De Broglie Waves

7/27/2019 De Broglie Waves

1/13

LOUIS DE BROGLIE'S CONCEPT OFMATTER WAVES

Louis de Broglie asked himself "doesnature respect symmetry?" He wasconvinced that it does. He postulated thatjust as light has dual character, so doesmatter. So for a photon,

E = h.(1)also E = p c .(2)

(for a particle of zero rest like Photon)Equating (1) and (2)

p c = h or p = h/c =h/p = h/ ..(3)


2/13

The above equation was interpreted byde Broglie as the correlation betweenparticle (momentum p) and wave (wave

length ) characteristics. He simply invertedthe equation and wrote it as

= h/ p ..(4)and declared that particle of matter ofmomentum p must have its dual wavecharacter of wave length . The wavesassociated with matter are called matter-waves. Matter-waves cannot be consideredas oscillations of some field (s) like electro-magnetic waves. They are also calledprobability waves.


3/13


4/13

A beam of electrons acceleratedthrough the potential V were allowed to

strike a nickle crystal. Measurements weremade to count the number of electronsscattered by the crystal.


5/13

Davisson and Germer further investigatedwith properly oriented crystals thatelectron behave as waves of all wave

lengths () as given by De-Broglie'shypothesis. They calculated the wavelength of electron from the knownaccelerating potential V by applying the

relation:=12.24/V1/2 Ao

OBSERVATIONS

Davisson and Germerreported unexpected

results that the electronsreflected very strongly atcertain angles only andnot at other directions.


6/13

vm

h

m

eVv

or

vmeV

2

2

1 2Kinetic energygained by theelectron when

acclerated through Vvolts

De-Broglie

wavelength is givenby

Substituting for v

e-charge on electronh-Plancks constant

mo-rest mass of

electronA

VeVm

h

or

m

eVm

h

vm

h

24.12

2

2


7/13


8/13

PHASE VELOCITY AND GROUP VELOCITY

The de Broglie wave velocity/phase velocity isgiven by

vp=..(1)

where is the frequency is the de Broglie

wavelength=h/mv.

Equating the quantum expression E=h with therelativistic total energy E=mc2 to obtain h = mc2or = mc2 /h.

The de Broglie wave velocity is therefore

vp = = (mc2 /h)(h/mv)

vp =c2/v..(2)


9/13

As vc! violating special relativity;de Broglie introduced the concept of groupvelocity vg and showed that a group of waves

need not have the same velocity as the wavesthemselves to avoid the violation.

To begin with consider a group of waves

formed by combining the two waves differingslightly in frequency and wave number as

Y1=Acos(t-kx) .(1)

and

Y2=Acos[(+)t-(k+ k)x] .(2)

The resultant displacement Y at any time t andany position x is the sum of Y1 and Y2


10/13

With the help of trigonometric identity

cosA + cosB = 2 cos[(A+B)/2] cos[(A-B)/2] and

cos(-A) = cos(A) .(3)

We find

Y=Y1 + Y2

=2Acos[(2+)t/2 + (2k+k)x/2]*cos[(t- kx)/2] ..(4)

Since and k are small compared with andk respectively,

2+ 2 and 2k+k 2k and so

Y= 2Acos(t-kx) cos[(t- kx)/2] ..(5)


11/13

Equation (5) represents a wave of angularfrequency aand wave number k that has

superimposed upon it a modulation of angularfrequency /2 and of wave number k/2.

The effect of modulation is thus to producesuccessive wave groups as shown

+

=


12/13


13/13

Documents

De Broglie Waves