50468574 classes-5 cwinter11-5c113bid24-5c-lecture-notes-qm


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50468574 classes-5 cwinter11-5c113bid24-5c-lecture-notes-qm

  1. 1. 1 AbstractIn these notes I present an overview of electrodynamics and quantum mechanicswhich (together with statistical mechanics) are the foundation of much of today’stechnology: electronics, chemistry, communication, optics, etc.
  2. 2. CONTENTS1 Introduction: the Unity of Science 42 Quantum Mechanics 5 2.1 The puzzles of matter and radiation 6 2.1.1 Planck’s Black-body radiation 8 2.1.2 The photo-electric effect 12 2.1.3 Bohr’s atom 14 2.1.4 Adsorption, stimulated emission and the laser 16 2.2 Quantum Mechanical formalism 19 2.3 Simple QM systems 21 2.3.1 The chiral amonia molecule 21 2.3.2 The amonia molecule in a constant electric field 24 2.3.3 The amonia maser and atomic clocks 26 2.3.4 The energy spectrum of aromatic molecules 28 2.3.5 Conduction bands in solids 29 2.4 Momentum and space operators 31 2.4.1 Heisenberg uncertainty principle 34 2.5 Schroedinger’s equation 35 2.5.1 Diffraction of free particles 35 2.5.2 Quantum interference observed with C60 37 2.5.3 QM tunneling and the Scanning Tunneling Micro- scope 38 2.6 The correspondance principle 41 2.6.1 Gauge invariance and the Aharonov-Bohm effect 42 2.7 Dirac’s equation: antiparticles and spin 44 2.7.1 Angular momentum and spin 49 2.8 The Hydrogen atom and electronic orbitals 52 2.8.1 Spin-orbit coupling 54 2.8.2 Many electron systems 55 2.8.3 The periodic table 56 2.9 The chemical bond 59 ¨ 2.9.1 Huckel’s molecular orbital theory 63 2.9.2 Molecular vibrational spectrum 65 2.9.3 Molecular rotational spectrum 67 2.10 Time independent perturbation theory 68 2.10.1 The polarizability of atoms in an electric field 71 2.10.2 Atom in a constant magnetic field: the Zeeman effect 73 2.10.3 Degenerate eigenstates 76 2.11 Time dependent perturbation theory 77 3
  4. 4. 2 QUANTUM MECHANICSQuantum mechanics (QM) is a theory of matter and its interactions with force fields(here we will only care about electromagnetic fields). While classical mechanics andelectromagnetism are intuitive (one has a direct experience of gravitation, light, elec-tricity, magnetism, etc.) quantum mechanics is not. The description of matter that arisesfrom the QM formalism is totally at odds with our daily experience: particles can passthrough walls, can be at two different places and in different states at the same time,can behave as waves and interfere with each other. Worse, QM is a non-deterministicdescription of reality: it only predicts the probability of observing events. This aspectdeeply disturbed Einstein who could not accept that QM was the correct final descrip-tion of reality (as he famously quipped: ”God does not play dice”). He and many otherscame up with alternative descriptions of QM introducing hidden variables (unknow-able to the observer) to account for its non-deterministic aspects. But in 1964 JohnBell showed that if hidden variables existed some measurements would satisfy certaininequalities. The experiments performed by Alain Aspect and his collaborators in the1970’s showed that the Bell inequalities were violated as predicted by QM, but not bythe hidden variable theories thereby falsifying them. Yet, for all its technical prowess Aspect’s experiment was only addressing a philo-sophical issue concerning the foundations and interpretation of QM. The theory itselfhad been amply vindicated earlier by its enormous predictive power: QM explains thestability of atoms, their spectra, the origin of the star’s energy and of the elements andtheir properties, the nature of the chemical bond, the origin of magnetism, conductivity,superconductivity and superfluidity, the behaviour of semi-conductors and lasers, etc.,etc.. All of today’s micro-electronic industry is derived from applications of QM (tran-sistors, diodes, integrated circuits, etc.), the development of the chemical industry is aresult of the QM understanding of the chemical bond and the nuclear industry would ofcourse been impossible without an understanding of the nucleus and the nuclear forcesthat QM provided. So, for all its weirdness Quantum Mechanics is the most successfull explanation ofthe World ever proposed by Mankind. It beats Platonicism, the Uppanishads, Kabbalah,Scholasticism, etc., yet is non-intuitive and cannot be understood except by followingits mathematical formalism to its logical conclusions. ”The great book of Nature iswritten in the language of mathematics”, Galileo’s quip is truer for QM more than forany other scientific theory. More recently one of the founder’s of QM, Eugene Wignerwrote in an article entitled ”the unreasonable effectiveness of mathematics in the naturalsciences”, that ”the miracle of the appropriateness of the language of mathematics forthe formulation of the laws of physics is a wonderful gift which we neither understandnor deserve”. 5
  5. 5. 6 QUANTUM MECHANICS It is with this mind set that I would like you to approach the study of QM. Like anapprentice sorcerer learning the tricks of his master without fully understanding them,yet always at awe confronting their power. As we have done with electromagnetism,we will approach QM by following as far as we can the historical narrative. We willsee why the radiation of a Black Body was such a puzzle that it prompted Max Plank tointroduce the idea that energy was quantized; why the stability of atoms and their spectraprompted Bohr, Sommereld and others to suggest that the energy levels of atoms werealso quantized; how the idea that particles could also have wavelike behaviour was firstsuggested by de Broglie and brought to fruition by Schroedinger, Heisenberg and Dirac.And how from then on, QM revolutionized the understanding of matter, the chemicalbond, magnetism, conduction, etc.2.1 The puzzles of matter and radiationAt the end of the 19th century, scientists disposed of a very succesfull theoretical frame-work that could explain many of the problems known at that time and which was tech-nologically revolutionary. Newtonian mechanics was amazingly successful in predict-ing the motion of celestial bodies. Its most striking success