prelude course structure quantum shock being photon

preludeCourse StructureQuantum Shock

Being PhotonThe Road to Planck’s Black Body Radiation

Quantum Mechanics - IThe Unmaking of Classical Physics

R I S H I K E S H V A I D Y [email protected]

Physics Department, BITS-Pilani, Pilani.

Rishikesh Vaidya QM-1

Anyone who is not shocked byquantum theory has notunderstood it.

Neils Bohr

The converse is of course nottrue. Don’t let your gradesprove to you.

Rishikesh Vaidya

Anyone who is not shocked byquantum theory has notunderstood it.

Neils Bohr

The converse is of course nottrue. Don’t let your gradesprove to you.

Rishikesh Vaidya

Outline

1 prelude

2 Course Structure

3 Quantum Shock

4 Being PhotonA twist in the spin

5 The Road to Planck’s Black Body RadiationSpecific Heat “feels the heat”Wein’s Law: Displacement in Despair

The place of quantum theory in overall scheme ofthings

Shaunak’s question to Angiras (Mundaka Upnishad 1.1.3)

Sir, what is it knowing which, everything else becomes known?

Shaunak’s question to Angiras (Mundaka Upnishad 1.1.3)

Sir, what is it knowing which, everything else becomes known?

I would like to view this question from a reductionistperspective. It admits the existence of elementary conceptualbuilding blocks to(from) which complex reality can bede(re)constructed. Quantum theory is then this science ofultimate reduction (as of today!).

Quantum Mechanics is different!

Classical physics is beautiful. It is –Fairly simple (taught in schools)Economical (governed by few simple laws)Explains most of the palpable phenomena that we can see(Mechanics, Optics, electrodynamics), hear (sAcoustics)and feel (thermodynamics)Confers to our classica reasoning and logic (this will besoon qualified and quantified!)

Relativity is surely crazy, only as long as you do notunderstand it. Makes immense sense once you get thehang of it.

Quantum Mechanics is truly bizarre!You cannot come to understand quantum mechanics in a wayyou make sense of classical physics and Relativity. It not onlyrefuses to answer our naively classical questions, it audaciouslyredefines what we can ask and what we cannot. Not only that,it challenges our naive classical reasoning and logic.

QM-1 is different as a course as well!

Newtonian Mechanics was, not surprisingly, developedsingle handedly by Newton. Its principles can be writtendown on a sheet paper and has far less ‘structure’ to it.

Analytical Mechanics was developed mainly D’Alembert,Lagrange, Hamilton, and Jacobi. Although it is equivalentto Newtonian mechanics, it has far more structure to it andhence is very powerful and versatile in its application.

The structural aspects of Quantum Mechanics cannot bedivorced from the structural aspects of classicalmechanics, electrodynamics, thermodynamics, andstatistical mechanics as it has to reproduce them inappropriate limits. Yet the whole (QM) is radically differentfrom the summation of the parts. It was developed over aperiod of 3 decades with fundamental contributions fromover 2 dozen physicists. The point is, you cannot doproblems in quantum mechanics without a deeperunderstanding of its structural aspects, unlike Newtonianmechanics which is a straight forward application of itsprinciples.

So what’s the fuss all about?

Quantum mechanics pulls the rug from under the mostcherished notions through which we make sense of thephysical reality around us. It trades

exclusive identity with wave-particle dualismdeterminism with uncertainty and probabilityobjectivity with subjectivity (caution !)continuum with discrete Quantum

Any one of this attribute in itself is bizzare enough. Put togetherit is outlandish, outragious, queer, and weird! How are wesupposed to make sense of all this?Or perhaps, mother nature may accurately describe us with justone adjective – pretentious!

The Course and the Evaluation

Books:Quantum Mechanics by Bransden and Joachain, PearsonQuantum Mechanics Vo. 1 Shin-itiro TomonagaQuantum Mechanics by Powell and Crasseman NarosaQuantum Mechanics by David Bohm Dover

The course structure in briefInevitability of the birth of QM ( 10 lectures, TomonagaChap. 1,2, Chap.1 BJ)Rudiments of Schrodinger’s wave mechanics ( 16 lectures,Chap.2,3 BJ)Applications to simple 1-dimensional systems ( 7 lectures,Chap.4 BJ)Rudiments of the formal structure of QM ( 7 Lecture, Sec.5.1-5.4, BJ)

Evaluation components: Tutorials (Best 10 from 13, 27 %),Midsem (30 %), Comprehensive ( 43 %)Rishikesh Vaidya QM-1

Waves OR Particles: Cracking the electron identity

Particle: Localized bundles of energy and momentum,described by q(t), q(t) (or q(t),p(t)), evolve according to 2nd

order equations of motion.Wave: A disturbance spread over space and time, described bya wave-function ψ(~r , t) which characterizes the disturbance atthe point ~r at a time t . Time evolution described by 2nd orderwave equation.

