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The Blind Men and the Quantum:The Blind Men and the Quantum:A Quick Tour To Quantum WorldA Quick Tour To Quantum WorldThe Blind Men and the Quantum:The Blind Men and the Quantum:A Quick Tour To Quantum WorldA Quick Tour To Quantum World

K.P.SATHEESHK.P.SATHEESHPrincipal,

Govt. College

Ambalapuzha

Lecture @ S.H.College, Thevara 17-2-2005

QuantumMechanics

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The Blind MenThe Blind Menand the Elephantand the Elephant

by John Godfrey Saxe (1816by John Godfrey Saxe (1816--1887)1887)

The Blind MenThe Blind Menand the Elephantand the Elephant

by John Godfrey Saxe (1816by John Godfrey Saxe (1816--1887)1887)

It was six men of Indostan, To learning much inclined, Who went to see the Elephant,

(Though all of them were blind), That each by observation, Might satisfy his mind. .

The First approached the Elephant, And happening to fall, Against his broad and sturdy side, At once began to bawl:

God bless me! but the Elephant, Is very like a wall!

The Second, feeling of the tusk, Cried, Ho! what have we here, So very round and smooth and sharp? To me tis mighty clear,

This wonder of an Elephant, Is very like aspear!

The Third approached the animal, And happening to take, The squirming trunk within his hands, Thus boldly up and spake:

I see, quoth he, the Elephant, Is very like asnake!

The Fourth reached out an eager hand, And felt about the knee. What most this wondrous beast is like, Is mighty plain, quoth he;

Tis clear enough the Elephant, Is very like a tree!

The Fifth, who chanced to touch the ear, Said: Een the blindest man, Can tell what this resembles most; Deny the fact who can,

This marvel of an Elephant, Is very like afan!

The Sixth no sooner had begun, About the beast to grope, Than, seizing on the swinging tail, That fell within his scope,

I see, quoth he, the Elephant, Is very like a rope!

And so these men of Indostan, Disputed loud and long, Each in his own opinion, Exceeding stiff and strong,

Though each was partly in the right, And allwere in the wrong!

Moral: So oft in theologic wars, The disputants, I ween, Rail on in utter ignorance, Of what each other mean,

And prate about an Elephant, Not one of them has seen!

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What is Quantum Mechanics?What is Quantum Mechanics?

Quantum mechanics is a theory. It is ourcurrent standard model for describingthe behavior of matter and energy at thesmallest scales (photons, atoms, nuclei,quarks, gluons, leptons, ).

Like all theories, it consists of amathematical formalism, plus aninterpretation of that formalism.

However, quantum mechanics differs from other physical theoriesbecause, while its formalism of has been accepted and used for 80years, its interpretation remains a matter of controversy and debate.Like the opinions of the 6 blind men, there are many rivalQMinterpretations on the market.

QuantumMechanics

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Classical Particles

Classical Particles Newtonian Mechanics.

It answers questions, such as: What is the trajectory of a satellite around the earth? If a missile is fired, where will it land? If a billiard ball is hit at a certain angle, what will the outcomebe?

Description of a classical particle:Mass (m), charge (q), position (r(t)), velocity (v(t))

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Given the force, F, equation of motion:

Fdt

trdm

TT

!22 )(

If force and initial position/velocity are given, the positionand velocity at anytime in the future can be predicted withcomplete certainty.

Physical observables of a particle, such as energy, variescontinuously.

m

pKE

prl

vmp

2

2

!

v!

!

:energyKinetic

:momentumAngular

:MomentumTT

TT

(1.1)

(1.2)

(1.3)

(1.4

)

Classical Particles

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Classical Wave Phenomena

Examples of waves: Ocean waves, sound waves, waves inviolin strings, electromagnetic waves (light),

Description of waves: Amplitude, velocity, frequency,

wavelength , but not mass and position as for particles.

Wave Equation:x

x

x

x!

2

2 2

2

2

10

] ]( , ) ( , )x tx v

x t

t

amplitude

velocity

A general solution of the Equation:

] [ [( , ) exp[ ( )] exp[ ( )]x t A i k x t B i k x t!

(3.5)

(3.6)

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where A and B are amplitude of the waves, k is calledwavenumber and w the angular frequency.

Wavelength:

Frequency: f =

Velocity: v =k

PT

[T

[P

!

!

2

2

k

f

-8

-6

-4

-2

0

2

4

6

8

0 0.01 0.02 0.03 0.04 0.05

A

l

(1.7)

(1.8

)

(1.9)

Classical Wave Phenomena (contd)

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Wave Interference:

When monochromatic light passes through two slits, a striped

interference pattern is produced.

constructiveinterference

destructiveinterference


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Wave Diffraction:

(a) Light casts a sharp shadow when the opening is large compared tothe wavelength of the light. (b) Diffraction is apparent and theshadow is fuzzy when the opening is small. (c) Intensity of diffracted

light through a thin slit.

Plane waves passes through openings of various sizes. Thesmaller the opening, the greater the bending of the waves atthe edges.

c


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Wave Groups and Dispersion:

] [( , ) exp[ ( )]x t A i k x t

!

Velocity: v =k

p [P! fPhase

Plane wave:

Wave Packet:Or wave group

Group velocity: v =dk

gd[

What is the velocity, vg???

(1.10)


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Construction of a wave packet:

= +

+ +

=

A ik x1 1exp( )

A ik x2 2exp( )A ik x

3 3exp( )

FourierFourier

transformtransform(3.11)

g

g

! dktkxikAtxf )(exp[)(),( [

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Wave dispersion:

If thenthephase volecity depends on k (wavelength)

and the medium is called dispersive.

ddk

22 0[ { ,

In a dispersive medium, individual plane waves travel with

different phase velocities and the shape of the wave packetchanges.

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The Origins of the Quantum Theory

Emax

o Data

Slope = h

o Data

Theory

Planck (1900)

Einstein(1905)

Bohr(1911)

E = h f

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What is quantum mechanics?

It is a framework for the development of physical theories.

It is not a complete physical theory in its own right.

Quantumelectrodynamics (QED)

Operating system

Applications software

Quantum mechanics

Specific rules

Newtons laws of motion

Newtonian gravitation

QM consists of four mathematical postulates which lay theground rules for our description of the world.

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How successful is quantum mechanics?

It is unbelievably successful.

No deviations from quantum mechanics are known

Most physicists believe that any theory of everythingwill be a quantum mechanical theory

Not just for the small stuff!

QM crucial to explain why stars shine, how the Universeformed, and the stability of matter.

A conceptual issue, the so-calledmeasurement problem, remainsto be clarified.

Attempts to describe gravitationin the framework of quantummechanics have (so far) failed.

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Timeline - Modern Physics

Modern Physics was a sudden revolutionstarting around 1900, and ending ????

Einstein

200019501900

Michelson

Planck

ThomsonRutherford

Bohr

SpecialRelativity

GeneralRelativity

QuantumMechanics

De BroglieSchrodingerHeisenberg

TransistorInvented

All theQuarks

discovered

LaserInvented

Nuclear EnergyReleased

Expansionof Universediscovered

Neutron Starsdiscovered

Start ofQuantumMechanics

Curie

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General Comment Quote from the famous modern physicist,

Richard Feynman:

If we were able to pass along only one bit of scientificknowledge to future generations, what would be the mostimportant one piece of information to choose?

Feynmans answer: That matter is made of atoms

What could this mean? How could this fact be so important?

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The Appeal of Atomism It is natural to try to explain the vast diversities that

we see in terms of the arrangements andinteractions of a small number of fundamentalbuilding blocks: atoms!

Atomism in Ancient Greece: Democritus: There are only atoms and the void. Apparent

qualities are result of shape, arrangement, and position of atoms.Atoms remain unaltered.Gave us the name: atom - indivisible

Explains the basic properties of matter

Changes but is never created nor destroyed (in our ordinaryexperience)

Solid: Atoms linked together

Liquid: Atoms flowing around each other

Gas: expands to fill any container because the atoms are inmotion

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The Periodic Table

Question: Do the properties of atoms (elements)

indicate that there are more than 100 differentflavors of these fundamental pieces? Or do the properties indicate a pattern of substructure??

Atomic # = # of protons

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Are Atoms Indivisible?

Marie Curie (1897) discovers immense radiation energy fromelement she named Radium.

Surprising? Yes!

If the radiation comes from the atom, it could indicate that anatom had been transformed into another kind of atom!

If atoms are not immutable, then it makes sense to ask whatare atoms made of?

X-rays discovered in 1895 byRoentgen - World Wide sensation!

Unknown ray produced from electricdischarge that penetrates matter!

J.J. Thomson discovers the electronin 1897.

Henri Becquerel (1896) tries to

produceX

-rays from naturalsources.

Finds radiation (less penetrating than X-rays) given off from ore containingUranium.

