lecture 6 de broglie waves_2

7/25/2019 Lecture 6 de Broglie Waves_2

1/13

Lecture 6. De Broglie Waves

Outline:

The de Broglie Hypothesis

The Davisson-Germer Experiment

The Electron Interference Experiment


2/13

The Need for a New Mechanics

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one

sentence passed on to the next generation of creatures, what statement would contain the

most information in the fewest words? I believe it is the atomic hypothesis that

All things are made of atoms-little particles that move around in perpetual motion,

attracting each other when they are a little distance apart, but repelling upon being

squeezed into one another.

In that one sentence, you will see, there is an enormous amount of information about the

world, if just a little imagination and thinking are applied.

Oops! Classical physics cannot explain existence of stale atoms!

2 2

2

e vF m

R R = 610 /v m s - non-relativistic motion

2

2

eK U R+

2 2 2 4

3 3 2

2 2

3 3

e a e vI c c R =

- poer emitted y an

accelerated charge

32 3 210

2 4 10 !!!

K U e c R R cs

I R e v v v

+ = =

The lifetime of

a "classical# atom$


3/13

- de Broglie wavelength

p- the o%ect&s moment'm

De Broglie Hypothesis

The e(-m( aves can e descried 'sing the lang'age of quantumparticles

)photons*( Can particles ehave as aves+

De Broglie ),./* s'ggested that a

plane monochromatic ave is associatedith a freely moving particle$

( ) ( )

0

i t kx

x e

=

This ave )its phase* travels ith thephasevelocity

This is a sol'tion of the ave e0'ation in one dimension$

2 22

2 2v

t x

=

vk

=

article! properties Wave! properties

,E

i pc

r,i k

c

r- oth the time-li1e and space-li1e

components of these 2-vectors sho'ld

transform 'nder 3

Th's4 e&ll re0'ire p k=r

h

The phase is a orentz-invariant quantity4

the )scalar* prod'ct of to 2-vectors$

t kr ( ),ict rr

,i kc

r t kr r

2 h

k p

= =

5e&ll apply the same logic hich helped 's to estalish therelationship eteenpand for photons$

E =h and


4/13

De Broglie Wavelength

- depends on the momentumrather then energy)e(g(4 for an o%ect

at rest4 " *

2 h

k p

= =

Compare ith Compton

avelength of the particle C

h

mc =

- formally spea1ing4 C( avelength can e

considered as the dB avelength that corresponds

to the moment'm e0'al to the length of 2-vector)i!"c,p*


5/13

#$a%ples&. 5hat is the de Broglie avelength of the charge carriers in a typical metal+ The 1inetic

energy of charge carriers )"cond'ction# electrons* in metals is of an order of a fe e6 )78 e6

foru*9 it&s called the :ermi energy4 !#(

nmeVhccKm

hc

Km

h

p

h

ee

1240,22

2

nmeVeV

nmeV55.0

10552

1240

5

'.; 'c1eyall )f'llerene* is a large molec'le comprised of


6/13

#$a%ple

5hat o'ld e the 1inetic energy of each electron in a eam of electrons having a

de Broglie avelength of micro6olts( :or comparison4 at room temperat're the 1inetic energy of a free electron

is

( )2319

1.5 1.4 10 / 30030.04

2 1.6 10 /B

J K KK k T eV

eV J

= = =


7/13

hase and (roup )elocities

Thisphasevelocity

2 2

p

E mc cv c

k p mv v

= = = = >

- no limitations on the phase velocity4

)phase of a plane ave does not

carry any information*

The oservale is the group velocity )the velocity ofpropagation of a ave "pac1et# or ave "gro'p#( 3et&s

consider the s'perposition of to harmonic aves ith

slightly different fre0'encies )4 11*$

( )1 cosy A t kx=

( ) ( )2 cosy A t k k x = + +

( ) ( ){ } ( )

( )