was the prediction by LeVerrier in 1846 of the existence of the planet Neptune. Analysing some anomalies inthe motion of Uranus, he predicted Neptune’s precise location in the sky, a predictionwhich was immediately confirmed by German astronomers. In 1861, Maxwell unifiedelectricity, magnetism and optics opening the area of electrical appliances and wirelesscommunication: Edison invented the light bulb in 1879 and founded ”General Electric”in 1892 while Marconi established the ”Marconi Wireless Telegraph Company” in 1897.Finally, thermodynamics was sustaining the advance of the industrial revolution as ther-mal engines were driving industry and railways. In spite of terrible social inequalities(as described by C.Dickens, E.Zola and others) this was a time of peace, prosperity andoptimism, illustrated by the nascent Impressionist movement. Yet, many fondamental scientific questions remained unsolved and paradoxical. Thechemical properties of the various elements were not understood. The periodicity ofthese properties as a function of the mass of the elements as determined in 1869 byMendeleev in his famous Periodic Table of the Elements was a mystery. Nonethelesson the basis of his ad-hoc classification Mendeleev predicted the existence of two newelements, Gallium and Germanium, which were duly discovered in 1875 and 1886 andare essential in today’s semiconductor industry! The existence of atoms (indivisibleparticles of matter characteristic of each element) postulated by Dalton to explain theproperties of molecules was not generally accepted. Because of the successful applica-tions of continuum mechanics (in the design of bridges, buildings (e.g. the Eiffel tower),etc.) and fluid dynamics (in explaining the tides, water waves, etc.), matter was gener-ally believed to be some sort of continuum akin to a gel not a swarm of particles. It wasEinstein who in 1905 finally managed to convince the scientific world of the existenceof atoms and molecules by showing that the erratic motion exhibited by dust particleson the surface of water (first observed by the botanist R.Brown in 1827) was due to theshocks of the water molecules. The continuum pre-conception also sustained the inter-pretation of electromagnetic waves. Since all known waves at the times were observed
  6. 6. THE PUZZLES OF MATTER AND RADIATION 7F. 2.1. The emission spectrum in the visible range for a few elements. Notice the fine spectral lines and the different spectral characteristics for the different elements. This was one of the puzzles that QM solved.to propagate in a continuum medium (such as water, air, etc.) at a velocity v = κ/ρ(where κ is the compressibility and ρ the density of the medium), the electromagneticwaves predicted by Maxwell and discovered by Hertz were assumed to propagate insome continuum medium: the ether which properties determined their velocity. How-ever all attempts to detect the ”wind” of ether resulting from the motion of the Earth inthat medium proved negative. This prompted Einstein to formulate his theory of rela-tivity which postulated the constancy of the speed of light and got rid of any notion ofether, see Appendix. Then there were questions related to the emission and absorption spectra of elementsthat exhibited discrete lines rather than an undifferentiated continuum of absorption oremission, as was the case for sound and water waves and as we have seen for scatteredand refracted light. Not only did the elements exhibit specific adsorption lines but thosediffered from element to element. These observations did not fit with the then prevailingconception of matter as a continuum. Finally there was the problem of the radiation from a Black Body, a material (such asto a good approximation graphite) which adsorbs radiation uniformly at all frequenciesand which can therefore also emit radiation uniformly at all frequencies. Notice how-ever that many bodies (e.g. the elements just mentioned) are not black-bodies as theyadsorb/emit only at certain frequencies. At a given temperature, the radiation inside ablack body cavity is at thermal equilibrium with the walls of the cavity that absorb andre-emit it. When computing the electromagnetic radiation energy emitted by a black-body at a given temperature, one found its energy to diverge because the number ofmodes at high frequencies diverged. This was not only absurd but also in contradiction
  7. 7. 8 QUANTUM MECHANICSwith the experiments which studied the energy distribution inside the cavity by measur-ing the energy leaking out of the cavity (for example through a small hole) as a functionof frequency. To see how this comes about, consider a square cavity (an oven) of size a at tempera-ture T . As we have seen, a body at a given temperature emits electromagnetic radiation,see Fig.??. Imagine that the walls of the cavity are made of small oscillators emitting ra-diation at frequency ω (like the oscillators we considered when studying the frequencydependence of the refraction index). Stationary waves of the form sin k · r cos ωt willbe present in the cavity if its walls are reflecting (though for the energy to equilibratebetween oscillators the walls cannot be 100% reflective). To satisfy the boundary con-ditions on the walls we shall require that: k x = πl/a, ky = πm/a, kz = πn/a (n, m, l ≥ 0).Hence we have: k2 = (l2 + m2 + n2 )(π/a)2 ≡ (πρ/a)2 . The number of modes dNlnm withsuch wavelength is: 4πρ2 ka adk k dk dNlnm = 2 dρ= π( )2 = 8πa3 ( )2 8 π π 2π 2πThe factor 2 results from the two possible polarizations of the fields, while the factor(4πρ2 /8)dρ counts the number of modes in a shell in the positive octant (n, m, l ≥ 0).According to the equipartition theorem of statistical mechanics (see below) the averageenergy of each oscillatory mode is: < E >= kB T . Using the relations: k = ω/c ≡ 2πν/c(ν like f is the frequency), the energy density of the emitted radiation du = dNlnm <E > /a3 becomes: 8πkB T 2 du = ν dν (2.1) c3hence the total energy, the integral of the energy density over all frequencies, diverges asν3 . This divergence became known as the Jeans’ (or ultra-violet) catastrophy. While thedata agreed with that formula at low frequencies, it differed at high frequencies (smallwavelengths).2.1.1 Planck’s Black-body radiationRather than questioning the equipartition theorem which was verified in other contextsor the possibility of atoms to emit light of arbitrarily high frequencies, Planck suggestedin 1900 that light was emitted by the cavity walls in very small discrete quantities,quantas of energy: e = hν, where h, the Planck constant is: h = 6.626 10−27 erg sec = 4.135 10−15 eV sec, so that light of energy En is made up of n quantas: En = nhν. In that case the averageenergy emitted at frequency ν is the sum over all possible energies En , weighted by theirBoltzmann probability (see below the Chapter on Statistical Mechanics): e−En /kB T P(En ) = −En /kB T neSo that the average energy is:
  8. 8. THE PUZZLES OF MATTER AND RADIATION 9 hν < E >= En P(En ) = n ehν/kB T − 1When hν kB T , one recovers the previous result: < E >= kB T , however at largeemission frequencies the average energy decays as < E >∼ hν exp(−hν/kB T ). Plancktherefore suggested to modify the previous result, Eq.2.1 to yield, see Fig.2.2: 8πhν3 1 ρ(ν) ≡ du/dν = (2.2) c3 ehν/kB T − 1Where ρ(ν) is known as the spectral density of radiation. Identifying the smallest quantaof energy with a light particle (a photon of energy hν), Eq.2.2 states that the density ofphotons in a Black-body is: dN p ρ(ν) 8πν2 1 = = 3 hν/k T (2.3) dν hν c e B −1Notice that the total energy density in the cavity is now finite: ∞ 8π(kB T )4 ∞ x3 8π5 k4 4 Utot = ρ(ν)dν = dx = B T (2.4) 0 h3 c3 0 ex − 1 15h3 c3 ∞Where we used the equality: 0 dx x3 /(e x − 1) = π4 /15. The total radiated power perunit area trough a small hole in the cavity becomes, 1 cUtot cUtot Irad = k · ndΩ = ˆ ˆ cos θd(cos θ) = σS B T 4 (2.5) 4π 2 0which is known as Stefan’s law and where the Stefan-Boltzmann constant: 2π5 k4 σS B = B = 5.67 10−5 erg sec−1 cm−2 ◦ K−4 = 5.67 10−8 W m−2 ◦ K−4 15h3 c2Therefore by measuring the total intensity of the radiation leaking out from a cavity(for example an oven) one can measure the temperature of that cavity. One can testthe validity of Planck’s law (actually how close to a black-body the cavity really is)by measuring the dependence of the intensity on the radiation wavelength. From thewavelength λmax at which the intensity is maximal an other estimate of the temperaturecan be deduced: kB T hc/5λmax . For example at 300K (which corresponds to a thermalenergy kB T 25 meV), the maximum of emission is at λmax ∼ 10µm. The thermal cam-eras that visualize humans and warm animals (see Fig.??) must therefore be sensitive tofar-infrared light. The temperature of the Sun and the Earth The Sun is to a very good approximation a black body, see Fig.2.2. The radiationsemitted by the fusion reactions occuring at its core (at temperature of 13 106 K) are
  9. 9. 10 QUANTUM MECHANICSF. 2.2. The emission spectra of the sun and the universe. The sun emission spectra is pretty well fit by the spectrum of a lack body at 5770K, however notice the existence of some specific adsorption bands in the visible and UV spectrum. The universe on the other hand presents a spectrum that is perfectly matched by a black-body at 2.726K.at thermal equilibrium with the reacting nuclear particles and diffuse out to the Sun’ssurface which is much cooler. By fitting the spectrum of the sunlight to Planck’s for-mula one can determine the Sun’s surface temperature: T S = 5770K. The total powergenerated at the Sun’s core and emitted at its surface is: PS = 4πR2 σS B T S = 3.85 1026 W S 4where RS = 6.96 108 m is the Sun’s radius. Since the radius of the Sun’s core is es-timated to be ∼ RS /5 the volume of the core is: Vcore = 1.13 1025 m3 and the averagepower per unit volume generated in the Sun’s core is: PS /Vcore = 34 W/m3 . This is lessthan the power generated by our body to keep warm!! There are a few ways to verify that. Let us assume an average daily calory intake of3000kcal 1.2 107 J, which comes to a power consumption of 150W. Approximatinga man as a cylinder of height L = 2m and radius r = 0.2m, the power consumption perunit volume is 600 W/m3 of which about half goes to metabolic activity. Alternativelyone can use Stefan’s law to estimate the losses between a body at 37◦ C (T b = 310◦ K)and an environment at 27◦ C (T e = 300◦ K) (this is a crude estimate since other effectssuch as perspiration regulate our temperature): ∆I = σS B (T b − T e ) 64W/m2 which 4 4yields a power per unit volume 640W/m3 . From these consistent estimates we de-duce that our power consumption per unit volume is much larger than the Sun’s!! Whatmakes the Sun so bright and hot is its huge mass, not its rather inefficient thermonuclearreactions. Let us now estimate the temperature of the Earth T E resulting from its adsorption of
  10. 10. THE PUZZLES OF MATTER AND RADIATION 11the Sun’s radiation and its own radiation at T E . The sunlight impinging on the Earth ata distance from the Sun RS E = 1.496 1011 m has an intensity: IE = PS /4πR2 E = (RS /RS E )2 σS B T S = 1.37 kW/m2 S 4Of that radiation a fraction (known as the Earth’s albedo) α ∼30% is reflected, mostlyby the clouds, snow and ice-caps. The Sun radiation power arriving at the surface of theEarth is thus about 1 kW/m2 . It is an important number to remember when designingsolar energy plants: its sets the maximal power per unit area available from the Sun.Notice that by measuring the radiation arriving on Earth and the angle sustained bythe Sun: θS = (RS /RS E ) one can also get an estimate of the Sun’s temperature: T S =(IE /σS B θS )1/4 . The energy absorbed by the Earth heats it and is reradiated (to a good 2approximation) like a black body at temperature T E . We can compute the temperatureof the Earth by a simple energy balance. At steady-state the energy radiated is equal tothe energy absorbed: σS B T E 4πR2 = (1 − α)IE πR2 4 E EFrom which we get T E = ((1 − α)IE /4σS B )1/4 = 255K = -18C. The Earth is actuallyslightly warmer because of the green house effect that reflects part of the emitted energyback to Earth. The Universe as a perfect black-body While it is difficult to design a perfect black-body, since as we shall see belowbound electrons adsorb at their resonance frequency (as is the case for the