However, Waves are waves and particles are particles !But we need operational definitions that would stand againstthe results of some experiment.

A Double Slit Experiment with Bullets

Imagine a Drunkard spraying bullets with a Gun randomly in alldirections.(All images in this section from: http://www.upscale.utoronto.ca/PVB/Harrison/DoubleSlit/DoubleSlit.html)

Slit 1 open:What we canmeasure is theprobability P1 offinding a bullet adistance x fromthe center.

Slit 2 open:Now we measurethe probability P2of finding a bulleta distance x fromthe center.

both the slitsopen:We measure theprobabilityP12 = P1 + P2 offinding a bullet adistance x fromthe center.

Definition of Particle NatureA physical entity has a particle nature if its probabilitydistribution adds in the manner of bullets when it is subjected totwo slit experiment.

Ripple Tank Experiment to Demonstrate Wave Nature

Consider tapping the surface of a water filled tank and just asbefore we have two slits.

Slit 1 open:We can measure theIntensity distribution of waveenergy arriving at thebackstop I1 = |h1|2 at anydistance x from the center.

Slit 2 open:Now we measure theintensity I2 = |h|2 at anydistance x from the center.

Both the slits open:We measure the intensityI12 = |h1 + h2|2I12 = |h1|2 + |h2|2

+2|h1||h2|cosδat a distance x from thecenter.Notice theinterference pattern – it is thesignature of wave nature

Constructive InterferenceWaves are in phase

Destructive InterferenceWaves are out of phase

Definition of wave natureA physical entity has a wave nature if its intensity distributionshows interference pattern when it is subjected to two slitexperiment.

Cracking the Identity of Electrons:Two Slit Experiment with Electrons

Consider a “thought” experiment with electrons fired from anelectron gun and passing through two slits as before

Every TV houses an Electron Gun

“Everybody knows” electrons are particles. They after all havemass ∼ 10−30Kg. Lets crack the identity once and for all.

Probing the shady behavior of electrons

Flash animation (shown separately) from David Harrison(university of Toronto) to demonstrate that interference patternis produced even if we allow only one electron at a time.

Interpretation 1: May be the electrons passing throughthe two slits are conspiring to produce the interferencepattern.This does not seem to be the case as even when you allowone electron at a time, after waiting for sufficiently longtime, they still produce the interference patternEmploy Detectives: Track their path and observe theirshady behavior.

No interference pattern produced. This is very very strange andweird !

Double-slit: Not just a thought experimentTonomura at Hitachi labs actually observedinterference

Heisenberg’s Uncertainty:Position and Momentum are at Loggerhead

In order to pin down which hole the electron went through,we need to measure its position fairly accurately.To resolve the position accurately, we need to use a probe(light) of very small wavelength, i.e., high frequency andhence high energy. This would impart large momentum tothe electron and make its momentum very uncertain.To measure momentum fairly accurately we need to uselight of low frequency and hence low energy. Unfortunatelysuch a probe will have large wavelength and henceposition cannot be resolved accurately.

A twist in the spin

Light on light

Transverse EM waves.EM fields oscillate in XY plane if wave is travelling alongz-axis. Light is plane (linearly) polarized when ~E isoscillating in a fixed direction. ~E = Ex i + Ey j .Wave nature of light cannot explain photoelectric effect.Ejection of electrons from the metal surace when we shinex-rays, can be explained only when we assume that lightconsists of a stream of particles called photons.When plane polarized light is incident on a metal electronsare ejected in a preferred direction. Thus polarizationproperties are clearly connected with particle properties oflight and we must be a state of polarization to photons.

A twist in the spin

Light on light

A twist in the spin

Light on light

A twist in the spin

Light on light

A twist in the spin

Polaroid Filters: Sorting unpolarized light

Light from sun is unpolarizedA polaroid filter (such as tourmaline crystal) allows light ofonly certain fixed polarization (one that is normal to itsoptic axis)Let us obtain a polarized beam by passing unpolarizedlight through polaroid filter.If we now let this light beam incident on another tourmalinecrystal such that its polarization makes at an angle α to theoptic axis of crystal then a fraction sin2 α will go through.So far so good.When you think of this in terms of photon picture you are indeep trouble!