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Atomic Models Conclusion: Atoms contain electrons.

Questions: How are they arranged? Since atoms

are neutral, where is the positive charge? Two models:

Plum pudding: Electrons are embedded in continuum ofpositive electricity like plums in a pudding.

Planetary model: Electrons orbit a small nucleus of positivecharge like planets orbit the Sun.

Electrons

Positive Charge

Or

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Atomic Models How to distinguish between these models?

Ernest Rutherford had discovered that certain rays

given off by radioactive material were E rays -positive particles (ions) with the mass He atom.

Used to study the atoms itself! Observe how Eparticles (Helium ions) scatter from a Gold foil.

U

Au

Ev

Count the number of times an Eparticle scatters through an angle

U, for different angles U.

What do you expect? Plum Pudding: only small deflections since E particles muchheavier than electrons. Planetary: can occasionally get large deflections if most of themass of the atom resides in the nucleus.

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Scattering Experiments

probe

probeFor plum pudding: expectonly small angle scattering.

For planetary model: may

see small angle or largeangle scattering.

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Scattering Experiments

probe For plum pudding: expectonly small angle scattering.

probe

For planetary model: may

see small angle orlargeangle scattering.

Rutherford sawlarge anglesplanetary model!

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The Problem of the atom Experiments supported the picture that an atom is

composed oflight electrons around a heavy

nucleus

Problem: if the electrons orbit the nucleus, classicalphysics predicts they should emit electromagneticwaves and loose energy.

If this happens, the electronswill spiral into the nucleus!

The atom would not be stable!

What is the solution tothis problem?

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Bohrs Revolutionary Idea

Can the new quantumtheory explain the stabilityof the atom?

If the energies can take ononly certain discrete

values, i.e., it is quantized,there would be a lowestenergy orbit, and theelectron is not allowed tofall to a lower energy!

What is the role ofPlancks Constant h?

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Plancks Constant h and the atom Bohr (and others) noted that the combination

a0 = (h/2T)2/ me2

has the units oflength about the size of atoms

Bohr postulated that it was not the atom thatdetermined h, but h that determined the properties

of atoms!

Since the electron is bound to the nucleus byelectrical forces, classical physics says that the

energy should beE = - (1/2) e2/a0

If the radii are restricted to certain values, the the

energy can only have certain values

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The allowed orbits are labeled by theintegers: n = 1, 2, 3, 4.

The radii of these orbits can bedetermined from the quantizationcondition:radius = n2 a0 = n

2 (h/2T)2/ me2

The energy can only have the valuesEn= E1/n

2, E1 = - (1/2)(e2/ a0)/n

2

The spectra are the result of

transitions between these orbits,with a single photon (f= E/h)carrying off the difference in energyE between the two orbits.

The Bohr Atom (NOT Correct in detail!)

1

23

4

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Ideas agree with Experiment Bohrs picture:

The only stable orbits of the electrons occur at definiteradii.

When in these orbits, contrary to classical E&M, theelectrons do not radiate.

The radiation we see corresponds to electrons moving from

one stable orbit to another. Experiments (already known before 1912)

Experiment: Balmer had previously noticed a regularity inthe frequencies emitted from hydrogen:

f= f0

( (1/n2) - (1/m2)) where n and m are integers.

Bohrs Theory: Fits exactly using the value of h determinedfrom other experimentsPhoton carries energy (=hf) = difference of stable orbits.

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Hydrogen Spectrum: Balmer series

Balmer Formula:f= f0 ( (1/n2) - (1/m2))32.91 ( 1/4 - 1/9 ) = 4.571

32.91 ( 1/4 - 1/16 ) = 6.171

32.91 ( 1/4 - 1/25 ) = 6.911

32.91 ( 1/4 - 1/36 ) = 7.313

32.91 ( 1/4 - 1/49 ) = 7.556

frequency (1014 Hz)

4.571

6.171

6.912

7.314

7.557

IT WORKS!

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Demonstration:Spectra of different atoms

frequency (1014 Hz)

4.571

6.171

6.912

7.314

7.557

Observe spectra of different gases Individual grating for each student

Using interference - wave nature of light - to separate thedifferent frequencies (colors)

Hydrogen

Neon - strong line in Red

Sodium - strong line in yellow (street lights)

Mercury - strong lines in red, blue (street lights)

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Blackbody Radiation The true beginnings of the quantum theory lie in a strange

place: the frequency spectrum emitted by a solid when it isheated (blackbody radiation).

Experimental measurements: the frequency spectrum waswell determined.. a continuous spectrum with a shape thatdepended only on the temperature (light bulb, )

Theoretical prediction: Classical kinetic theory predicts the

energy radiated to increase as the square of the frequency(Completely Wrong! - ultraviolet catastrophe).

frequency

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Plancks Solution Max Planck (1901): In order to describe the data Planck made

the bold assumption that light is emitted in packets orquanta,each with energy

E = h f, where f is the frequency of the light. Some texts use the notation R for frequency.

The factorh is now calledPlancks constant, h = 6.626 (10-27) erg-sec.

o DataTheory

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E =h f

The two most important formulas in modern physics

E = mc2

(Einstein special relativity - 1905)E = h f (Planck quantum mechanics - 1901)

Planck initially called his theory an act of

desperation. I knew that the problem is of fundamental significance forphysics; I knew the formula that reproduces the energydistribution in the normal spectrum; a theoretical interpretationhad to be found , no matter how high.

Leads to the consequence that light comes only incertain packets orquanta

A complete break with classical physics where allphysical quantities are always continuous

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Photoelectric Effect Einstein took Plancks hypothesis seriously

in order to explain the photoelectric effect.

Effect: Shining light on a metal canliberate electrons from its surface.

Experimental facts:

Easy for UV light (high frequency) hard

for red light (low freq).

Energy of the electrons depends on frequency of light

Increasing intensity of light increases numberof electronsemitted, but not the energy of each electron

Cant be explained by

wave behavior oflight.

If light is generated in quantized units,Einstein reasoned it would also arrive withquantized amounts of energy

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Photoelectric Effect: The Theory Einsteins explanation: Suppose the energy in the

light is concentrated in particle-like objects (now wecall them photons) whose energy depend on thefrequency of the light according to Plancksequation: E = hR.

Prediction:Maximum energy of electrons liberated =

energy of photon - binding energy of electron.Emax = hf - hf0

Experiment: done accurately by Millikan in 1916:

Emax

o Data

Slope = h

Frequency f

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Light is Quantized! We referred to light as a wave.

We did experiments to show that light behaves like a wave.

Recall: Waves continuously transmit energy, they do not transmit

matter.

Blackbody radiation and the photoelectric effectindicate that the energy transmitted by light comesin packets!!

Light doesnt behave like a wave.

The energy light carries is quantized, which means it comes intiny bursts. The amount of energy per burst is determined bythe frequency and Plancks constant h:

E=hf

Light can behave like a particle. Any chance aparticle can behave like a wave?

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The Two-Slit Experiment We will first examine an experiment which Richard

Feynman says contains all of the mystery of

quantum mechanics. The general layout of the experiment consists of a

source, two-slits, and a detector as shown below;

source detector

x

The idea is to investigate three different sources (a classical particle(bullets), a classical wave (water), and a quantum object (electron or

photon)). We will study the spatial distribution (x) of the objects which

arrive at the detector after passing through the slits.

slits

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Classical Particles Classical particles are emitted at the source and arrive at the

detector only if they pass through one of the slits.

Key features: particles arrive in lumps. ie the energy deposited at the

detector is not continuous, but discrete. The number of particlesarriving per second can be counted.

The number which arrive per second at a particular point (x) withboth slits open (N

12

) is just the sum of the number which arriveper second when only the top slit is opened (N1) and the numberwhich arrive per second when only the bottom slit is opened (N2).

x

N

only bottomslit open

only topslit open

x

N

Both slitsopen

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Classical Waves Classical waves are emitted at the source and arrive at the

detector only if they pass through the slits.

Key features: detector measures the energy carried by the waves. eg for water waves,

the energy at the detector is proportional to the square of the height of thewave there. The energy is measured continuously.

The energy of the wave at a particular point (x) with both slits open (I12) isNOTjust the sum of the energy of the wave when only the top slit is

opened (I1) and the energy of the wave when only the bottom slit isopened (I2). An interference pattern is seen, formed by the superpositionof the piece of the wave which passes through the top slit with the pieceof the wave which passes through the bottom slit.

x

I

only bottom slit openonly top slit open

x

I

Both slits open

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Quantum Mechanics

Particles act like waves!