Sun’s spec-trum for example), the Universe as a whole turned out to be the best known exampleof a black-body, see Fig.2.2. The Universe is bathed in a uniform radiation field ofvery low frequency whose spectral distribution is perfectly matched by a black-bodyat 2.726K. This phenomena was predicted by George Gamow in 1948 and observedserendipitously by Arno Penzias and Robert Wilson in 1964 when measuring the noiseof a microwave antenna they had built. It was higher than they had expected as theywere actually detecting the 3K radiation of the Universe. This background radiation isthe most striking evidence for the existence of the Big-Bang. According to this scenario,the Universe began as a big explosion of matter and radiation. At the beginning lightand matter interacted continuously and were in thermal equilibrium (as they are in theSun’s core). But then as the Universe expanded it cooled. When it reached a temper-ature of ∼ 3000K Hydrogen atoms started to form that could not absorb non-resonantlight: radiation decoupled from matter. At that point the radiation spectrum was that of ablack body at the temperature of decoupling. It is the relics of that original radiation thatwe are observing today as an isotropic cosmic background radiation. Let us see why itexhibits a black-body spectrum at a temperature of 2.7K. Since once hydrogen atoms formed radiation largely stopped to interact with matter,the number of photons at frequency ν (see Eq.2.3): a3 dN p remained constant. But as theuniverse continued to expand to a size a > a so did the radiation wavelength (recallthat in a box k = 2π/λ is a multiple of π/a), i.e. the frequency of the radiation decreased
  11. 11. 12 QUANTUM MECHANICSby the expansion factor αe = a /a. So that the energy density du of the backgroundphotons at frequency ν obeys now: 8πν2 1 (a )3 du = hν a3 dN p = hν a3 3 ehν/kB T − 1 dν c 8πν 2 1 = (a )3 α−3 hν α3 3 hν /k (T/α ) e e dν (2.6) c e B e − 1which is the energy density of a black-body at a temperature T , smaller than the tem-perature at decoupling T by the expansion factor αe : T = (T/αe ): 8πhν 3 1 du = 3 hν /kB T − 1 dν c eBecause the Universe expanded by a factor αe ∼ 1100 since the decoupling time, oneobtains a current temperature for the background radiation of T = 2.72◦ K. The preciseagreement on the value of that temperature is not very important as is the observationthat the cosmic background radiation is the best Black-Body ever observed. It is alsohighly isotropic in the rest frame of the Universe. As our galaxy the Milky Way movesat about 600 km/sec with respect to the Cosmic background, the Doppler effect red-shifts the radiation in one direction and blue-shifts it in the opposite one. This effect canbe subtracted from the measured distribution of radiation intensities. One also needs tosubtract the contribution from the stars in the galaxy (which fortunately emit at muchhigher frequencies, in the visible mostly). The measured variations in the temperature ofthe Universe at different angular positions are then smaller than 10−5 K, yet these smallfluctuations served as the nucleation points for the galaxies and can account for theirobserved distribution, see Fig.2.3. As E.Wigner wrote it is a ”miracle ... that we neitherundersand nor deserve” that a theory devised to explain (approximatively) the radia-tion of hot bodies has turned out to provide such an amazingly precise and powerfulldescription of the Universe!2.1.2 The photo-electric effectBesides the emission spectrum of atoms and the black body radiation, an other ex-periment stood in apparent contradiction with Maxwell’s electromagnetic theory: thephoto-electric effect which observed that electron were emitted from a conducting ma-terial with an energy that depended on the color (the frequency) of the radiation not onits intensity. This was at odds with Maxwell’s electromagnetic theory that asserted thatthe energy of radiation was related to its intensity (see Eq.??) not its frequency! Ein-stein knew the solution for Black-body radiation for which Planck had to assume thatthe radiation emitters in the walls’ cavity could only emit light in small quantas. In 1905Einstein went further and assumed that all light actually comes in small bunches, pho-tons, which energy is proportional to their frequency: E = hν. When such a photon isabsorbed by an electron its energy is used to tear the electron from the binding potentialΦ of the material and move it at velocity v: 1 2 hν = mv + Φ (2.7) 2
  12. 12. THE PUZZLES OF MATTER AND RADIATION 13F. 2.3. The temperature of the cosmic microwave background measured across the sky by the COsmic Background Explorer (COBE) satellite. The top image is the raw data which is red/blue shifted due to the movement of our galaxy through the universe at ∼ 600km/s. Correcting for this Doppler shift yields the middle image which is still ”polluted” at the equator by the light emitted from the stars in our galaxy. Substracting that measurable emission yields the bottom image where the temperature fluctuations of the microwave background across the Universe are as small as 10µ◦ K. These small fluctuations nonetheless served as nucleation points for the formation of the stars and galaxies shortly after the decoupling time.Hence an electron can only be observed if light of high enough frequency is used toremove it from the material. The kinetic energy of the electron increases then linearlywith the illumination frequency. The current emitted is however proportional to thenumber of adsorbed photons, i.e. to the light intensity. At the time Einstein proposalwas revolutionary since it assumed that energy came in discrete packets that could not beinfinitely divided and it appeared to contradict Maxwell’s equations. It took 16 years andconfirming experiments to establish the validity of his model, for which he got the Nobelprize in 1921 (and not for his more profound and revolutionary theories of relativityand gravitation). Incidently from Einstein’s relation between energy and momentum:E 2 = (mc2 )2 + (pc)2 (see Appendix) one deduces that if the energy of the photon (ofmass m = 0) is quantized so must its momentum be: p = E/c = h/λ ≡ k ( ≡ h/2π).