A twist in the spin

Being photon!

A plane polarized photon when confronted with tourmalinecrystal it is thorroughly confused

to pass through unscathedtotally submit and get absorbeddvide into half and let the half of him pass by and the otherhalf be absorbedJust toss a coin and decide

Someone please shed light on light!

A photon who sheds light on everything else finds himselftotally in dark.

A twist in the spin

Being photon!

A plane polarized photon when confronted with tourmalinecrystal it is thorroughly confused

to pass through unscathedtotally submit and get absorbeddvide into half and let the half of him pass by and the otherhalf be absorbedJust toss a coin and decide

Someone please shed light on light!

A photon who sheds light on everything else finds himselftotally in dark.

A twist in the spin

Outline

1 prelude

2 Course Structure

3 Quantum Shock

A twist in the spin

Breaking News!–Indiscrete spinning (h)-barred!

Charged electron orbiting the nucleus forms a current loopi.e., a tiny magnetic dipole µElectrons also possess spin and hence µsAin’t it electron’s discretion how fast to spin and where toorient itself.If a certain Mr. B (magnetic field) pops up,shouldn’t electron decide how cozy (what angle) it can getwith him?Alas, electron can only be a friend (parallel) or afoe(anti-parallel). Proof: When in the magnetic field B, theµs · B interaction splits the atomic energy levels.Experiment by Stern and Gerlach (1921) directlydemonstrated this beyond any doubt.

A twist in the spin

The Classic Stern-Gerlach Experiment

They sent a collimated beam of Ag atoms through theinhomogeneous magnetic field.

Riding on each of the Ag atom was an unpaired (47th) electronwhose spin provides the angular momentum s and hence amagnetic moment µ = −es

Interaction energy: −µ · BForce in the inhomogeneous B: Fz = − ∂

∂z (µ · B) = −µz∂Bz∂z

Ag atoms experience upward or downward force dependingupon their spin orientation.

A twist in the spin

∂z (µ · B) = −µz∂Bz∂z

A twist in the spin

∂z (µ · B) = −µz∂Bz∂z

A twist in the spin

Angular Momentum orientation is quantized !

For s = 1/2 particles, only two (2s + 1) projections about z-axisare allowed and not a continuum of values possibles.

A twist in the spin

Sequential Stern-Gerlach Experiment

A twist in the spin

No big deal!

You have blocked Sz−, so obviously you won’t find them in thesecond SG apparatus. It is elementary sir!

A twist in the spin

This is weird!From where on earth did you pull out the Sx+ and the Sx−components. Well, may be the 50% of Sz+ atoms from the firstSG apparatus are Sz+ and Sx+ and the remaining 50% are Sz+and Sx−. Sounds weird but looks like the only explanation. Notelementary anymore though!

A twist in the spin

Phew!Wasn’t the Sz− component gobbled up by the first apparatusalready? Is this Physics or paraphysics? Neither elementarynor satisfactory!

A twist in the spin

Welcome to QMech-1Nature unfortunately never confirmed to our intuition before shewrote her most beautiful songs. You just had your first brushwith the non-commuting observables. In QM you cannotdetermine Sx and Sz simultaneously. Measuring one willdestroy the other completely and beyond any memory.

A twist in the spin

A Quantum Paradigm Shift: Peeping Through theQuantum Veil

The Quantum Wonderland lies beyond the boundariesof classical objective reality, certainty, anddeterminism. It seems like nature is hiding her realityin a “mysterious and mind reading quantum veil” thatseems to know when you are watching and what youare wanting ! The quantum world sure redefines the“classical” task of Physics. Neils Bohr has famouslysaid, “The business of Physics is not to describe hownature is but what we can say about nature.”

A twist in the spin

Coming to Terms with the Quantum Shock:Reality Check!

What is reality after all? Isn’t our notion of reality builtfrom classical perceptions? Are we justified in ourunqualified extrapolations down to the bottom-mostlayer of nature? Isn’t the classical world “sensible”because the Quantum world is “Weired”? Just to putour weired expectation and extrapolations in place –given the details of marks of every student we cansure construct the average; but given the “average” wecan never construct the distribution. From thisperspective, the quantum shock was as much ourabsurd expectation of unqualified extrapolation asmuch it was nature’s mystery.

