de broglie wave & heisenberg uncertainty principle

8/17/2019 De Broglie Wave & Heisenberg Uncertainty Principle

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De Broglie Waves & Heisenberg Uncertainty Principle

DE BROGLIE WAVE

De Broglie!s "ypot"esis#

We know interference and diffraction of light can only be explained if we assume that the light behavesas a wave. The photoelectric effect and Compton scattering can be understood only if light behaves as aparticle. Is this dual particle-wave nature a property only of light or of material objects as well In a boldand daring hypothesis in his !"#$ doctoral dissertation% de &roglie suggested that associated with any

material particle moving with momentum p there is a wave of wavelength λ% related to p according to

]1[, p

h=λ

The wavelength λ of a particle computed according to '(. )!* is called its de &roglie wavelength.

The de &roglie wavelength of a nonrelativistic particlemv

h

p

h==λ can also be expressed in terms of the

particles kinetic energy. +or example% consider an electron freely accelerated from rest at point a to pointb through a potential increase ,b ,a ,ba . The work done on the electron /e,ba0 e(uals its kineticenergy 1.

Therefore% baee

ba eV m pm

peV 2,

2

2

=∴= and the de &roglie wavelength of the electron is

baeeV m

h

p

h

2==λ .

The speed of a particle is mhv λ = and the kinetic energy is mhmhmmv K 2222 2]][2[

2

1λ λ === .

This result shows that for a given wavelength% the kinetic energy is inversely proportional to the mass.2ence% the proton with a smaller mass% has more kinetic energy than the neutron.

E$a%ple #

Compute the de &roglie wavelength of the following3

)!* 4 !555-kg automobile traveling at !55m6s.

)#* 4 !5 gm bullet traveling at 755 m6s.

)8* 4 smoke particle of mass !5-" gm moving at !cm6s.

)$* 4n electron with a 1. ' of !ke,.

)7* 4n electron with a 1.' of !55 9e,.

ol'tion#

)!* m x smkg

s J x

mv

h

p

h 3934

106.6]/100][1000[

.106.6 −−

====λ

)#* m x smkg

s J x

mv

h

p

h 342

34

103.1]/500][10[

.106.6 −−

−

====λ

[3] m x smkg

s J x

mv

h

p

h 20212

34

106.6]/10][10[

.106.6 −−−

−

====λ


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[4] nmeV x

nmeV

pc

hc

p

h24.1

100.1

.12403

====λ

[5] fmm xeV

nmeV

pc

hc

p

h4.12104.12

10

.1240 158

===== −

λ

:ote that the wavelengths computed in parts )!* to )8* are far too small to be observed in the laboratory.;nly in the last two cases% in which the wavelength is of the same order as atomic or nuclear si


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&ecause the particle velocity v must be less than the velocity of light c% the de &roglie waves alwaystravel faster than lightB In order to understand this unexpected result% we must look into the distinctionbetween phase velocity and group velocity. hase velocity is what we have been calling wave velocity. Itmay be noted that a group of waves need not have the same velocity as the waves themselves.

We know that a pure sine wave is completely unlocali


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we get

2

][cos

2

])2()2[(cos2

21

kxt xk k t A

y y y

∆−∆∆+−∆+=

+=

ω ω ω

Eince ∆ω and ∆k are small compared with ω and k respectively%

k k k 22

22

≈∆+

≈∆+ ω ω ω

and so

]22

cos[]cos[2 xk

t kxt A y ∆

−∆

−= ω

ω

:ow the phase velocity vp and group velocity vg are defined as

dk

d vand

k v g p

ω ω ==

The angular fre(uency and wave number of the de &roglie waves associated with a body of rest massm5 moving with the velocity v are

][,222

][,22

bh

p

phk

ah

E

π π

λ

π

π πν ω

===

==

+rom )a*%

ω ω π

==2

h E

and from )b*%

k k h

p ==π 2

The group velocity of the de &roglie wave can then be expressed as

dp

dE

dp

dE

dk

dp

dp

dE

dE

d

dk

d v g ==== ]][][

1[]][][[

ω ω

+or a classical particle having onlym

p E K

2.