  13. 13. 14 QUANTUM MECHANICS Einstein’s understanding of the photo-electric effect has had enormous technologi-cal impact. All digital cameras are based upon it. These CCD (Charge Coupled Device)cameras consist of an array of small capacitors (a few micron in size) each defininga pixel (= picture element). When light (with frequency in the infrared or higher) im-pinges on a given pixel it kicks off an electron from one side of the capacitor to the otherand charges it with an amount which is proportional to the light intensity. The chargesin a given row of capacitors are then read out by transfering them from one capacitor tothe next along the line like in a ”bucket brigade”. Thus is the image read and stored. Bycovering the array of pixels with a mask-array that filters different colors (Red, Green orBlue) the device can be transformed into a color camera where adjacent pixels respondto different colors. Similarly all of today’s solar cells are based on the photo-electric effect using lightto generate a current by transfering electrons in a semiconductor material from the so-called valence band (and leaving a positively charge ”hole” behind) into the conductionband (on which more below).2.1.3 Bohr’s atomFollowing on the footsteps of Planck and Einstein who proposed that energy and mo-mentum were quantized: E = n ω and p = n k, Niels Bohr suggested in 1913 thatthe angular momentum of electrons in an atom was similarly quantized: L ≡ mvr = nthereby explaining their paradoxical stability (see above). Indeed given the balance ofelectrostatic and centrifugal forces: mv2 /r = e2 /r2 and the assumed quantization of an-gular momentum one deduces that in the hydrogen atom the orbits of the electron arequantized with a radius: r = n2 2 /me2 ≡ n2 r0 (r0 ≡ 2 /me2 = 0.53Å is known as theBohr radius of Hydrogen), velocity v = e2 /n and energy: 1 2 e2 me4 e2 1 13.6eV En = mv − =− 2 2 =− =− (2.8) 2 r 2 n 2r0 n2 n2Thus the energy to ionise a hydrogen atom, i.e. kick off its electron from its ground stateat n = 1 is 13.6 eV. Because of energy quantization an electron orbiting the nucleus willnot radiate continuously, but emit (or absorb) radiation in quantas of energy: 1 1 hνnm = Em − En = 13.6eV( 2 − 2) (2.9) n mHence the emission or adsorption spectra of atoms consists of discrete lines correspond-ing to electronic transitions between states with different quantum numbers. For the hy-drogen atom only the lowest energy transitions to n = 2 (the so-called Balmer series) liein the visible range with wavelengths in the red: 656.3nm (m = 3); in the blue 486.1nm(m = 4) and in the UV range: 434.1nm (m = 5) and 410.2nm (m = 6), see Table 2.1 andFig.2.1. The explanation of the stability of atoms and the emission lines of hydrogenwas a major success of the Bohr model for which he was awarded the Nobel prize in1922.
  14. 14. THE PUZZLES OF MATTER AND RADIATION 15 m= 2 3 4 5 6 7 8 Lyman series (n=1) 121.6 102.6 97.2 94.9 93.7 93.0 92.6 Balmer series (n=2) - 656.3 486.1 434.1 410.2 397.0 388.9 Pashen series (n=3) - - 1870 1280 1090 1000 954Table 2.1 The major emission lines in the hydrogen atom. The wavelength (in nm) istabulated for various values of the initial (m) and final state (n) Bohr’s approach to the hydrogen atom was generalized in 1915 by Arnold Som-merfeld who proposed that for any bound particle (atom, harmonic oscillator, etc.) aquantity known in classical mechanics as the action was quantized: p · dq = nh (2.10)where p, q are the momentum and coordinate of the particle. The Wilson-Sommerfeldquantization rule, Eq.2.10, reduces to Bohr’s quantization of the angular momentum inthe case of the hydrogen atom since the integral pdq = 2πmvr. But the same rule alsoexplains why the oscillators of frequency ω assumed by Planck to exist in the walls ofa back-body would emit radiation in quantas of energy ω. For a harmonic oscillator √with frequency ω = k/m, the energy is: p2 kq2 p2 mω2 q2 Eosc = + = + 2m 2 2m 2from which one derives: qm qm pdq = 2 2mEosc − (mωq)2 = 2mω q2 − q2 dq = πmωq2 m m −qm −qmwith q2 ≡ 2Eosc /mω2 . Hence Eq.2.10 implies: Eosc = n ω, which is the assumption mmade by Planck. The Wilson-Sommerfeld quantization rule can also be used to find the energy levelof a quantum rotator rotating at frequency ω about one of its major axes with momentof inertia I . In that case the angular moment Iω = l (l = 0, 1, 2, ...) and the angularenergy is El = Iω2 /2 = 2 l2 /2I. In 1924, Louis de Broglie (in his Ph.D thesis!) generalizing Sommerfeld’s idea sug-gested that all matter are described by waves. So, just as a photon possesses a momen-tum p = k, thus an electron is characterized by wavevector: k = p/ . The Wilson-Sommerfeld rule is thus equivalent to the requirement that the electron wave in a boundsystem interfers constructively to form a standing wave. As we shall see in the followingde Broglie’s analogy opened the way for the formal development of quantum mechan-ics from its analogy with optics: classical mechanics becoming to quantum mechanicswhat geometrical optics is to electromagnetic waves.