A twist in the spin

The agony irony and the ecstasy of being an electron

I am here I am thereWhen I am freeI am everywhereTry and find me (You peeping Toms)And I collapse anywhere.

When I am bang onAm momentously uncertainBut with a focused momentumAm all over for certain.

Try and cage meAnd I hit hard on the wallsThe smaller your cageHarder I hit.

A twist in the spin

Inside of a tiny prison h-barAm never bored and lonesomeAs flabbergast you watchAn army of electronsMerrily dancing and gigglingArms around the waistWith the hottest of positrons.

Bewildered and stung in disbelief you askWhat on earth is happeningIt sure takes two to tangoOh! but you can’t have too much funFlirting with a lonesome electronSqueezing it to size h-barWith c-through walls.

A twist in the spin

As you wonder and deliberateIf I am a reality-check on your fantasyOr a disillusioned physicist’s fantasyI embrace all uncertaintiesWith best of my potentialAt best a probability

Doomed in anonymityAm a waving particlePerhaps a particulate waveNeither Gods of any religionNor their best childrenCould settle on my coordinatesOr give me a decent name.

A twist in the spin

It is my agony and an ironyAm the stuff that stuffs all the stuffBut God only knows the stuff I amOr may beShe is clueless too!

Specific Heat “feels the heat”Wein’s Law: Displacement in Despair

Outline

1 prelude

2 Course Structure

3 Quantum Shock

An advice that may have “aborted” Quantum Physicsbefore conception!

Young man, replied Jolly. Why do you want to ruinyour life? The theoretical physics is practicallyfinished, the differential equations have all beensolved. All that is left now is to consider individualspecial cases involving variations of initial boundaryconditions. Is it worthwhile taking up a job which doesnot hold prospects for the future?

Philip Jolly, Max Planck’s teacher and mentor, onbeing consulted by Max Planck about career in

Physics

An advice that may have “aborted” Quantum Physicsbefore conception!

Young man, replied Jolly. Why do you want to ruinyour life? The theoretical physics is practicallyfinished, the differential equations have all beensolved. All that is left now is to consider individualspecial cases involving variations of initial boundaryconditions. Is it worthwhile taking up a job which doesnot hold prospects for the future?

Philip Jolly, Max Planck’s teacher and mentor, onbeing consulted by Max Planck about career in

Physics

A Hike to the Summit

Instead of the customary helicopter areal survey, we shall takea slightly more detailed climb to the summit of Planck’s formulafor the following reasons:

All your future courses shall only do a lip service, since ittruly falls in the domain of “modern physics“.It is not sufficiently modern for typical 21st century modernphysics courses and hence hurriedly gobbled up.To me however it truly demonstrates unity and synthesis ofvarious branches of physics like mechanics,electrodynamics, thermodynamics and statistical physics.Nature knows no compartments. What is more it is abrilliant example of how the whole sometimes is just notthe sum of the parts – an uneasy calm before the storm.

The Problem of Blackbody Radiation (BBR)

Blackbody: An idealized physical body that absorbs (anddoes not reflect) all the incident radiation regardless of thefrequency and the angle. In thermal equilibrium it also emitslight in all frequency and in all directions (isotropic).

The Problem of Blackbody Radiation (BBR)

Blackbody: An idealized physical body that absorbs (anddoes not reflect) all the incident radiation regardless of thefrequency and the angle. In thermal equilibrium it also emitslight in all frequency and in all directions (isotropic).

image source:hephaestusaudio.com

The ProblemTheoretically explain how theenergy density of the radiation isdistributed in different frequenciesat different temperature.

We will first try and understand adifferent problem to appreciate thisproblem and its resolution.

A Different Problem: Apportioning of Total Energy

Imagine an object immersed in a thermal reservoir is in a stateof thermal equilibrium.

Though there is no apparent thermal exchange between theobject and the reservoir, the actual energy of the molecules ofthe object fluctuate around some average value.

Using Boltzman’s statistical mechanics, we can say somethingabout this fluctuations, i.e., what is the distribution of molecularspeeds.That is, how is the total energy of the object partitionedamongst its constituent molecules and atoms

More Precisely

The state of a dynamical system with f DOF is specified bygiving the coordinates q1,q2, ...qf and corresponding momentap1,p2, ...pf . All these, as well as the energyE(q1,q2, ...qf ,p1,p2, ...pf ) of the system varies with time (sinceit is exchanging energy with the reservoir at microscopic level).