Experiment shows that particles (likeelectrons) also act like waves!

x

I

only bottom slit openonly top slit open

x

I

Both slits open

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The de Broglie Wavelength Big question: How can we quantify deBroglies

hypothesis that matter can sometimes be viewed as

waves? What is the wavelength of an electron?

de Broglies idea: define wavelength of electron sothat same formula works for light also, when

expressed in terms of momentum! What is momentum of photon? This is known from relativity:

p = E / c (plausible since: E = mc2 and p = mc E = pc)

How is momentum of photon related to its wavelength?

from photoelectric effect: E = hf pc = hf

change frequency to wavelength: c =Pf c/f=P

p P= h P= h / p

P i l A Lik W !

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Particles Act Like Waves!

P = h / p

SchrodingersEquation

De Broglies

Matter Waves

T d U d di

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Towards Understanding

Bohr atom Quantized energy levels, allowed orbits

deBroglie waves Particle acts like wave, wavelength depends upon momentum

Obviously related, but unclear exactly how

Erwin Schroedinger pulled it all together in 1926

Th S h di E i

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The Schrodinger Equation In 1926 Erwin Schrodinger proposed an equation

which describes completely the time evolution of the

matter wave =( - (h2/ 2m) 2 + V) =!i h (d=/dt)

where m = characteristic mass ofparticleV = potential energy function to describe the forces

Newton: SchrodingerGiven the force, find motion Given potential, find wave

F = ma = m (d2x/dt2) (- (h2/ 2m) 2 + V) =!i h (d=/dt)

solution: x = f(t) solution: = = f(x,t)

Note: Schrodingers equation is more difficult to solve, but it isjust as well-defined as Newtons. If you know the forces acting,you can calculate the potential energy V and solve the

Schrodinger equation to find =.

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Key Results of Schrodinger Eq.

The energy is quantized

Only certain energies are allowed Agrees with Bohrs Idea in general

Predicts the spectral lines of Hydrogen exactly

Applies to many different problems - still one ofthe key equations of physics!

The wavefunction is spread out

Very different from Bohrs idea

The electron wavefunction is not at a given radiusbut is spread over a a range of radii.

Wh t i = #

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What is = # Our current view was fully developed by Bohr from

an initial idea from Max Born.

Borns idea: = is a probability amplitude wave!= tells us the probability of finding the particle at agiven place at a given time.

Leads to indeterminancy in the fundamental laws ofnature goodbye Newtonian worldview!

Uncertainty principles

Not just a lack of ability to measure a property - but a

fundamental impossibility to know some things

Einstein doesnt like it:

The theory accomplishes a lot, but it does not bring uscloser to the secrets of the Old One. In any case, I am

convinced that He does not play dice.

P b bilit i t t ti f =

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Probability interpretation for = The location of an electron is not determined by =.

The probability of finding it is high where = is large,and small where = is small.

Example: A hydrogen atom is one electron around anucleus. Positions where one might find theelectron doing repeated experiments:

Nucleus

Higher probability

to find electronnear nucleus

Lower probability

to find electronfar from nucleus

S

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Summar

y Near the turn of the 20th century, a second revolution was inthe works.

Experiments were probing very small distance scales, learning aboutelectrons, atoms, nuclei

Max Planck (1900) had the idea that blackbody radiation couldbe explained if light was emitted in quanta with E=hf

Einstein (1905) reasoned that this would also explain the

photoelectric effect (light transfers quanta of energy to emittedelectrons)

Light can behave like a particle!

deBroglie (1923) proposed that matter could behave as a wave

Scattering experiments showed this to be true!

The quantum theory is born.

Nature is not continuous as Newton thought.

It is discrete. Energy comes in packets.

This explains how atoms behave as well

Th U t i t P i i l

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Werner Heisenberg proposed that the basic ideas onquantum mechanics could be understood in termsof an Uncertainty Principle

The Uncertainty Principle

where (p and (x refer to the

uncertainties in the measurement ofmomentum and position.

Similar ideas lead to uncertainty in time and energy

(p (x u h/2T = h/2

The constant h-bar has the approximate value

h = 10 -34 Joule seconds

(E (t u (1/2) h/2T = (1/2) h

U t i t P i i l d M tt W

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The uncertainty principle can be understood from

the idea of de Broglie that particles also have wavecharacter

What are properties of waves

Waves are patterns that vary in space and time

A wave is not in only one place at a give time - it is

spreadout

Example of wave with well-defined wavelength P andmomentum p = h/ P, but is spread over all space, i.e.,

its position is not well-defined

Uncertainty Principle and Matter Waves

P

-1.2

-0.8

-0.4

0

0.4

0.8

B

1.2

-20 80 180 280 380

A

-1.2

-0.8

-0.4

0

0.4

0.8

B

1.2

-20 80 180 280 380

A

Th N t f W ti d

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Example of wave with well-defined position in space

but its wavelengthP

and momentum p = h/P

is notwell-defined , i.e., the wave does not correspond to adefinite momentum or wavelength.

The Nature of a Wave - continued

0

Position x

Most probable position

Q ant m T nneling

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Quantum Tunneling In classical mechanics an object can never get over a barrier

(e.g. a hill) if if does not have enough energy

In quantum mechanics there is some probability for theobject to tunnel through the hill!

The particle below has energy less than the energy neededto get over the barrier

Ene

rgy

tunneling

Example of Quantum Tunneling

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Example ofQuantum Tunneling The decay of a nucleus is the escape of particles bound

inside a barrier

The rate for escape can be very small. Particles in the nucleus attempt to escape

1020 times per second, but may succeed in escaping onlyonce in many years!

RadioactiveDecay

Ene

rgy

tunneling

Example of Probability Intrinsic to

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Example of Probability Intrinsic toQuantum Mechanics

Even if the quantum state (wavefunction) of the nucleus iscompletely well-defined with no uncertainty, one cannotpredict when a nucleus will decay.

Quantum mechanics tells us only the probability per unittime that any nucleus will decay.

Demonstration with Geiger Counter

RadioactiveDecay

Energy

tunneling

Heisenbergs Uncertainty Principle

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Heisenbergs Uncertainty Principleinvolving energy and time

If our measurement lasts a certain time (t, thenwe cannot know the energy better than anuncertainty (E

Imagine the Roller Coaster

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Normally, the car can only get as far as C,before it falls back again

But a fluctuation in energy could get it over

the barrier to E!

Imagine the Roller Coaster ...

Quantum Tunnelling

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Quantum Tunnelling

A particle borrows an energy (Eto get over abarrier

Does not violate the uncertainty principle,provided this energy is repaid within a certaintime (t

The taller the barrier, the less likely tunnellingwould occur

Example of Quantum Tunnelling:

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Example ofQuantum Tunnelling:Radioactivity

Concept of Half Life

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Concept of Half-Life

13N has a half-life of 10 min

Consider a sample of13N

After 10 min, half of the 13N atoms would have decayed and halfwould not have decayed

After another 10 min, half of the remaining13N atoms wouldhave decayed and half would not

Probabilistic process: can never predict exactlywhena given atom would decay

Applications of Quantum Tunnelling

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Applications ofQuantum Tunnelling

Scanning tunnelling microscope

Tunnel diode

Josephson junction

Scanning Tunnelling Microscope

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Scanning Tunnelling Microscope

Tungsten STM tip(photo taken with an SEM)

Iron Atoms on Copper

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Iron Atoms on Copper

35 Xenon Atoms on Nickel

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35 Xenon Atoms on Nickel

Discourse on Quantum Weirdness

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Discourse on Quantum Weirdness

Einsteins moon

Schrdingers cat

EPR paradox

Paradox 1 (non-locality):Paradox 1 (non-locality):

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Paradox 1 (non locality):Einsteins Bubble


Situation: Aphotonis emittedfrom anisotropic source.


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Situation: Aphotonis emittedfrom anisotropic source.Its sphericalwave function=

expands like aninflating bubble.


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Question(AlbertEinstein):If a photon is detected at Detector A, how does the photonswave function = at the location of Detectors B & C know that itshould vanish?

Situation: Aphotonis emittedfrom anisotropic source.Its sphericalwave function=

expands like aninflating bubble.It reaches a detector, and the =bubble pops and disappears.


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Itis as ifonethrows a beer bottleintoBostonHarbor. It disappears, and itsquantum ripples spread allovertheAtlantic.

Thenin Copenhagen,the beer bottlesuddenly jumps ontothe dock, and the

ripples disappeareverywhereelse.Thats whatquantum mechanics sayshappens toelectrons andphotonswhenthey move fromplacetoplace.



P d 2 ( ll )P d 2 ( ll )

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Experiment: A cat is placed in a sealed boxcontaining a device that has a 50% chance ofkilling the cat.

Question1: What is the

wave function of the catjust before the box isopened?

When does the wave function collapse?

Paradox 2 (= collapse):Schrdingers Cat


1 1

2 2( dead + alive ?)= !

P d 2 ( ll )P d 2 ( ll )

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Experiment: A cat is placed in a sealed boxcontaining a device that has a 50% chance ofkilling the cat.

Question1: What is the

wave function of the catjust before the box isopened?

