Historical Overview of QM

Quantum Mechanics is now over 100 years old, and isone of the most successful scientific theories evercreated. We believe it to be the underpinning of allphysical laws. But at ordinary human scales, its effectsare almost totally masked. Only by looking atphenomena at very short length and time scales can wesee quantum behavior.

By the 1890s, classical physics—Newtonian mechanicsplus Maxwell’s electromagnetic theory and Boltzmann’sstatistical mechanics—seemed capable of explainingvirtually all physical phenomena. But a number ofseemingly minor puzzles proved to be gaps that wouldcompletely overthrow the classical structure of physics.

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Planck’s law

The first of these puzzles was the attempt by MaxPlanck to derive the proper distribution of thermalenergy for an electromagnetic (EM) field. The model heused was a closed box at temperature T , empty exceptfor whatever electromagnetic radiation it contained.

A well-known result from thermodynamics is theequipartition theorem: if a system is in thermal equilibriumat temperature T , every independent degree of freedomcontains an average energy of kBT/2. For EM radiationin a box of size L, the degrees of freedom are the normal

modes, with wavelengths λ = 2L/n for all values ofn = 1, 2, . . . and frequencies f = cn/2L. Since there areinfinitely many normal modes, equipartition implies thatthe EM radiation in the box has infinite energy.

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Needless to say, this is not what is observed.

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In 1900, Planck came up with a derivation for a finite result.He assumed that the energy in each normal mode came indiscrete chunks, proportional to the frequency E = hf , witha constant of proportionality h. Instead of each modecontaining energy kBT/2, it contained m chunks with aprobability proportional to exp(−mhf/kBT ). By choosingthe proportionality constant h appropriately, he derived adistribution law which closely matched experiment. This lawis Planck’s Law for black-body radiation:

E(f) =8π

c3hf3

ehf/kBT − 1.

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The constant of proportionality h is the incredibly tiny

h = 6.6261× 10−34 kgm2/s,

which has units of action. It is now known as Planck’s

constant. Planck called the discrete chunks of energy quanta,and they gave the name to quantum mechanics.

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The Photoelectric Effect

Confirmation of the discrete nature of light came in1905, when Albert Einstein solved another puzzle: thephotoelectric effect.

When light shines on a metal, electrons can be emitted.This is how photovoltaic cells work. The energy of theemitted electrons (the voltage produced) is proportionalto the frequency of the light, but not to the intensity. Belowa certain frequency (known as the work function W ), noelectrons are emitted at all. Above it, they are. Thenumber of electrons emitted (the current) is proportionalto the intensity of the light, but not the frequency.

Einstein’s solution to this puzzle was to take Planck’squanta seriously.

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If light does come in discrete chunks with energy hf , thenthe photoelectric effect makes sense. Each electron isemitted by being “struck” by a single quantum. The workfunction W gives the minimum energy needed to pull anelectron free from the metal. The energy (hf −W ) of theemitted electron is the excess over this minimum, and ishence proportional to the frequency. The intensity of thelight gives the total number of quanta, and hence the numberof electrons emitted.

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Einstein won the Nobel Prize for his explanation of thephotoelectric effect, not for Special or General Relativity orany of his other work. Ironically, in later years he neveraccepted quantum mechanics, which he had had such animportant role in founding.

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The Stability of Atoms

Another puzzle for classical physics was the stability ofatoms. If atoms consisted of electrons which orbited anucleus (the Solar System Model), then why are theystable?

As the electrons whirl around, they should emitelectromagnetic radiation and lose energy, causingthem to spiral in towards the nucleus.

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One attempt to get around this problem is the Plum

Pudding Model, in which negatively-charged electronsnestle inside a spongy, positively charged nucleus likeplums in a pudding.

Unfortunately, this model could not explain thescattering experiments of Ernest Rutherford; andscientists outside of Britain found it hard to believenature was built out of plum pudding.