2

= % we can find

vm

p

m

p

dp

d

dp

dE === ]

2[

2

which is the velocity of the particle.

We can also find group velocity in the following way3

4


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( )

υ

υ

=

=

=

+

=

+∂

∂=

∂

∂=

2

2

2

42

0

22

2

42

0

22

mc

mvc

E pc

cmc p

pc

cmc p p p

E g

Therefore%

vdp

dE v g ==

Therefore% the de &roglie wave group associated with a moving body travels with the same velocity as

the body.

E$a%ple ,#

4n electron has a de &roglie wavelength of #.55 pm#.55x!5-!# m. +ind its 1.'% phase velocity and groupvelocity of its de &roglie waves.

ol'tion#

keV eV xm x

sm x seV xhc pc 6201020.6

1000.2

]1000.3][.10136.4[ 512

815

====−

−

λ

Fest energy of the electron '5 7!! ke,

Therefore%

keV keV keV keV E pc E E E E K 292511)600()511()(. 22022

00 =−+=−+=−=

The electron velocity can be found from

22

0 1/ cv E E −= or% cc E E cv 77.0]803511[11 2220 =−=−=

Therefore%

vv

cc

cvcv

g

p

=

=== 30.177.0

22

HEIE3BERG! U34ER5AI356 PRI34IPLE

To regard a moving particle as a wave group implies that there are fundamental limits to the accuracywith which we can measure such particle properties as position and momentum.

To make clear what is involved% let us look at the wave group of +ig.! below3

5

∆x


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The particle that corresponds to this wave group may be located anywhere within the group at a given

time. ;f course% the probability density |ψ|# is a maximum in the middle of the group% so that it is most

likely to be found there. :evertheless% we may still find the particle anywhere where |ψ|# is not actually

+ig. #a?.2owever% the wavelength of the waves in such a narrow packet is not well definedG there are not enough

waves to measure λ accurately. This means that since mvhH = % the particle@s momentum mv is not a

precise (uantity.

;n the other hand% a wide wave group% such as that in +ig. #b% has a clearly defined wavelength. Themomentum that corresponds to this wavelength is therefore% a precise (uantity. &ut where is the particlelocated The width of the group is now too great for us to be able to say exactly where it is at a giventime.

Thus we have the 'ncertainty principle3

“It is impossible to know both the exact position and exact momentum of an object at the sametime”

This principle% which was discovered by Werner Heisenberg in !"#% is one of the most significant of physical laws.

:ow consider what happens when we add to our original wave another wave of slightly different

wavelength ) i.e. different k*. When we had a single sine wave% ∆k was


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Ksing )b* in )a* we get

≥∆∆ x

p x J. J. J. )c*

where the x subscript has been added to the momentum to remind us that '(. )c* applies to motion in agiven direction and relates the uncertainties in position and momentum in that direction only. Eimilar and

independent relationships can be applied in other directions as necessaryG thus

,≥∆∆ y

p y or% ,≥∆∆ z p z

4nother form of uncertainty principle is sometimes useful. =et us consider emission of electromagnetic

radiation% where energy ' is being emitted in the time interval ∆t in an atomic process. If the fre(uency of

the electromagnetic wave is ν then% there is a possibility that we will make an error of at least one cycle

in counting the fre(uency > number of waves?. Eo the error i.e. uncertainty in fre(uency would be

∆ν ≥ t ∆

1

The corresponding uncertainty in energy is

∆' h∆ν

and so% ∆' ≥ t

h

∆ or% ∆' ∆t ≥ h

a precise for of calculation based on the nature of wave groups changes this result to

∆' ∆t ≥ 2

The above expression gives the uncertainty principle in terms of energy and time.

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de broglie wave & heisenberg uncertainty principle

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