When does the wave function collapse?



Question 2: IfweobserveSchrdinger,whatis hiswave function duringtheexperiment? When does itcollapse?

1 1

2 2( dead + alive ?)= !

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Thequestionis,when andhow does thewave function

collapse.

Whateventcollapses it?

How does thecollapsespread to remotelocations?

P d 3 ( ti l )P d 3 ( ti l )

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Paradox 3 (wave vs. particle):Wheelers Delayed Choice


A source emits one photon.Its wave function passesthrough slits 1 and 2, makinginterference beyond the slits.

The observer can choose to either:

(a) measure the interference pattern atplane Wrequiring that the photon travelsthrough both slits.

or(b) measure at plane W which slit image it

appears in indicating thatit has passed only through slit 2.

Theobserverwaitsuntilafterthephotonhaspassed the slits todecidewhich

measurementto do.

*

**

P d 3 ( ti l )P d 3 ( ti l )

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Thus,thephoton does notdecideifitis aparticleor a

waveuntilafteritpassesthe slits,eventhough aparticlemustpass throughonlyone slit and a wave mustpassthrough both slits.

Apparentlythe measurementchoice determineswhetherthephotonis aparticleor a waveretroactively!



P d 4 ( l l )P d 4 ( l l )

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Paradox 4 (non-locality):EPR ExperimentsMalus and Furry


An EPR Experiment measures the correlatedpolarizations of a pairof entangled photons, obeyingMalus Law [P(U

rel) = Cos2U

rel]


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Paradox 4 (non-locality):EPR Experiments

Malus and Furry

Paradox 4 (non-locality):EPR Experiments

Malus and FurryAn EPR Experiment measures the correlated

polarizations of a pairof entangled photons, obeyingMalus Law [P(Urel) = Cos2Urel]

The measurement gives the same resultas if both filters were in the same arm.

P d 4 ( l li )P d 4 ( l li )

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An EPR Experiment measures the correlatedpolarizations of a pairof entangled photons, obeyingMalus Law [P(Urel) = Cos2Urel]

The measurement gives the same resultas if both filters were in the same arm.

Furry proposed to place both photons inthe same random polarization state.This gives a different and weaker correlation.

P d 4 ( l li )P d 4 ( l li )

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Apparently, the measurement on the right side ofthe apparatus causes (in some sense of theword cause) the photon on the left side to be inthesame quantum mechanical state, and this

does not happen until well after they have leftthe source.

This EPR influence across space time workseven if the measurements are light yearsapart.

Could that be used for FTL signaling? Sorry, SFfans, the answer is No!

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ThreeThreeInterpretationsInterpretations

of Quantum Mechanicof Quantum Mechanic

The Copenhagen InterpretationThe Copenhagen Interpretation

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Heisenbergs uncertainty principle:Wave-particle duality,conjugate variables,e.g.,xandp, Eand t;Theimpossibilityof simultaneous conjugate measurements

Borns statistical interpretation:

The meaningofthewave function] asprobability:P = ]]*;

Quantum mechanicspredicts onlytheaveragebehaviorof a system.

Bohrs complementarity:

The wholeness ofthe system and the measurement apparatus;Complementarynatureofwave-particle duality: aparticle OR a wave;Theuncertaintyprincipleispropertyofnature,notof measurement.

Heisenbergs "knowledge" interpretation:

Identificationof] with knowledgeof anobserver;] collapse and non-locality reflectchanging knowledgeofobserver.

Heisenbergs positivism:

Dont-ask/Donttell aboutthe meaningor reality behind formalism;

Focus exclusivelyonobservables and measurements.

The Copenhagen InterpretationThe Copenhagen InterpretationQuantumMechanics

he Many-Worlds Interpretationhe Many-Worlds Interpretation

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RetainHeisenbergs uncertainty principle andBorns statistical interpretationfrom the CopenhagenInterpretation.

No Collapse.

Thewave function] nevercollapses; it splits intonewwave

functions that reflectthe differentpossibleoutcomes ofmeasurements. The splitoffwave functions resideinphysicallydistinguishable worlds.

No Observer:

Ourpreceptionofwave functioncollapseis becauseourconsciousness has followed aparticularpatternofwave functionsplits.

Interference between Worlds:

Observationofquantum interferenceoccurs becausewave functionsin several worlds havenot been separated becausetheylead tothesamephysicaloutcomes.

he Many-Worlds Interpretationhe Many-Worlds InterpretationQuantumMechanics

Transactional Interpretation (JGC)Transactional Interpretation (JGC)

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Heisenbergs uncertainty principle andBorns statistical interpretation arenotpostulates, becausetheycan be derived from theTransactionalInterpretation..

Offer Wave:

Theinitialwave function] is interpreted as a retarded-waveofferto form aquantum event.

Confirmation wave:

The responsewave function] (presentintheQM formalism)is interpretedas an advanced-waveconfirmationtoproceed withthequantum event.

Transactio

n the Qua

ntum Ha

ndsha

ke:

A forward/back-in-time] ] standingwave forms,transferringenergy,momentum, and otherconserved quantities, and theevent becomes real.

No Observers:

Transactions involvingobservers areno different from othertransactions;Observers and their knowledgeplayno special roles.

No Paraoxes:

Transactional Interpretation (JGC)Transactional Interpretation (JGC)

Summary of QM InterpretationsSummary of QM Interpretations

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Summary of QM InterpretationsSummary of QM Interpretations

CopenhagenManyWorlds

Transactional

Uses observer knowledge toexplainwave functioncollapse and non-locality.Advises dont-ask/donttell about reality.

Uses world-splitting toexplainwavefunctioncollapse. Hasproblems withnon-locality. Usefulinquantum computing.

Uses advanced-retarded handshake toexplainwave functioncollapse and non-locality. Provides

a wayof visualizing quantum events.

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The TransactionalThe TransactionalInterpretationInterpretation

of Quantumof QuantumMechanicsMechanics

Listening to the FormalismListening to the Formalism

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gof Quantum Mechanics

gof Quantum Mechanics

Consider a quantum matrix element:

= v ]S]dr3 =

a ]* - ] sandwich. What does this suggest?

Hint: Thecomplexconjugationin] is theWigneroperator fortime reversal. If] is a

retarded wave,then] is an advanced wave.If ]!%ei(kr-[t) then ]!%ei(-kr+[t)

(retarded) (advanced)

M lls El t tiM lls El t ti

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Maxwells ElectromagneticWave Equation (Classical)

Maxwells ElectromagneticWave Equation (Classical)

Fi !cx2Fi xt2

This is a 2nd order differential equation,which has two time solutions, retarded

and advanced.

Wheeler-Feynman Approach:Use retarded and advanced(time symmetry).

Conventional Approach:Chooseonlytheretarded solution(a causality boundarycondition).

A Classical Wheeler FeynmanA Classical Wheeler Feynman

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A Classical Wheeler-FeynmanElectromagneticTransactionA Classical Wheeler-FeynmanElectromagneticTransaction

The emitter sends retarded andadvanced waves. It offersto transfer energy.

Th Qu ntumTh Qu ntum

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The QuantumTransactional Model


Step 1: The emitter sendsout an offer wave =.

The QuantumThe Quantum

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Step 1: The emitter sendsout an offer wave=.

Step2: The absorber respondswith a confirmationwave=*.

The QuantumThe Quantum

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Step 1: The emitter sendsout an offer wave =.

Step2: The absorber respondswith a confirmationwave=*.

Step3: Theprocess repeatsuntilenergy and momentumis transferred and thetransactionis completed(wave functioncollapse).

The Transactional Interpretation andThe Transactional Interpretation and

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The Transactional Interpretation andWave-Particle Duality

The Transactional Interpretation andWave-Particle Duality

The completed transactionprojects out only that partof the offer wave that had beenreinforced by the

confirmation wave.

Therefore, the transactionis, in effect, a projectionoperator.

This explains wave-particleduality.

ti l I t t ti d th B Pti l I t t ti d th B P

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tional Interpretation and the Born Prtional Interpretation and the Born Pr

Starting from E&M and the Wheeler-Feynmanapproach, the E-fieldecho that the emitter receives

from the absorber is the productof the retarded-wave E-field atthe absorber and the advanced-wave E-field at the emitter.

Translating this to quantummechanical terms, the echo

that the emitter receives fromeach potential absorber is ==*,leading to the Born Probability Law.

Th R l f th Ob iTh R l f th Ob i

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The Role of the Observer inthe Transactional Interpretation

The Role of the Observer inthe Transactional Interpretation

In the Copenhagen interpretation,observers have a special role as the collapsers ofwave functions. This leads to problems, e.g., inquantum cosmology where no observers arepresent.

In the transactional interpretation, transactionsinvolving an observer are the same as any othertransactions.

Thus, the observer-centric aspects of theCopenhagen interpretation are avoided.