What is the probability for q1,q2, ... to take values between q1and q1 + dq1, q2 and q2 + dq2, ...and, at the same time for thep1,p2 ...to take some values between p1 and p1 + dp1, p2 andp2 + dP2, ... respectively?

More Precisely

Boltzman’s answer:

P(q1q2...qf p1p2...pf )dq1dq2...dqf dp1dp2...dpf

= A exp(− 1

KTE(q1q2...qf p1p2...pf )

)dq1dq2...dqf dp1dp2...dpf

A =1∫ ∫

...∫

exp(− 1

KT E(q1q2...qf p1p2...pf ))

dq1...dqf dp1...dpf

An Important Conclusion

As long as KE of each DOF is proportional to square ofmomentum, i.e.,:

Ekin = α1p21 + α2p2

2 + ...+ αf p2f

Law of Equipartition of Energy

Average value of the kinetic energy per degree of freedom inthe reservoir at temperature T is:

< αsp2s >=

An Important Conclusion

As long as KE of each DOF is proportional to square ofmomentum, i.e.,:

Ekin = α1p21 + α2p2

2 + ...+ αf p2f

Law of Equipartition of Energy

Average value of the kinetic energy per degree of freedom inthe reservoir at temperature T is:

< αsp2s >=

Energy of Unit Mole (N = 6.02× 1023) of a Substance

Ideal Monatomic Gas

U = Ekin = ΣNn=1

(p2nx + p2

ny + p2nz)

kT (3N ∗ Average Value per DOF)

RT (R = Nk = 1.98 cal/oK : Gas constant)

Specific Heat: The necessary heat C to raise the temperatureby 1o is:

R = 2.97 cal/oK

Ideal Diatomic Gas: 5 degrees of freedom (3 trans. + 2 rot.)

R = 4.95 cal/oK

Crystalline SubstanceAtoms on lattice perform random small oscillations.Any arbitrary mode of oscillation consists of superpositionof normal modes of oscillations.No. of normal modes = No. of DOF.Energy of the sth normal mode is: Es = αsp2

s + βsq2s .

Law of Equipartition holds: < αsp2s >=< βsq2

s >= 12kT .

Crystalline SubstanceSo for f normal modes:

< E > = fkTU = 3NkT = 3RT f = 3NC = 3R = 5.85 cal/oK

Specific Heat “feels the heat”!

source: Quantum Mechanics by Tomonaga

Molar specific heat for hydrogen.

Molar specific heat for lead(Pb).

Cracks Appear: Fall of the Law of Equipartition !The figures amply show that agreement is great atcomparatively high temperature but theory fails badly at lowtemperatures. This could mean that at low temperatures notevery degree of freedom receives enough energy to satisfy thelaw of equipartition of energy.

What does it cost to heat “Vacuum”!

At low temperatures energy is not equitably sharedamongst all the DOF of a system.This is even more striking in case of “vacuum”!Let us immerse “vacuum” in our reservoir; that is a hollowcavity surrounded by walls of temperature T .Despite being hollow, the cavity contains randomoscillations of electromagnetic fields and hence it hassome thermal energy. We can then consider specific heatof “vacuum” itself.Any arbitrary mode of oscillation can be written as asuperposition of normal modes. Thus the EM fields insidethe cavity can be considered to be a collectionindependent harmonic oscillators.

Whatever happened to Stefan’s law (U = σT 4)!

Mathematically, we can describe the state of EM field in termsof normal coordinates q1,q2,q3, .. and correspondingp1,p2,p3, .... The energy of a state is given as a sum of termsEs = αsp2

s + βsq2s .

We can then apportion the total energy amongst each of thesef DOF using Law of equipartition. Thus E = fkT .

But how many DOF (normal modes) does a (continuum)system of EM fields has? Infinite! A cavity contains infinitelymany normal modes up to arbitrary small wavelengths.

Since f =∞, E =∞ and hence Specific heat of “vacuum” is∞too! Hollow cavity will endlessly “drink heat” from the reservoir,feeding into normal modes of shorter and shorter wavelengths.Rishikesh Vaidya QM-1

s + βsq2s .