1 2

1 1

2 cos 2 2 cos2 2

2 cos cos2 2

y y y A t k k x t k x

kA t kx t x

= + = + +

( ) ( )1 1

cos cos 2cos cos2 2

+ = +

"envelope#

ave gro'p

fast oscillations

ithin the ave

gro'p

The velocity of propagation of

the ave pac1et$

g

dv

dk

= -the groupvelocity

2" "

k


8/13

(roup (elo%*ang


9/13

(roup )elocity of de Broglie waves

g

d dEv

dk dp

= = ( ) ( )

22 2E pc mc= +

( ) ( )

2 2 2

222 2

1 2

2

dE pc pc mv c vdp E mc

pc mc

= = = =+

gv v=

2

g pv v c=

- the gro'p velocity of de

Broglie aves coincide ith

the particle&s velocity

+,&

f-t

&

;periodic processes$ contin'o's spectr'm )represented as :o'rier integral*

( )

1, 0

sinc sin, 0

==

eriodic processes$ discrete spectr'm ):o'rier series*(

t


10/13

/nterference at Low /ntensities

To-dimensional array of small

particleFphoton )"quanton#* detectors

"trictly-ave# model$ smooth

oscillating variations of intensity)the n'mer of particles*(

"trictly-particle# model$ discrete

events 't no oscillations of

intensity(

The ave model descries correctly the statistical distributionof 0'anton arrivals4 the

particle model descries the interaction of each individ'al 0'anton ith a detector

)"collapse of the avef'nction in the process of meas'rement#*(

%mplications:

ehavior of individ'al 0'antons is not deterministic )"netonian#* each individ'al 0'anton "1nos# ao't oth slits

any attempt to cond'ct "hich-ay# experiment 1ills the interference

neither the particle nor ave models are ade0'ate

0tatistical /nterpretation of de Broglie Waves)ax Born*$ de Broglie ave the ave of

proaility4 the intensity of dB ave at a given location is proportional to the proaility to

detect the particle at this location - to e disc'ssed later(

The statistical properties can e st'died only if one can repeat the same experiment ith

identical particles many times )or oserve many identical particles in identical conditions at

the same time*(


11/13

#arlier -incorrect /nterpretation of de Broglie Waves

Earlier ideas )chrdinger*$ particle the ave gro'p( In favor$ the gro'p velocity of de

Broglie aves coincide ith the particle&s velocity( Hoever4 the ave pac1et o'ldn&t live

for a long time eca'se of the dispersionof de Broglie aves in vac''m$

( )2

22E p mcc

=

( )( ) ( )

22 2 2 2

1 1p

pc mcE mc mcv k c c

k p p p k

+ = = = = + = + h

( ) ( )pk v k k = for light in vac''m c k= - no dispersion )cc)1**

In general

( ) ( ) ( )22 2 2ck mc =h

( )

22

22 mcck

= + h

Th's4 a particle %& 'O( the gro'p of de Broglie aves!

Deformation of a ,D ave gro'p4 m!me), Bohr=(=8/nm4 time 'nits "/Fmee

2.(2,=-,>s*


12/13

0tate,of,the,art1 detection of single TH2 photons

;:IJ

The energy resol'tion s'fficiently

high for detecting single =(,-TH?

photons K =(,A and ,-TH?

photons K =(/A(

cold )2A* antenna to

red'ce photon "noise#

nanostr'ct'res at 'ltra-

lo # to increase

sensitivity

( ) ( )34 12

19

6.6 10 1 10

0.0041.6 10 /

J s Hz

E eVeV J

= =

Detection of the visile-range photons$ not a ig

deal4 the photon energy is s'fficient to generate

photoelectrons )the photoelectric effect*

photom'ltiplier

( )1 3phE eV

- comparale ith the energy

gap eteen the valence and

cond'ction ands in typical

semicond'ctors

This tas3 *eco%es %ore challenging at lower photon energies...


13/13

article,Wave Dualis%

)onclusion$ all forms of matter )oth particles and fields* exhiit aveli1e aspects(

De Broglie&s e0'ations2 h

k p

= =E =h e0'ally apply to particles and photons

The wave-li*echaracter of an o%ect ecomes more apparent at lo 1inetic energies as its

de Broglie avelength increases$ it is m'ch easier to oserve interference ith visile light

than ith electrons(

( )characteristic dimensions of the exp. set-p

lecture 6 de broglie waves_2

Documents