C I t t tiC I t t ti

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Can Interpretationsof QM be Tested?

Can Interpretationsof QM be Tested?

The simple answer is No!. It is theformalism of quantum mechanicsthat makes the testable predictions.

As long as an interpretation is consistent with the formalism, it will

make thesame predictions as any other interpretation, and noexperimental tests are possible.

However, there is a new experiment (Afshar), which suggests that theCopenhagen and Many-Worlds Interpretations may be inconsistentwith the quantum mechanical formalism.

If this is true, then these interpretations can befalsified. The Transactional Interpretation is consistent with the Afshar results

and does not have this problem.

Wheelers DelayedWheelers Delayed

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Wheeler s DelayedChoice ExperimentWheeler s DelayedChoice Experiment

One can choose to either:

Measure at W1 the interference pattern, giving thewavelength and momentum of the photon, or

Measure at W2 which slit the particle passed through, givingitsposition.

Wheelers DelayedWheelers Delayed

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Wheeler s DelayedChoice ExperimentWheeler s DelayedChoice Experiment

Thus, one observes either:

Wave-like behavior with the

interference patternor

Particle-like behavior in determiningwhich slit the photon passed through.

The Afshar ExperimentThe Afshar Experiment

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Put wires with 6% opacity at the positions of the interferenceminima at W1, and

Place detector at 2 on plane W2 and observe the particles passing

through slit 2. Question: What fraction of the light is blocked by the grid and

not transmitted? (i.e., is the interference pattern still there whenone measures particle behavior?)


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Copenhagen-influenced expectation:The measurement-type forces particle-like behavior, sothere should be no interference, and no minima.Therefore, 6% of the particles should be intercepted.


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Many-Worlds-influenced expectation:The universe splits, and we are in a universe in which thephoton goes to 2. Therefore, there should be no interference,and no minima. Consequently, 6% of the particles should beintercepted.


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Transactional-influenced expectation:The initial offer waves pass through both slits on their way topossible absorbers. At the wires, the offer waves cancel in firstorder, so that no transactions can form and no photons can beintercepted by the wires. Therefore, the absorption by thewires should be very small(

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Grid + 1 Slit

6% Loss

Grid + 2 Slits

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fshar st su tsfshar st su ts

Copenhagen ManyWorlds

Transactional

Predicts nointerference. Predicts nointerference.

Predicts interference, as does theQM formalism.

Afshar Test ResultsAfshar Test Results

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ff

Transactional

T

hus,it appears thattheT

ransactionalInterpretationis theonlyinterpretationofthethree discussed thathassurvived theAfshartest. It also appears thatotherinterpretations onthe market(Decoherence, Consistent-H

istories,etc.

) failtheA

fsharT

est.

However,quantum interpretationaltheorists are fairlyslipperycharacters. It remains to be seeniftheywillfind somewayto savetheirpetinterpretations.

Einsteins Moon

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Does the moon exist if nobody is lookingat it?

Copenhagen interpretation:NO! The moon exists only in terms of probabilitywave functions

Only when observed do these wave functionscollapseto definite states

Conflict between objective and subjective realities

Schrdingers Cat

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One atom of13N and a detector

If atom decays, detector activates a hammerwhich breaks a glass containing poison gas

Everything inside a box, together with a cat,and seal it

After 10 minutes has passed,j t b f th b

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just before we open the box...

Copenhagen interpretation: The cat existsin a probabilistic state of being 50% aliveand 50% dead

Other Interpretations to the Rescue?

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Bohm: Quantum potential

Wigner, Penrose: Human consciousness

von Neumann, Wheeler: Participatory universe

Everett, Deutsch: Many-worlds interpretation

Recall: Double-slit experimentwith electron gun and detector

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with electron gun and detector

Can we try to trick the electron, by turning onthe detector only as it is passing through the wall?

Delayed Choice Experiment

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Expt carried out now will determine whether each photon,billions of years ago, behaved like a particle or wave

Many-Worlds Interpretation

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Each quantum process will split theuniverse into two or more universes

All the possible universes exist, butMany-worlds FAQ

Einstein-Podolsky-Rosen (EPR) Paradox

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Decay of pion

Electron and positron have opposite spins

EPR Paradox (contd)

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Let the electron and positron fly very far apart

Measure the spin of one of them, say the electron

This would instantaneously determine the spin ofthe positron

Experimentally verified by Aspect (1982)

Non-Locality ofQuantum Mechanics

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Events in Region Binstantaneouslydependent upon

events in Region A

Events in Region Ainstantaneouslydependent upon

events in Region B

widely separatedregions

Einstein: Spooky action at a distance

Does this contradict special relativity?

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In other words, can this be used to transmitmessages faster than light?

NO! Because outcome is completely probabilistic

We would never know in advance whether theelectron is going to be spin up or down

Ni l B h

Quotes to ponder

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Niels Bohr:

Anyone who is not shocked by quantum theory hasnot understood it.

Richard Feynman:

I think I can safely say that nobody understandsquantum mechanics.

J H Q t Phil h

Further Reading

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J. Horgan, Quantum PhilosophyScientificAmerican (July 1997)

R.E. Crandall, The Challenge of Large NumbersScientificAmerican (Feburary 1997)

Worldview

Q t h i h i

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Quantum mechanics has given us: Probability waves: we cant know exactly where a particle is at

nor can we know exactly what its momentum is. Tunneling effects: a particle is permitted to tunnel through a

barrier. We can know the likelihood (probability) it will tunnel,but we cant know when it will tunnel!

Recall the Newtonian worldview:

If we knew the state of the universe at some time, Newtonianphysics fully explained how the universe would evolve. Thisled to a deterministic universe.

The Newtonian worldview is annihilated by thequantum theory.

Every single interaction is now random! We can calculate theprobability for an event to occur, but we cant guarantee it willoccur!

Philosophical consequences of quantum theory run very deep,in part because of our inability to comprehend it.

Quantum Mechanics

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Quantum mechanics vs.

ClassicalMechanicsFormulated toexplainthe behaviorof microscopic systems.

Formulated toexplainthe behaviorofmacroscopicobjects.

Newtons second law:

Integratetwice x(t).

Twoconstants ofintegration two additionalpieces ofinformation required touniquely definethe stateofthe system (e.g.xo and vo).

The stateofthe system is defined by: FORCES,POSITIONS, VELOCITIES

Knowledgeoftheinitial stateofthe system canpredict future statesprecisely.

2

2

dtxdmmaF !!

Quantum Mechanics

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Designed to describethe behaviorobserved for microscopicparticles and systems.

Behavior:

discrete ratherthancontinuous energylevels

photons,electrons and nucleiexhibit bothwave andparticle nature.

History:

1925 WernerHeisenberg,Max Born and Pascual Jordonintroduced matrix-basedmathematical formalism to describeobserved quantum mechanicalphenomena

1926 ErwinSchrdingerintroduced a differentialequation and its solutionthatequivalently describes equationobserved quantum mechanicalphenomena

Q

=,!=

tdt

tdiJ

QM analogytoNewtons 2nd law

The Wavefunction

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Stateofthe system is defined by a wave function:

=(t) = f + ig f and g are real functions ofcoordinates and time

An abstract,complex quantity but related tophysically measurable quantities

Stateis dependentoncoordinates (spatial and spin) and time

Thetime-dependentSchrdingerequation:

A singleintegrationwith respecttotimeis required toobtain=(t), sothatonlyoneconstantofintegrationis required topredict future states ofthe system.

=,!=

tdt

tdiJ

QM analogytoNewtons 2nd law

The Wavefunction

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Howcanwe describethe stateof a quantum mechanical system such as nuclear

spins?Complexwavefunction:

=(t) = =(X,t) descriptionof all knowableinformation aboutthe stateofthesystem

Whatifwewantto knowifthe system is in a given state=(t) attimet? Theprobability thatthe system is inthe stategiven by=(t) attimetis:

P = =*(t)=(t) = =2

Forexample, for a singleparticle attimet,thewavefunctionis =(x,y,z,t), and theprobabilitythattimet,theparticleis in a given volumeof space(dxdydz)is givenby:

=(x,y,z,t)2dxdydz

Spatial and spincoordinates(independentoftime)

T

ime dependence

probability density

dxdydz

Complex Conjugate

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p j gComplex Conjugate=*:

= = f + ig

=* = f ig (replaceiwith i)

=*= = (f + ig)(f ig) = f2ifg + ifg (i2)g = f2 + g2

real,non-negative(as P should be!)

Normalization Condition

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Sincethe system mustexistin some state attime,t,ifweintegrateover all

coordinates ofthe system (X represents thegeneralized coordinates,which mayinclude spatialcoordinates and spin state),theprobability densityis 1.