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If electrons can orbit the nucleus at any distance, theyshould absorb and emit radiation at any frequency.However, this is not the case. Atoms absorb and emit EMradiation at only a discrete set of frequencies. ForHydrogen, the simplest atom, these frequencies obey anexact mathematical law:

f = cRH

(

1

n2−

1

m2

)

,

where n = 1, 2, . . . and m = n+ 1, n + 2, . . .

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Niels Bohr suggested that this could be explained if onlycertain discrete orbits were allowed: orbits whoseactions are an exact multiple of Planck’s constant ofaction (plus, for technical reasons, 1/2). Thefrequencies corresponding to these orbits turned out toexactly match the above law; and moreover, theyexplained why atoms were stable, since every atom hasa definite lowest possible orbit, the ground state.

The frequencies emitted correspond to energy differences

between the allowed orbits—each orbit is labeled by aninteger m (or n), called a quantum number.

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Matter waves and interference

Another piece of the puzzle was provided by Louis deBroglie in his Ph.D. thesis. Since light, which usuallymanifests itself as waves, had been shown tosometimes behave like particles, De Broglie speculatedthat particles might sometimes behave like waves. Inparticular, he proposed that every particle had anassociated wavelength λ = h/p proportional to Planck’sconstant and inversely proportional to the momentum p.

This guess was based on a well known analogybetween the paths of particles in a potential and rays ina refracting medium. De Broglie’s proposed theory wascalled wave mechanics.

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The hypothesis was verified by firing beams of electrons atpairs of slits in a screen. The beam produced aninterference pattern on the other side, characteristicbehavior for waves. Even more peculiar, the patternpersisted even if the beam intensity was so low that theelectrons arrived one at a time. It seemed that electronscould interfere with themselves.

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New Quantum Theory for Old

The solutions to all of these problems—black bodyradiation, photoelectricity, the Lyman alpha spectrumand two-slit experiment—were found in an ad hoc way,making changes to classical mechanics one at a time.

These changes, however, did not provide a completetheory to replace classical physics. That waited for thework of Schrödinger, Heisenberg, Dirac, von Neumann,Pauli, and others in the 1920s and 30s. The early daysare now called the old quantum theory, and the revolutionthey led to is known as quantum mechanics.

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Properties of Quantum Mechanics

What are the revolutionary properties of quantummechanics? Any quantum theorist can make his or herown list of quantum peculiarities. Here is mine:indeterminism, interference, uncertainty and complementarity,

discrete spectra for bound systems, superposition (linearity), and

entanglement.

We will use many of these properties in this course, andcome to understand them in a technical sense. But letus first get a qualitative picture of what they mean.

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Indeterminism

The most fundamental distinction between classical andquantum mechanics is that classical mechanics is adeterministic theory: given perfect knowledge of thecurrent state of a system, its state at all past and futuretimes is, in principle, calculable. In classical mechanics,probabilities are used only to describe situations whereone’s knowledge is incomplete.

By contrast, quantum mechanics makes statementsonly about probabilities. If the same measurement isperformed on several identically prepared systems, onecannot in general expect the same outcome. This is notbecause we lack information about the systemsdescribed; rather, it is because the outcome of themeasurement is inherently unpredictable.

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Interference

Probabilities in quantum mechanics are not calculateddirectly, but from probability amplitudes∗, which are complex

numbers. The probability is the square of the probabilityamplitude. Their relationship is similar to that betweenthe amplitude of a wave and its intensity.

For example, in the two-slit experiment, the amplitudefor a particle to hit a particular point on the screen is thesum of the amplitude to go through slit A and hit thepoint, and the amplitude to go through slit B and hit thepoint. The probability to hit the point is then

p = |αA + αB|2.

Because the amplitudes can either add or cancel eachother out, this system exhibits interference fringes.

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Some parts of the screen will not be hit by particles, eventhough there is a nonzero amplitude to reach that part of thescreen from each slit. Other parts will be hit at a higher rate.

∗The probability interpretation of QM was presented first byMax Born. It was in a footnote to this pioneering paper,added in proof, that he realized that probabilities were notequal to the amplitudes but to their squares.