Counting the “Caged” Normal Modes

Figure source: Quantum Mechanics by Tomonaga (vol.1)

Assignment 1(a): Consider a string of length L fixed at twoends. If the velocity of elastic waves on this string is c, showthat the number of normal modes contained in frequency rangebetween ν and ν + dν is

Z (ν)dν =2Lc

Assignment 1(b):Following the same method as above, 3-Doscillations in a cube of side L can be specified by sets of threepositive integers, sx , sy , s and sz . Show that for the case of EMwaves number of oscillations with frequencies between ν andν + dν is give by:

Z (ν)dν =8πL3

c3 ν2dν

A “hole” in the argument!

Boundary conditions: We assumed the ends of strings rigidlytied at the ends. This implies that string exchanges no energywith the surroundings. Analogously we must assume perfectlyreflecting mirrors of the cavity wherein our EM fields are caged.How do we achieve thermal equilibrium then?

We drill a small hole in the cavity that is big enough to allowthermal equilibrium via exchange with reservoir but smallenough to not disturb our calculation of number of normalmodes.

Following equipartition we obtain total energyE(ν)dν = Z (ν)dν × kT

The First Base Camp

Rayleigh-Jeans Formula

E(ν)dν = Z (ν)dν × kT

=8πL3

c3 ν2dνkT

U(ν)dν =8πkT

c3 ν2dν

Although we have considered EM waves caged in cube of sideL, it is applicable more generally, without regard to the natureand shape of walls. Integration over ν diverges as expectedfrom the law of equipartition of energy. However, for smallfrequency Rayleigh-Jeans (RJ) formula does accuratelydescribe the black body curve.Rishikesh Vaidya QM-1

Reflections for the Climb Ahead

The agreement of RJ formula extends to higherfrequencies for higher temperatures.According RJ formula the spectral distribution of light isindependent of temperature but the total intensity isproportional to it. This implies that color of the light fromthe cavity does not depend on the temperature. This iscontrary to experience.Although energy equipartition does not hold in general butit still accounts for apportioning of energy to the lowfrequencies oscillations, widening its application withincreasing temperature.This could mean that with decreasing temperature,degrees of freedom corresponding to large frequenciesstart violating the equipartition law and “die”; i.e., energy isnot duly distributed to these oscillations.Rishikesh Vaidya QM-1

Outline

1 prelude

2 Course Structure

3 Quantum Shock

Notion of Adiabatic Invariants

Adiabatic InvariantsAdiabatic Invariants are approximate constants of motion(Invariants) of a given dynamical system, which are preserved(constant) during a process of very slow change of system’sparameters.

Examples:(a) An oscillating pendulum whose string is gradually shortenedor (b) radiation in a hollow cavity sealed from the reservoir andwhose walls are adiabatically compressed. The frequency νand amplitude (and hence energy) of the oscillations willnaturally change, however, the ratio E/ν remains constantduring slow deformation.

Notion of Adiabatic Invariants

Adiabatic InvariantsAdiabatic Invariants are approximate constants of motion(Invariants) of a given dynamical system, which are preserved(constant) during a process of very slow change of system’sparameters.

Examples:(a) An oscillating pendulum whose string is gradually shortenedor (b) radiation in a hollow cavity sealed from the reservoir andwhose walls are adiabatically compressed. The frequency νand amplitude (and hence energy) of the oscillations willnaturally change, however, the ratio E/ν remains constantduring slow deformation.

An Example from MEOW!

Image source: QuantumMechanics by Tomonaga

Problem: Consider an oscillatingpendulum as shown in figure.When the string is being pulledvery slowly (on a time scale muchlarger than the period T ), showthat average tension< T >= mg + 1

4mgθ20. Also show

that E/ν is constant.

stop here

Quiz-2

A. (a) A collimated beam of Ag atoms from the owen is passedthrough SG-x . Draw and label the output from SG-x . (b) Each ofthe output from the previous experiment is subjected to a SG-z.Label the output. (c) Any one of the output (your choice) frompart(b) is subjected to a SG-y . Label the output. (d) What canyou conclude from above observations (one sentence).

B. (a) What is the property of a polarized material? (b) A beam oflight is incident at an angle of 45o to the optic axis of a polaroidmaterial. What fraction of light is transmitted? (c) Output beamin part (a) is fed into another polaroid material whose optic axisis at an angle α to the optic axis in part(a). What fraction of lightis seen in the output. (d) What can you say when a singlephoton was made to incident in part (a).

prelude course structure quantum shock being photon

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