Normalizationcondition:

=*(t)=(t)dX = 1

i.e.theprobabilityof findingtheparticlesomewhere in spaceis one.TheSchrdingerequation describes theevolutionin time of a given system:

=!=

tHdt

tdiJ

TheSchrdingerEquation

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TheHamiltonian,H, represents the forces actingonthe system,whichcan betime-dependentortime-independent.

g q

=!= tHdt

tdiJ

Hamiltonian = FORCES

constant,oftenomitted

Thetime-independentSchrdingerequation

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IftheHamiltonian is time-independent,thentheSchrdingerequationcan be

solved by separationof variables.=(t) = ](X)J(t)

TheHamiltonian acts onlyonthegeneralized coordinatepartof=(t),](X),sinceHis independentoftime(i.e.J(t) acts as a constant). Choosingunits sothat = 1:

Multiplying both sides by=*(X) dX

tdttdi JX]JX] ,!

tt

ddt

d

di ,! XX]X]J

XX]X] **

timeindep.

time-dep.

= 1


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Notingthatthewave functionis normalized and multiplying by i:

To simplifythings,lets define:

Sothat

Nowthegoalis to solvethis differentialequation.

tt didt

d JXX]X]J ,! *

! XX]X] dE H* tt iE

dtd JJ !


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The solutionis:

This can be seen by differentiatingwith respecttot:

Nowwecanwrite

iEtet !J

!!

tiEiEedt

tdiEt J

J

iEtet t !!=

X]JX]


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Differentiatingwith respecttotime:

Multiplying both sides byi:

Comparingtotheoriginal form oftheSchrdingerequation,weget

Whichis thetime-independent Schrdinger equation.Thetime dependencecan

bethoughtof as aphase factorthatcancels whentheprobability distributioniscalculated:

=!X]!= tiEeiEdt

td iEt

=!=

tEdttd

i

=!= EH

1* !!! iEtiEtiEt eee ttt JJJ

=!=

tHdt

td

i


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Wehave just shownthat:

His anoperator

Eis anenergy

= is thewave function

=!= EH

ProbabilisticnatureofQM

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If= is a solutiontotheSchrdingerequation,then sois c= (c = arbitrary

constant), and ci=i is also a solution,whereeach=i is apossible stateofthesystem.

= isnt really aphysicalwave. Itis an abstract mathematicalentitythatyieldsinformation aboutthe stateofthe system.

= gives informationontheprobabilities forpossibleoutcomes of measurements of

the systemsphysicalproperties.

Quantum mechanics says a lot, but does not reallybringus anyclosertothe secrets ofthe Old One. I, at

any rate, am convinced thatHe does notthrow dice.Albert Einstein

Eigenvalues and Eigenfunctions

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Eigenvalueequations:

Anoperator acts on a functiontogive another function:

f(x) = g(x)

Whereis anoperator.

Forexample, anoperatorcan be defined where represents multiplying by a:

f(x) = af(x)

Dependingupontheoperator,thenew functioncan be very different from theoriginal function. However,in a specialcase,thenew functionis a multipleoftheoriginal function:

f(x) = Pf(x)

Inthis case, f(x)is said to be aneigenfunction ofwiththe associated eigenvalue,P. InthecaseoftheSchrdingerequationwith a time-independentHamiltonian,His theoperator,= is theeigenfunction and Eis theeigenvalue:

H = = E=

Operators

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Foreveryobservablequantity,A,thereis an associated Hermitianoperator,A, such

that:A] = P]

In fact,ifthere aren allowed states thenthere areneigenfunctions,]i,that satisfy:

A]i = Pi]i

Hermitian Operators

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The adjointof anoperator(A) satisfies theequation:

]*A

=P

*]*Hermitianoperators are self-adjoint(A = A):

This

has s

ev

era

lim

plication

s:

Eigenvalues forHermitianoperators are real

Eigenfunctions forHermitianoperators form a completeorthonormal set:

]i*]jdX =]j*]idX = Hij = Kronecker delta

Hilbert space: a complete setofNorthonormal functions whichconstitutes a basisset:

*d*d*

-

X]J!XJ] AA

timeondependcanthatnumberscomplexarenN

1nnn cc ]!=

!

Quantum Mechanics

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A]i = Pi]i

The resultof making a measurementofA is oneoftheeigenvalues ofA.Thatis,only a limited setofoutcomes arepossible(discretenatureofquantum mechanics).

Whatis the valuewe mightexpectto measure? Expectation Value

Theexpectation valueis the average magnitudeof aproperty sampled over anensembleofidenticallyprepared systems. Theexpectation value,,is thescalarproductof= and A=:

= =*A=dX

Ifthewavefunctionis aneigenfunctionoftheoperator(= = ]n):

= =*A=dX = ]n*A]ndX = Pn]n*]ndX = Pn

Expectation Value

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If a system is in state=(X,t),the averageof anyphysicalobservable C attimetis:

=]*]dXIfone makes a largenumberof measurements of C withidenticalinitial state=(X,0),thenoneobtains a setof values C1, C2, , CN. The averageof C is givenbythe rule:

Apostulateofquantum mechanics is thattheintegral and summation aboveprovidethe same value,whichis theexpectation value.

!!

N

1i

iC

N

1C

Quantum Mechanics

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Ingeneral= ]n, but

Hence:

!

]!=N

1i

iic

j

N

1jjj

N

1jj

*j

ijj*ij

*ij

N

1jj

*i

N

1i

jjjj*i

N

1j

j*i

N

1i

j

N

1jj

*i

N

1i

*i

j

N

1j

ji

N

1i

i

2ccc

ji1ji0

d:notingdcc

A:notingdcc

dcc

dc*c

d*

P!P!

!

{!H!X]]X]]P!

]P!]X]]!

X]]!

X]]!

X==!

!!

!!

!!

!!

!!

-

-

-

-

A

A

A

AA

probabilitythatcj is obtained in a single measurement

Quantum Mechanics

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What does this mean?

WhenAis measured for a single memberof anensemble,the resultis oneoftheeigenvalues ofA, butwhichonecannot bepredicted in advance.

The result means thattheeigenvaluePj will beobtained in a single measurementwiththeprobabilityofcj2.

So, for a single measurement,there are specified values ofAthat arepossible, butover anensemble,theexpectation valuecan be a continuous value.

Commutator

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AB = BA

[A,B] AB BA

Ingeneral,AB BA, so[A,B] 0e.g.in matrix multiplication,order matters!

If[A

,B] = 0,thencan moveA

and B withrespecttoeachother

EXAMPLE

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Example: Time-dependentexpectation value forthe magnetic momentof

a single spin (I = ):Sinceitis a spin nucleus,there aretwopossible states:

E = + and F =

Thewavefunction forthe spinin a static magnetic field is:

FF

EE

FFEE ]

]

!]]!=

-

- tiEebti

Eeacc

Realnumbersa2 + b2 = 1

Energies Stationary states(eigenfunctions)

Expectation value forQz

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= =*z=dX = K=*Iz=dX z = KIz

= K(a exp(iEEt)]E* + b exp(iEFt)]F*)Iz(a exp(iEEt)]E + b exp(iEFt)]F) dX

Wehave fourterms to multiply:

1) = K (a exp(iEEt)]E*)Iz(a exp(iEEt)]E) = Ka2exp(iEEt)exp(iEEt) ]E*Iz]EdX

= Ka2 () ]E*]EdX= Ka2/2 note:Iz]E= ]Eand ]E*]EdX= 1

2) = K (b exp(iEFt)]F*)Iz(a exp(iEEt)]E) = Kba exp(iEFt)exp(iEEt) ]F*Iz]EdX

= Kba ()exp(i(EEEF)t ]F*]EdX= 0 note: ]F*]EdX= 0 (orthogonal)

3) = K (a exp(iEEt)]E*)Iz(b exp(iEFt)]F) = 0 (orthogonal)

4) = K (b exp(iEFt)]F*)Iz(b exp(iEFt)]F) = Kb2/2 note:Iz]F= ]FGiving:

= (K/2)(a2b2)

Expectation valueofQ

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A similar approachcan beused to determine and :

= Kab cos([ot)

= Kab sin([ot)

Thethreeequations represent a vectorofconstant magnitudeprecessing around thez axis with angular momentum,[o. This is equivalenttothe Bloch formulation.

eia = cosE + isinEand e-ia = cosE isinE

So:eia e-ia = cosE + isinE (cosE isinE) = 2isinE

eia + e-ia = cosE + isinE + cosE isinE = 2cosE

= =*x=dX = K=*Ix=dX x = KIx

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= K(a exp(iEEt)]E* + b exp(iEFt)]F*)Ix(a exp(iEEt)]E + b exp(iEFt)]F) dX

Wehave fourterms to multiply:

1) = K (a exp(iEEt)]E*)Ix(a exp(iEEt)]E) = Ka2exp(iEEt)exp(iEEt) ]E*Ix]EdX

= Ka2 () ]E*]FdX= 0 note:Ix]E= ]Fand ]E*]FdX= 0 (orthogonal)

2) = K (b exp(iE

Ft)]F*)Ix(a exp(

iE

Et)]E) = Kba exp(iE

Ft)exp(

iE

Et) ]F*Ix]EdX= Kba ()exp(i(EEEF)t ]F*]FdX= Kba ()exp(i(EEEF)t

3) = K (a exp(iEEt)]E*)Ix(b exp(iEFt)]F) = Kba ()exp(i(EEEF)t

4) = K (b exp(iEFt)]F*)Ix(b exp(iEFt)]F) = note:Ix]F= ]EGiving:

= Kab ()(exp(i(EEEF)t) + exp(i(EEEF)t))

= Kab cos((EEEF)t) = Kab cos([ot)

Matrix representationofI

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Pauli spin matrices (matrix representationofthe angular momentum operator for a

single spin system):

|E> and

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Operator manipulations:

E!