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Uncertainty

For a classical particle, complete information is given by theposition of the particle and its velocity (or momentum). Fora quantum particle, this is not the case. As famouslyrealized by Heisenberg, a measurement of a particle’sposition disturbs its momentum, with the size of thedisturbance related to the precision of the measurement.Similarly, a measurement of the momentum disturbs theparticle’s position. This constraint on the precision withwhich position and momentum can be measured isquantified by the inequality

(∆x)2(∆p)2 ≥~2

4, ~ = h/2π.

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This constraint could be seen as just a practical limitationon the precision of measurements; we might imaginethat particles really do have precise positions andmomenta, which we are unable to exactly determine.

However, in quantum mechanics an even more radicalexplanation holds: in fact, the two quantities are not even

simultaneously well-defined. As we will see, quantum statesthat give both a precise position and momentum (or anyother pair of incompatible quantities) do not exist.

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Complementarity

This idea—that different ways of describing a system maybe mutually exclusive—is called complementarity. For positionand momentum, this means that we can write downamplitudes for every possible position of a particle, OR forevery possible momentum, but not both, because thosequantities cannot be simultaneously measured. If ψ(x) isthe wavefunction giving the amplitude to be at every point x,

we can also write down ψ(p) (the Fourier transform) whichgives the amplitude for every point p. But there is no similarfunction ψ(x, p). For variables other than position andmomentum, similar restrictions hold. In particular, for thekind of discrete variables used in quantum informationtheory, uncertainty and complementarity still apply (thoughthey take a somewhat different form).

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Discrete spectrum

For the Bohr atom, only certain discrete orbits wereallowed, with discrete values of the energy. Thesevalues are called energy levels. Discrete spectra arecommon for bound systems in quantum mechanics.

This discreteness is useful in quantum informationtheory, because it matches the discreteness assumedin quantifying classical information. For instance, thesimplest quantum system would be one with only twodistinct levels. This is analogous to a classical bit, whichcan take one of two possible values. In quantuminformation theory, most of the systems we will dealwith have only a finite number of discrete levels.

However, while the number of energy levels may bediscrete, the possible states are continuous. This isbecause of another property of quantum mechanics:linearity. – p. 23/35

Superposition

Suppose that ψ and φ are two valid states of a quantumsystem. (That is, two possible wavefunctions.) Thenaψ + bφ, where a and b are complex numbers, is also avalid state of the system. This is an example of asuperposition.

The reason such superpositions are possible isbecause quantum mechanics is a linear theory. The setof all states forms a complex vector space (or Hilbertspace). The evolution equation for states, the Schrodinger

equation, is a linear differential equation. The variousphysical quantities in the theory are represented bylinear operators (matrices) on the Hilbert space.

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It is linearity which makes possible the famousSchrödinger’s Cat paradox, in which a cat issimultaneously alive and dead.

(More strictly speaking, it is in a superposition of beingalive and dead.)

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Entanglement

This last property of quantum mechanics is one of the mostdifficult to explain; but it plays a crucial role in quantumcomputation and quantum information. If a quantum systemconsists of multiple subsystems—for instance, of severaldistinct particles—it is possible for the joint system to have adefinite state ψ, while none of the subsystems has awell-defined state. In this situation, the subsystems are saidto be entangled.

ψ(x, y) 6= ψ(x)ψ(y).

While this may sound like a strange and exotic situation, infact it is not. Almost all states of multiple subsystems areentangled. But the effects of entanglement are masked atlarger scales.

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At the quantum level, entanglement behaves very muchlike classical correlation: measurement outcomes ondifferent subsystems are correlated. But thesecorrelations can be stronger than any possible classicalcorrelation. This result was proven theoretically by JohnBell in the 1960s, and experimentally demonstrated byboth Clauser and Aspect in the 1970s and 1980s.

Much has been made of entanglement, includingvarious wild assertions that quantum mechanics isnonlocal: that once in contact, quantum systemscontinue to influence each other even when far apart.These assertions are very overstated. But it is true thatentanglement is rather different from any phenomenonwhich occurs in classical physics.

