with a velocity v.

according to de Broglieshypothesis, there is a characteristic

frequency (or wavelength)f0(or "0) associated with the particle,

and, in the reference frame of the particle, we have

2

00 cmEhf ==

Then, in the same reference frame, it is possible to write theequation of a stationary vibration associated with the particle

(internal vibration), e.g.,

002sin tf!" =


10/13

Lecture 19

Wave in the Laboratory Frame

Using Lorentz transformation to convert the internal vibration

to a reference frame at rest (with respect to which the particle

travels), i.e.,

$& '= 20 cvx

tt (

we have

!#

$& '(

$%& '==

w

xtf

cvxtftf

)

*))+

2sin

2sin2sin2000

where

v

cw

ff

2

0

=

= !


11/13

Lecture 19

De Broglies Hypothesis Recovered

vm

h

cmv

hc

hfv

hc

fv

c

f

w

02

0

22

0

22

0

22

1

11

!

"

""#

=

$

=

$

=

$

==

p

h=!"

h

mv

v

c

h

p

v

cwf

22

===

! h

Ef =!


12/13

Lecture 19

An Alternative Approach

From the point of view of an observer (at rest), however, we have

the following scenarios:

The mass of the particle mis greater than its rest mass, so

the characteristic frequency is now given by

2

2

0

1 !"=

cmhf 0

2

0

1

ff

f !"

=

#

=

Decrease of the frequency of the internal vibration, due totime dilation:

0

0

1

Tf =

2

02

0

1 1

1/

1!

!"=

"

= fT

f


13/13

Lecture 19

de Broglies Solution

In the observers frame of reference, a wave is described by

tf

w

v

ft

w

vttf

w

x

tf

12sin

12sin

2sin

2sin

!

!

!

!"

#$%&

'()

*=

+,-

./0

$%&

'() *=

$%&

'()

*=

( )

2

22

2

1 11

c

v

w

v

ffw

v

f

==

!==

"#$

%&'

!

(

(

2cvw=!

phys 342 - lecture 19 notes - f12

Documents