!

!F

F!!

!E

E!!!F

F!!!E

F!

!

!F

E!!

!E

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

i21

0

i

21

1

0

0i

i0

21

I

i

2

1

i

0

2

1

0

1

0i

i0

2

1I

21

0

1

21

1

0

01

10

21

I

2

1

1

0

2

1

0

1

01

10

2

1I

21

1

0

21

1

0

10

01

21

I

2

1

0

1

2

1

0

1

10

01

2

1I

y

y

x

x

z

z

DensityMatrix

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The density matrixis a toolto describethe stateofthe spins and theirevolutionin

time. Ittreats the behaviorof a largeensembleof spins.Consideringonenucleus,A,theexpectation value forthe magnetic momentis:

= = KA

IxA is theoperatorofthex-componentofthe angular momentum. Considering anensembleof spins:

MxA = No = NoKA

whereNo is thenumberof spins in aparticular volume.As wehave discussed,thewavefunctioncan beexpressed as a linearcombinationofeigenfunctions, sothatwecanplacein matrix form.

DensityMatrix

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!!

v!vv!==

!!=!=

-

-

-

-

-

-

-

-

m nmm*n

nm mnm*n

n

mmnm

mmm2

mmm1

*n

*2

*1

n

2

1

nn2n1n

n22221

n11211

*n

*2

*1xA

nn2n1n

n22221

n11211

xA*n*2*1

n

2

1

IcccIc

cI

cI

cI

ccc

c

c

c

III

III

III

cccI

III

III

III

Iccc

c

c

c

/.

/

.

///

.

.

.

.

///

.

.

./

DensityMatrix

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Nowconsideringtheentireensemble:

whereInm are matrixelements oftheoperator,IxA. Thetime-dependent variablesareinthe averagedproducts:

Theycan be arranged in a matrixto form the DensityMatrix:

Thus,the DensityMatrixis Hermitian.

*nm

mnm

nAoxA ccINM K!

*nmcc

*mnnm

*nmmn

nn2n1n

n22221

n11211

ddandccdwhere

ddd

dddddd

D !!!

-

.

///

.

.

DensityMatrix

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Nowthe macroscopic magnetizationcan bewritteninterms ofthe DensityMatrix:

Thisprovides the formula forcalculatingtheobservable magnetization: MultiplyeachelementofIxAbythecomplexconjugateofthecorrespondingelementoftheDensityMatrix and add theproducts. Multiplythis byNoKA.

*nm

mnm

nAomn

mnm

nAoxA dINdINM K!K!

Important Quantum Effects in Our WorldI Lasers

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I Lasers

Usually light is emitted by an excited atom isin a a random direction - light from many atomsgoes in all directions direction and energyhave uncertainty for light emitted from any one atom

Excited Atoms

Photons

What is special about a Laser??

Important Quantum Effects in Our WorldI Lasers - continued

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Lasers work because of the quantum propertiesof photons -- one photon tends to cause another tobe emitted one photon cannot be distinguishedfrom another

If there are many excited atoms, the photons cancascade -- very intense, collimated light is emittedforming a beam of precisely the same color light

Excited AtomsMany PhotonsOne Photon

Important Quantum Effects in Our WorldI Lasers - continued

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Since photons cannot be distinguished, which atomemitted a given photon is completely uncertain

But that means:The direction and energy can be very certain!

If there are many excited atoms, the photons cancascade -- very intense, collimated light is emittedforming a beam of precisely the same color light

Excited AtomsMany Photons

One Photon

SuperconductivityDiscovered in 1911 by K Onnes

Important Quantum Effects in Our World

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Discovered in 1911 by K. Onnes

Completely baffling in classical physics

Explained in 1957 by Bardeen, Cooper And Shriefferat the Univ. of Illinois. (Bardeen is the only person

to win two Nobel Prizes in the same field!)Due to all the electrons acting together to form asingle quantum state -- electrons flow around a wirelike the electrons in an atom!

Current flowing without loss-- flows forever!

wire

High - Temperature SuperconductorsDiscovered in 1987 (Nobel Prize)

Demonstration

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Discovered in 1987 (Nobel Prize)

(Still not understood!)

Magnet

Superconductorlevitated above

magnet - repelleddue to currents insuperconductor

Summar

y Niels Bohr (1912) realized the significance that the quantization

could explain the stability of the atom

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y Schrodinger (1926): Equation for wave function =(x,t) for a

particle --- Still Today the Basic Eq. of QuantumMechanics.Explains all of Chemistry!

( =(x,t) ) is probablity of finding the particle at point x andtime t. More about this later.

Heisenberg showed that quantum mechanics leads touncertainty relations for pairs of variables

Quantum Theory says that we can only measure individualevents that have a range of possibilities

We can never predict the result of a future measurement withcertainty

More next time on how quantum theory forces us to reexamineour beliefs about the nature of the world

(p (x u h/2 (E (t u h/2

(Extra) Example: Harmonic Oscillator Classical situation: Mass attached to a spring.

The spring exerts a force on the mass which is proportional to thedi t th t th i i t t h d d Thi f th

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distance that the spring is stretched or compressed. This force thenproduces an acceleration of the mass which leads to an oscillating motion

of the mass. The frequency of this oscillation is determined by thestiffness of the spring and the amount of mass.

Quantum situation: suppose F is proportional to distance, thenpotential energy is proportional to distance squared.Solutions to Schrodinger Eqn:

What is shown here?Possible wave functions =(x) at afixed time t!

How does this change in time?They oscillate with the classicalfrequency!

What distinguishes the differentsolutions?

The Energy! (Classically thiscorresponds to the amplitude of theoscillation) Note: not all energies arepossible! They are quantized!

E = 3/2 h[

E = 5/2 h[

E = 7/2 h[

E = 1/2 h[

(Extra) Example: Hydrogen atom Potential Energy is proportional to 1/R (since Force

is proportional to 1/R2). What are the solutions to

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p p )Schrodingers equation and how are they related to

Bohrs orbits?

The Bohr orbits correspond tothe solutions shown which havedefinite energies.

The energies whichcorrespond to these wavefunctions are identical to Bohrsvalues! For energies above the groundstate (n=1), there are more than

one wave function with the sameenergy. Some of these wave functionspeak at the value for the Bohrradius for that energy, butothers dont!

Radial Wavefunctions forthe Hydrogen Atom

( vertical lines m Bohr radii )

The structure of quantum mechanics

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linear algebra

Dirac notation

4 postulates ofquantum mechanics

1. How to describe quantum states of a closed system.

2. How to describe quantum dynamics.

3. How to describe measurements of a quantum system.

4. How to describe quantum state of a composite system.

state vectors and state space

unitary evolution

projective measurements

tensor products

, , A] J

Example: qubits(two-level quantum systems)

1

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0

1

E F0 1

E F !2 2| | | | 1

Normalization

0 and 1 are thecomputational basis states

photonselectron spinnuclear spinetcetera

All we do is draw little arrows on a piece of paper - that's all.- Richard Feynman

Postulate 1: Rough Form

A i t d t t t i l t

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Quantum mechanics does not prescribe the state spacesof specific systems, such as electrons. Thats the job ofa physical theory like quantum electrodynamics.

Associated to any quantum system is a complex vector

space known as state space.

Example: well work mainly with qubits, which have state

space C2.0 1

EE F

F

| -

The state of a closed quantum system is a unit vector instate space.

A few conventions

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This is the ket notation.

We write vectors in state space as: ]

We always assume that our physical systems havefinite-dimensional state spaces.

0 1 2 1

0

1

2

1

0 1 2 ... 1

:

d

d

d] E E E E

E

E

E

E

!

! -

Quditd

C

(= )]&

nearlyv

Qu ntum n t t :

Dynamics: quantum logic gates

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Quantum not gate:

0 1 ; 1 0 .X X! !