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QIP: A Prehistory

As microelectronic components get smaller and smaller,computer chips are steadily approaching the pointwhere quantum effects must be taken into account.However, by the 1980s, some people were alreadystarting to ask if quantum mechanics could actually beexploited to make new information processingtechniques possible.

The first to propose an intrinsically quantum mechanicalcomputer was P. Benioff in 1980. Y. Manin andR. Feynman both proposed that a quantum computermight be able to efficiently simulate quantumsystems—something that ordinary classical computersfind very difficult. (This idea of quantum simulationremains one of the most important potentialapplications of a quantum computer.)

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Reversible Computation

But is a quantum computer even possible? As we willsee, such a computer must operate reversibly; that is, itcannot dissipate energy. Ordinary computers are highlydissipative, as anyone who has ever felt the heat theygive off has noticed.

In the 1970s, Charles Bennett of IBM showed that anycomputation can, in principle, be done reversibly, basedon work from the 60s by Rolf Landauer. That is, inprinciple, there is no requirement that a computerconsume power to operate (though it may take energyto start a computation). This paved the way for thepossibility of reversible quantum computers.

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Deutsch’s Algorithm

In 1985, David Deutsch presented a new idea. Becauseof linearity, a quantum computer can be in asuperposition of different computations. For instance, acomputer could simultaneously calculate the value of afunction f(x) for every possible input x in a single run.Deutsch called this possibility quantum parallelism, andspeculated that, just like ordinary parallelism, it wouldincrease computing power.

Naive applications of quantum parallelism add nothingto the power of the computer. But in 1988, Deutschfound a clever algorithm that exploited quantumparallelism indirectly to solve a problem more efficientlythan any possible classical computer. The problem wasartificial, but it was the first example where a quantumcomputer could be shown to be more powerful.

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Quantum Cryptography

Meanwhile, in 1984, Charles Bennett and GillesBrassard found another way in which quantumproperties could be exploited for informationprocessing. They exploited the uncertainty principle asa way to distribute a cryptographic key with perfectsecurity. Single quanta are used to send the bits of thekey, in one of two possible complementary variables. Ifan evesdropper tries to intercept the bits and measurethem, this automatically disturbs them in such a waythat it can always be detected.

This and similar schemes are called quantum cryptography

or quantum key distribution (QKD). This is the quantuminformation protocol which is closest to being realtechnology.

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Further Progress

Artur Ekert, in 1991, proposed another scheme forquantum cryptography, this one based on entanglementrather than uncertainty.

Bennett and collaborators found yet other uses forentanglement: quantum teleportation, in which separatedexperimenters, sharing two halves of an entangledsystem, can make use of the entanglement to transfer aquantum state from one to another using only classicalcommunication; and superdense coding, in which sendinga single quantum bit allows the transmission of two bits ofclassical information.

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Richard Josza and David Deutsch extended Deutsch’soriginal algorithm to a more general, but still artificialversion of Deutsch’s problem.

D.R. Simon found another problem, albeit still ratherspecialized, in which quantum computers outperformedclassical computers. The stage was being set for thereal breakthrough.

A mathematician named Peter Shor at AT&T ResearchLabs became interested in the potential of algorithmson quantum computers to solve computationally difficultproblems.

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Shor’s Factoring Algorithm

In 1994, Peter Shor published a paper showing that aquantum computer could decompose a large numberinto its prime factors in a time of polynomial order in thelength of the number.

The difficulty of factoring is the basis for the RSApublic-key encryption algorithms, which is the basis forsecure transactions on the world wide web.

Suddenly, it was known that quantum computers couldin principle solve a problem of importance in the realworld.

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From Prehistory to Today

Rather than being an obscure interest for a handful ofphysicists and computer scientists, quantuminformation processing was suddenly of interest toresearchers in many fields.

In the more than 20 years since the factoring algorithmwas discovered, the fields of quantum information andquantum computation have exploded.

In this course, we’ll see why.

Next time: the simplest quantum system.

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