quantum mechanics:uncertainty principle

7/28/2019 Quantum Mechanics:Uncertainty Principle

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A localized wave or wave packet:

Spread in position Spread in momentum

Superposition of waves

of different wavelengths

to make a packet

Narrower the packet , more the spread in momentum

Basis of Uncertainty Principle

A moving particle in quantum theory


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Heisenberg's Uncertainty Principle

___________________________________

The Uncertainty Principle is an important

consequence of the wave-particle duality of

matter and radiation and is inherent to thequantum description of nature

Simply stated, it is impossible to know both the

exact position and the exact momentum of anobject simultaneously

A fact of Nature!


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__________________________________

Uncertainty in Position :

Uncertainty in Momentum:

x

xp

2

hpx x


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- applies to all conjugate variables___________________________________

Position & momentum

Energy & time

2

hpx

x

2

h

tE


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Uncertainty Principle and the Wave Packet

___________________________________

p

h

2

hpx x

p

p

x


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Some consequences of the Uncertainty Principle

___________________________________

The path of a particle (trajectory) is not well-defined in

quantum mechanics

Electrons cannot exist inside a nucleus

Atomic oscillators possess a certain amount of energy

known as the zero-point energy, even at absolute zero.


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Why is nt the uncertainty principle apparent to

us in our ordinary experience?

Plancks constant, again!!___________________________________

Plancks constant is so small that the

uncertainties implied by the principle are alsotoo small to be observed. They are only

significant in the domain of microscopic

systems

J.10x6.634

h

2

h

px x


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Heisenberg Uncertainty Principle

The uncertainty principle states that the positionand momentum cannot both be measured,

exactly, at the same time.

Where h (6.6 x 10-34) is called Plancks constant. As h is so small, theseuncertainties are not observable in normal everyday situations

x p h

The more accurately youknow the position (i.e., thesmaller x is) , the lessaccurately you know the

momentum (i.e., the largerp is); and vice versa

hor2hor

For

Numerical

For

Applications

Historic importance


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Increasing levels of wavepacket localization, meaning theparticle has a more localized position.

In the limit 0, the particle'sposition and momentum become knownexactly. This is equivalent to theclassical particle.

p is less p is more
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The wave nature to particles means a particle is a wave packet,the composite of many waves

Many waves = many momentums, observation makes one

momentum out of many. Principle of complementarity: The moving electron will

behave as a particle or as a wave, but we can not observe bothaspects of its behavior simultaneously. It states thatcomplete description of a physical entity such as a

photon or an electron can not be made in terms ofonly particle properties or only wave properties, butthat both aspects of its behavior must be considered.

Exact knowledge of complementarities pairs (position, energy,time) is impossible.



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Same situation, but baseball replaced by an electron which has mass 9.11x 10-31 kg

So momentum= 3.6 x 10-29

kg m/s and p = 3.6 x 10-31

kg m/s

The uncertainty in position is then

Example A pitcher throws a 0.1-kg baseball at 40 m/s

So momentum is 0.1 x 40 = 4 kg m/s

Suppose the momentum is measured to an accuracy of 1 % , i.e.,p = 0.01p = 4 x 10-2 kg m/s

The uncertainty in position is then

No wonder one does not observe the effects of the uncertainty principle ineveryday life!


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Example: A free 10eV electron moves in the x-direction with a speed of1.88106 m/s. assume that you can measure this speed to precision of 1%.With what precision can you simultaneously measure its position?

the momentum of e- ispx = m vx = 9.1110

-31 kg 1.88106 m/s= 1.71 10-24 kg m/s

The uncertainty px in momentum is 1%

x h/4 px =3.1 n m

Example: A golf ball has a mss 45gm and speed of 40 m/s, which you canmeasure with a precision of 1%. What limits does the uncertainty principleplace on your ability to measure its position.