  15. 15. 16 QUANTUM MECHANICSF. 2.4. (A) The absorption of radiation by an atom in its ground state. (B) the stimu- lated emission of a photon in presence of radiation by an atom in its excited state: notice that this process is the time reversal of adsorption. (C) The spontaneous emis- sion of a photon (in absence of radiation) by an atom in its excited state.2.1.4 Adsorption, stimulated emission and the laserBased on the Bohr-Sommerfeld model, Einstein proposed in 1917 a simple theory oflight-matter interaction which could account for Planck’s formula and would be (40years later) the basis for the invention of the laser. First he pointed out that since micro-scopic processes are reversible the adsorption of a photon is indistinguishible from theprocess of stimulated emission, see Fig.2.5. In other words in presence of an externalelectro-magnetic field the transition rate B21 to state 2 from 1 should be the same as thetransition rate B12 to state 1 from 2. The transition being due to the interaction betweenradiation and matter, the overall rate T i j (i, j = 1, 2) will depend on the spectral densityof radiation at the transition energy ρ(ν = ∆E/h) and on the density of states 1 and2: n1 = N1 /V and n2 = N2 /V (where Ni is the number of atoms in state i and V thevolume): T i j = Bi j ρN j /V. Einstein also recognized that in abscence of interaction withthe external field, an atom in an excited state 2 could spontaneously return to the groundstate 1 by emission of a photon of energy hν. That process depends on the lifetime τ sof the excited state and occurs at a rate A12 = 1/τ s . To summarize in steady state thetransition rates to and from each state should balance: B21 ρN1 /V = B12 ρN2 /V + A12 N2 /VFrom which since B12 = B21 we derive: A12 /B12 ρ(ν) = N1 /N2 − 1Since at thermal equilibrium the probability of being in the excited state is smaller thanin the ground state by the Boltzmann factor N2 /N1 = exp(−∆E/kB T ) one obtains: A12 1 8πhν3 1 ρ(ν) = = B12 e hν/kB T − 1 c3 ehν/kB T − 1
  16. 16. THE PUZZLES OF MATTER AND RADIATION 17F. 2.5. Principle of operation of a laser. An amplifying medium is pumped by an external energy source (e.g. a flash lamp) to generate a higher density of excited states than of ground states (population inversion). The medium is placed in a cavity with reflecting mirrors, one of which lets a small fraction of the light out. The light reflected in the cavity is amplified by the stimulated emission of the excited states. When the threshold for lasing is achieved the losses in the cavity are balanced by the gain from the amplifying medium.with the identification: A12 /B12 = 8πhν3 /c3 or: c3 λ3 B12 = 3τ = 8πhν s 8πhτ s Einstein model could account for Planck’s Black-body radiation if atoms capableof absorbing radiation at all frequencies are uniformly present. It also made possibledecades later the invention of the laser, acronym for Light Amplification by StimulatedEmission of Radiation. A laser consists of a light amplifying medium inside a highlyreflective optical cavity, which usually consists of two mirrors arranged such that lightbounces back and forth, each time passing through the gain medium, see Fig.2.5. Typi-cally one of the two mirrors is partially transparent to let part of the beam exit the cavity.To achieve light amplification the excited state in the medium of a laser cavity emittingat frequency ν0 = ∆E/h has to be more populated than the lower energy state. Since atthermodynamic equilibrium low energy states are always more populated than higherenergy ones, energy must be injected in the medium to ”pump” (i.e. excite) atoms fromthe ground state into the light emitting state. Let us therefore consider light of intensity I(z, ν) = ρ(z, ν)c (0 < z < l) propagatingin a cavity of length l and cross section S . Due to stimulated emission of power dPemitin a volume dV = S dz the increase in the light intensity is:
  17. 17. 18 QUANTUM MECHANICS dI = dPemit /dV = hν0 (n2 − n1 )B12 ρ(ν)δ(ν − ν0 ) dz c2 = (n2 − n1 ) I(z, ν)δ(ν − ν0 ) (2.11) 8πν0 τ s 2For various reasons that we shall discuss later, the transition frequency between twostates is never infintely sharp and one replaces the δ-function in the preceeding equationby a function g(ν) which approximates it: g(ν)dν = 1. Quite often the response g(ν)of a resonant oscillator is appropriate (see Appendix on Fourier transforms): γ/π g(ν) = (ν − ν0 )2 + γ2One then obtains: dI c2 = (n2 − n1 ) g(ν)I(z, ν) ≡ γ(ν)I(z, ν) dz 8πν0 τ s 2As argued above, the light intensity grows exponentially when the population of atomsin the medium is inverted, i.e. when n2 > n1 . As light propagates in the cavity it sufferlosses (due to adsorption for example) of magnitude α cm−1 . Since light has to comeout of the cavity there are also losses due to the fact that only a portion R of the intensityis reflected back into the cavity. For a laser to operate the losses must equal the gain:R exp[(γ(ν) − α)l] = 1, which imply that the population inversion at threshold has tosatisfy: 8πτ s 1 (n2 − n1 )t = 2 (α − ln R) λ g(ν) lHence the longer the wavelength the smaller the required population inversion for las-ing. That is one of the reasons that masers (lasers in the microwave range) were thefirst to be invented while X-ray lasers, even though of great utility, have been difficultto develop. During steady-state laser operation the balance of losses and gain imply thatthe population inversion remains at threshold. The more the ground state is pumped, themore the excited state is induced to emit by the increased light density in the cavity, thuskeeping the difference between the density of the two states fixed at its threshold.