XInput qubit Output qubit

0 1

0

1

0 1

1 0X

! -

E F p0 1 ?E F E F p 0 1 1 0

Matrix representation:General dynamics of a closed quantum system(including logic gates) can be represented as aunitary matrix.

a b

Unitary matrices

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a bA

c d

!

-

Hermitian conjugation; taking the adjoint

* TA A!* *

* *

a c

b d

!

- A is said to be unitary if AA A A I! !

We usually write unitary matrices as U.

Example:

0 1 0 1 1 0XX

1 0 1 0 0 1I

! ! !

- - -

Nomenclature tips

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matrix=(linear) operator

=(linear) transformation

=(linear) map

=quantum gate (modulo unitarity)

Postulate 2

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The of a is describedevolution closed quantum system

unitary transforma

by a

tion.

' U] ]!

Why unitaries?

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Unitary maps are the only linear maps that preservenormalization.

' U] ]! implies ' 1U] ] ]! ! !

Exercise: prove that unitary evolutionpreserves normalization.

Pauli gates

1gate (AKA or )xX W W

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0 10 1 ; X 1 0 ; X

1 0X

! ! !

-

X

Y

Z

2Y gate (AKA or )yW W

0Notation: IW |

00 1 ; Y 1 0 ; Y

0

iY i i

i

! ! !

-

1 00 0 ; Z 1 1 ; Z

0 1Z

! ! !

-

3Z gate (AKA or )zW W

Measuring a qubit: a rough and ready prescription

0 1] E F!

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0 1] E F!

Quantum mechanics DOES NOT allow us to determine and .E F

We can, however, read out limited information about and .E F

Measuring in the computational basis

22(0) ; (1)P PE F! !

Measurement the system, leaving it ina state 0 or 1 determi

unavoidably disned by the outc

turbsome.

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More general measurements

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1Let ,..., be an orthonormal basis for .d

d

e e C

12

A gives re"measuremen st of in the basis , ..., "

(

ult

with probability .)d

j

je e

P j e

]

]! y

* *Reminder:E G

E G F HF H

y |

- -

Measurement the system, leaving it in a statedeter

unavoidablmined by t

y dhe

isturoutc

bsome.je

Qubit example

0 1] E F!

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] F

01

01Introduce orthonormal basis 2 2

! !

2

11Pr + =

12

E

F

y

- -

2

2

E F!

2

=2

E F

2

Pr( )2

E F !

Inner products and duals

Young man, in mathematics you dont understand

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The inner product is used to define the of a vectodual r .]

If lives in then the of is a function

defined bdu

ya

: l

ddC

C C

] ]

] ] J ] Jp | y

things, you just get used to them. - John von Neumann

Example: 1

0 0 1 =0

EE F E

F

! y - -

Simplified notation: ] J

**

Properties: , since , ,

, since , ,

a b b a a b b a

A b b A A b c b A c b A c

! !

m ! !

Duals as row vectorsSuppose = and = . Thenj jj ja a j b b j

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*

,

j j

j jja b a b a b! ! 1

* *

1 2 2

b

a a b

! - -

* *

1 2

identificatThis suggests the very useful of withthe row vector .

ion aa a -

Postulate 3: rough form

If i th l b i th

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12

If we measure in an orthonormal basis ,..., , then

we obtain the result with probability ) .( j

d

P j

e

j e

e

]

]

!

The measurement the system, leaving it in a statedetermined by the

disoutcome.

turbs je

The measurement problem

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Quantum system

Measuring apparatusRest of the Universe

Postulate 3Postulates 1 and 2

Research problem: solve the measurement problem.

Irrelevance of global phase

Suppose we measure in the orthonormal basis e e]

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1

2

Suppose we measure in the orthonormal basis ,..., .

Pr(Then ) .

d

j

e e

j e ]

]

!

21

2

Suppose we measure in the orthonormal basis ,..., .

Then Pr ) .(

id

ij jj e

e e

e

e

eU

U

]

]

]! !

global phase factor unobservabThe is thus , and we mayidentify the states and .

lei

iee

U

U] ]

Revised postulate 1

Associated to any quantum system is a complex inner

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Associated to any quantum system is a complex inner

product space known as state space.

The state of a closed quantum system is a unit vectorin state space.

Note: These inner product spaces are often calledHilbert spaces.

Multiple-qubit systems

00 01 10 1100 01 10 11E E E E

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00 01 10 11

E! 2( , ) | |xyP x yMeasurement in the computational basis:

General state of n qubits: _ a E 0,1 n xx x

Classically, requires 2 bits to describe the state.n

O

Hilbert space is a big place - Carlton Caves

Perhaps [] we need a mathematical theory of quantumautomata. [] the quantum state space has far greatercapacity than the classical one: [] in the quantum casewe get the exponential growth [] the quantum behaviorof the system might be much more complex than itsclassical simulation. Yu Manin (1980)

Postulate 4

The state space of a composite physical system is thet ns r pr duct f th st t sp c s f th c mp n nt s st ms

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tensor product of the state spaces of the component systems.

Example: 2 2 4Two-qubit state space is C C C !

Computational basis states: 0 0 ; 0 1 ; 1 0 ; 1 1

Alternative notations: 0 0 ; 0,0 ; 00 .

Properties

( ) ( )z v w z v w v z w ! !

1 2 1 2( )v v w v w v w !

1 2 1 2( )v w w v w v w !

Some conventions implicit in Postulate 4

Alice Bob

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If Alice prepares her system in state , and Bob prepareshis in state , then the joint state is .

ab a b

Conversely, if the joint state is then we say thatAlice's system is , and Bob's system iin the state in

the stat

s

e .

a ba

b

means"Alice thatappliesis applied to the joint syste

the gate to her system"m.

U IU

A B v w A v B w !

=i ia b e a e bU U

Suppose a NOT gate is applied to the second qubit of the state

Examples

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Suppose a two-qubit system is in thestate 0.8 00 0.6 11 . A NOT gate is applied to the

second qubit, and a measurement performed in thecomputational basis. What are the

Worked exe

probabilit

rcise

o

:

ies f

r thepossible measurement outcomes?

0.4

00 0.3

01

0.2 1

0 0.1 11 .

The resulting state is

0.4 00 0.3 01 0.2 10 0.1 11I X

0.4 01 0.3 00 0.2 11 0.1 10 .!

Quantum entanglement

Alice Bob

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00 11

2]

!

a b] {

Schroedinger (1935): I would not call[entanglement] one but rather the characteristictrait of quantum mechanics, the one thatenforces its entire departure from classical linesof thought.

0 1 0 1] E F K H! 00 10 01 11EK FK EH FH! 0 or 0.F Kp ! !

SummaryPostulate 1: A closed quantum system is described by aunit vector in a complex inner product space known as

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p p p

state space.Postulate 2: The evolution of a closed quantum system isdescribed by a unitary transformation.

' U] ]!

12

If we measure in an orthonormal basis,..., , then we obtain the result with probability

( ) .

Postulate 3 :d

j

e e j

P j e

]

]!

The measurement disturbs the system, leaving it in a state

determined by the outcome.je

Postulate 4: The state space of a composite physical systemis the tensor product of the state spaces of the componentsystems.

Domains of Physics

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Speed

Size

RelativisticQuantumMechanics

QuantumMechanics

Relativity

ClassicalMechanics

NewtonianCosmology

RelativisticCosmology

c

Nucleus(10-14 m)

Atom(10-10 m)

Galaxy(1020 m)

c/10

Paul Dirac (1902-84)

First to try to combine quantummechanics with special relativity

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p y

Obtained the relativistic version ofSchrdingers equation in 1928

Known as the Dirac equation:

The Dirac Equation Appears to have solutions with negative energies

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Vacuum consists of a sea of such negative-energy particles

Dirac identified holes in this sea as antiparticles,with opposite charge to normal particles

He (wrongly) identified the antiparticle of the

electron with the proton

Positron: Antiparticle of the Electron Discovered in cosmic rays by Carl Anderson in

1932

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Has the same mass asthe electron but positivecharge

Anderson saw a track in a cloud chamber left by somethingpositively charged, and with the same mass as an electron

Every particle has an antiparticle 1955: antiproton

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1960: antineutron

1965: anti-deuteron

1995: anti-hydrogen atom (only 9 produced!)

The photon is its own antiparticle!

f we bring a particle and antiparticle tog

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Particle AntiparticleAnnihilation / Creation A particle can annihilate with its antiparticle to

form gamma rays

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An example whereby matter isconverted into pure energy byEinsteins formula E= mc2

Conversely, gamma rays withsufficiently high energy can turninto a particle antiparticle pair

Particle antiparticle tracks in a bubblechamber

w much antimatter is there in the univers Cannot tell just by looking

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A possible cloud of antimatter at the centre of our galaxy?

But if there are antimatter stars and galaxies, thentheir boundaries with normal matter will besources of gamma rays

We do not detect such gammarays in significant amounts universe is mainly madeup of m

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