Calculations yields x 6 10-31 m. this is very small distance,

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Applications of uncertainty principle

1. Non-existence of electrons in the NucleusAssume that the electron is present in the nucleus. The radius of

the nucleus of any atom is of the order 5 fermi (1 fermi = 10-15 m). Forthe existence of electron in the nucleus, the uncertainty x in itsposition would be at least equal to the radius of the nucleus, i.e.uncertainty in the position

According to the uncertainty principle.

p h/4x= 1.05410-20 kg-m/sec

If this is the uncertainty in momentum of the electron then the

momentum of the electron must be at least of the order of its

magnitude, that is , p 1.05410-20kg-m/sec, an electron having somuch momentum should have a velocity comparable to the velocity

of light. Hence, its energy should be calculated by the relativistic

formula

E2

=p2

c2

+mo2

c4

15x R 5 10 m


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pc= 1.05410-203108 = 20 MeV

The rest energy of electron 0.51 MeV, is very small as compared topc. Hence second term in relativistic equation can be neglected.

Thus, if the electron is the constituents of the nucleus, it shouldhave an energy of the order of 20 MeV.

However, from experiment of decay it is found that theelectrons emitted from the radioactive element do not have morethan 2-3 MeV. Therefore, it is confirmed that electrons do notreside inside the nucleus.



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4. Particle in a Box Problem

Find the minimum energy for a particle confined to a box of size L

Macroscopic: 1 mg particle confined to 1 mm Emin

~ 10-29 eV

Microscopic: Electron confined to 0.1 nm Emin~ 4 eV

2 2

2 2

min

34 16

2

min 2

Using (Uncertainty Principle) and

where

(for 0),

2 2

1.05 10 J s or 6.58 10 eV s

2

p p p px

p pE

m m

EmL

x = L

xX=0 X=L

Energy



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Physical Origin of the Uncertainty PrincipleHeisenberg (Bohr) Microscope

The measurement itself introducesthe uncertainty

When we look at an object we see it

via the photons that are detected bythe microscope

These are the photons that are scattered

within an angle 2 and collected by alens of diameter D

Momentum of electron is changed

Consider single photon, this willintroduce the minimum uncertainty


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~(D/2)/L, L ~ D/2 isdistance to lens

Uncertainty in electron

position for small is

To reduce uncertainty in themomentum, we can eitherincrease the wavelength orreduce the angle

But this leads to increaseduncertainty in the position,since

h

pelectron2

electronx2sin 2



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electron

electron

electron

electron

2hp

hp

x2 x

( p )( x) h

( p )( x)


i l ill i f i i i l Si l li


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To see more clearly into the nature of uncertainty, we considerelectrons passing through a slit:

We apply the condition of

minima from single slitdiffraction,

and postulate that is the de

Broglie wavelength.

Momentumuncertainty in they component

Px=h/

Experimental illustration of Uncertainty Principle: Single slitdiffraction.

y y

x

y

y

sin =

for small sin tan

p ptan

p h /

pp h

h /

Since the electron can pass the slit

through anywhere over the width ,the uncertainty in the y position of theelectron is y=.

p y h

sin n


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which is in agreement with the uncertainty principle. If we try toimprove the accuracy of the position by decreasing the width of the slit,the diffraction pattern will be widened. This means that theuncertainty in momentum will increase.

The uncertainty principle is applicable to all material particles,from electrons to large bodies occurring in mechanics. In case of largebodies, however , the uncertainties are negligibly small compared tothe ordinary experimental errors.

y

p y h

H i b U t i t P i i l


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2

2 and:packetaveGaussian waFor

tExp

.1 then,andGiven xpkpxk

.1 then,andGiven tEEt

Particle is highly localized in space only if its momentum is undefined.

Particles energy is accurate only if measured for a long time.



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Example: Assume the position of an object is known soprecisely that the uncertainty in the position is onlyy=1.510-11 m. Determine the minimum uncertainty inthe momentum of the object and find the correspondingminimum uncertainty in the speed if the object in anelectron.

py=h/(4y)=(6.6310-34 Js)/(41.510-11 m)

py=3.510-24 kg m/s small

vy=py/m=(3.510

-24

kg m/s)/(9.110

-31

kg)

vy=3.9106 m/slarge

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quantum mechanics:uncertainty principle

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