  18. 18. QUANTUM MECHANICAL FORMALISM 192.2 Quantum Mechanical formalismThe early 20th century investigations by Planck, Einstein, Bohr, Sommerfeld, de Broglie, etc. revealed a picture of matter that was different from the infinitely divisible contin-uum of energy and momentum that prevailed untill then. Many of the properties ofmatter could be explained by assuming that these quantities were discrete rather thancontinuous. Thus could Planck explain the radiation spectrum of black-bodies, Einsteinthe photo-electric effect and Bohr the stability of atoms and the emission spectrum ofhydrogen (though not of other elements). These early efforts suggested that matter andradiation shared similar properties: light came as photons, particles of zero mass butpossessing definite energy and momentum. Similarly electrons had wave-like proper-ties and could interfere with themselves, as in the orbitals of Bohr’s atom. What wasmissing was a conceptual framework that would unite these observations and modelswith classical mechanics. The breakthrough came with the works of Werner Heisenbergand Max Born in 1925 and Erwin Schroedinger in 1926. The later in particular wrote anequation for the probability of finding a particle at a given position that was inspired bythe analogy pointed out by de Broglie between waves and particles. As we shall see laterthe eigenvalues of the famed Schroedinger equation yield the energy levels of a boundsystem, much as one determines the resonant modes of electromagnetic radiation in acavity (in both cases one solves a Helmholtz equation, see Appendix). Heisenberg proposed a matrix formulation of Quantum Mechanics that was latershown by Schroedinger to be equivalent to his own formulation. Heisenberg’s approachhowever inspired the mathematically rigorous and clear formulation of Quantum Me-chanics presented in 1930 by Paul A.M.Dirac in his landmark book (”the Principlesof Quantum Mechanics”, Clarendon press, Oxford). In the following we shall followDirac’s lead. According to Dirac, a physical system is characterized by its state |Ψ > (also calledwave-function by Schroedinger). These states are complex unit vectors in a so-calledHilbert space, defined so that their scalar product < Ψ|Ψ >= 1. When an observableO is measured, the system is perturbed and ends up in an eigenstate |n > of O witheigenvalue On : O|n >= On |n > (in linear algebra the eigenstates are the vectors thatdiagonalize the matrix O). As in linear algebra one can expressing the vector-state |Ψ >in terms of the eigenvectors |n > of O: |Ψ >= αn |n > (2.12) nSince |Ψ > is a complex vector its conjugate is: < Ψ| = n α∗ < n|. The physical ninterpretation of the amplitudes αn is at the core of Dirac’s formulation: the probabilityof observing a system in state |n > after the measurement of O is: |αn |2 . So that theprobability of measuring a value On when the system is in state |Ψ > is: P(On ) = | < n|Ψ > |2 = |αn |2 (2.13)
  19. 19. 20 QUANTUM MECHANICSNotice that |Ψ > being a unit vector: n |αn |2 = 1 as it should for |αn |2 to be interpretedas a probability. This is the physical interpretation of the wave-function and the intrinsicindeterminism of QM that so annoyed Einstein. The average value of O in state |Ψ > is: < Ψ|O|Ψ > = (< n|α∗ n )O(αm |m >) n,m = Om α∗ n αm < n|m > (2.14) n,m = |αn |2 On = P(On )On =< O > (2.15) n nwhere we assumed the eigenstates to be orthonormal < n|m >= δnm . Quantum me-chanics in Dirac’s formulation is thus reduced to linear algebra: the states are complexvectors and the observables complex matrices with real eigenvalues, i.e. Hermitian ma-trices satisfying Amn = A∗ . If the eigenstates of one operator (observable) A are also nmthe eigenstates of an other operator B then A and B commute: AB|n >= Abn |n >= an bn |n >= bn an |n >= bn A|n >= BA|n >If the operators commute they can be both measured simultaneously: they are diagonal-ized by (i.e they share) the same eigenstates. If on the other hand the eigenstates of Aand B are not identical, their simultaneous measurement is not possible. Let {|n >} bethe eigenvectors of A, then BA|n >= Ban |n >= an B|n >= an |m >< m|B|n >= an Bmn |m > m mOn the other hand: AB|n >= A|m >< m|B|n >= am Bmn |m > an Bmn |m >= BA|n > m m mhence the operators do not commute [A, B] ≡ AB − BA 0. We shall see later thatthe non-commutability of operators (which is quite common with matrix operations)is at the core of Heisenberg uncertainty principle which states that the position andmomentum of a particle cannot be simultaneously determined. The energy being an important observable in physics, the energy operator (or Hamil-tonian, H) plays an important role in Quantum Mechanics. Its eigenvalues are the pos-sible measured energies of the system (which can be discrete or continuous) and itseigenmodes are like the resonant modes of an oscillator or the specific orbits of theelectron in Bohr’s atom: H|n >= n |n >From Planck and Einstein, we know that there is a relation between energy and fre-quency: n = ωn , so that an eigenstate with given frequency ωn evolves in time asexp(−iωn t) = exp(−i n t/ ). Notice that if the initial state of the system is one of the
  20. 20. SIMPLE QM SYSTEMS 21eigenstates |n >, it remains there with probability P = | exp(−iωn t)|2 = 1. If however theinitial state is |Ψ(0) >= n αn |n > then: |Ψ(t) >= αn e−i n t/ t |n > nFrom which one derives Schroedinger’s equation: ∂ 1 1 |Ψ(t) >= αn n e−i n t/ t |n >= H|Ψ(t) > ∂t i n ior in the more common notation: ∂ i |Ψ(t) >= H|Ψ(t) > (2.16) ∂t This is essentially all of Quantum Mechanics: a definition of physical states as vec-tors in a complex space, measurements as matrix operations on these vector states, theoutcome of the measurements as eigenvalues of those matrices and a description of thetime evolution of the physical states by Schroedinger’s equation. The rest is applicationof this linear algebra formalism!2.3 Simple QM systems2.3.1 The chiral amonia moleculeOur first application of the QM formalism described above will be the amonia moleculeand the amonia maser (the microwave equivalent of the laser discussed earlier). We willconsider here the chiral amonia molecule NHDT (D and T stand for the isotopes of Hy-drogen: Deuterium and Tritium), rather than the achiral NH3 considered by Feynman (inVol3 of his ”Lectures on Physics”). This choice allows us not to care about rotationalmotions and it exemplifies the queer nature of QM better than the achiral molecule.NHDT is a tetrahedron with the four atoms sitting at the four apexes. The moleculepossesses distincts enantiomers, i.e. distinct states |1 > and |2 >, which are mirror im-ages of each other depending on whether the Nitrogen atom is on the right or the leftof the HDT plane, see fig.2.6. These states are not eigenstates of the Hamiltonian asthe Nitrogen in state |1 > can end up in state |2 > by passing through the HDT plane,like a left-handed glove can be transformed into a right-handed one by turning it insideout. Even though this energetically costly transition is classically impossible there is inQM always a small probability for such a process to happen (this tunneling through anenergetically forbidden zone (a wall, see below) is one of the oddities of QM). Due to the symmetry of states |1 > and |2 >, the Hamiltonian of this two-statesystem can thus be written as: E0 −A H0 = −A E0
  21. 21. 22 QUANTUM MECHANICSF. 2.6. The chiral amonia molecule, NHDT consist of a nitrogen atom bound to the different isotopes of hydrogen(H): deuterium(D) and tritium(T). This molecule exist with different chirality: left-handed or right-handed which are mirror images of each other. Since nitrogen is slightly more electrophilic (negatively charged) than the hydrogen isotopes the molecule possesses a small electric dipole moment µ.Its eigenvalues (eigen-energies) are: E I,II = E0 ± A and its eigenvectors are: 0 1 1 |I >= 1 √ |II >= 1 √ 2 −1 2 1You can check that H0 |I >= E I |I > and H0 |II >= E II |II >. In the eigenvector basis the 0 0Hamiltonian is diagonal: E +A 0 H0 = 0 D 0 E0 − AIt is a general result from linear algebra that the matrix of eigenvectors: 1 1 1 Λ= √ 2 −1 1diagonalizes the original matrix: H0 = ΛT H0 Λ. D Notice that the energy eigenstates for this chiral amonia molecule consist of a co-herent superposition of a left- and a right-handed state with probability 1/2! This is aclassically absurd situation akin to Schroedinger’s famous cat paradox, see Fig.2.7. Inthis gedanken experiment he proposed to couple a cat enclosed in a box to a two-stateQM system: the cat is dead if the system is in state |1 > and alive if in state |2 >. Ac-cording to QM, before one looks into the box the cat exists as a superposition of the twostates: dead and alive; just like our chiral amonia molecule is described as a superpo-sition of left and right-handed states. However once a measurement is made the cat iseither dead or alive; just as the chiral amonia molecule is -when observed- either left-or right-handed.
  22. 22. SIMPLE QM SYSTEMS 23F. 2.7. The gedanken (thought) experiment that Schroedinger proposed to test the va- lidity of QM. In a closed box isolated from the external world there is a cat and a radioactive source of say β−particles (electrons). If a particle is emitted and detected by a Geiger counter a poison vial is broken that kills the cat. Otherwise the cat is alive. As the state of the particle is a superposition of bound and emitted particle, one must consider the cat to be in a superposition of a live and dead animal. For Ein- stein that thought experiment demonstrated that QM was incomplete since it gave rise to absurd assumptions. Yet, when isolated from the external world (a tall order for macroscopic systems) all experiments so far are consistent with this ”absurd” superposition of states. So did Einstein ask: how was the cat ”really” before we looked into the box? Heclaimed that the superposition is non-sensical and the cat is either dead or alive. He thenargued that some unknown factors (hidden variables) in the description of the micro-scopic two-state sytem that determines the cat’s observed state results in the QM proba-bilistic and verified predictions. However as mentioned earlier, the experimental viola-tion of Bells’ inequalities suggest that Einstein was wrong and that such ”Schroedingercats” exist as a superposition of dead and alive states, even if we have no clue what thatmeans! We will discuss again that point more quantitatively below. A crucial ingredient enters into the QM picture and it is the measurement process.For a system to exhibit the interference effects resulting from QM state superpositionsit must not interact with the environment. These interactions are like independent mea-surements of the system and they destroy its coherence (e.g. the two-state superposition)by implicating (entangling them with) many external uncontrolled states. Now it is veryeasy for a macroscopic system like a cat to interact with the world outside the box (forexample through the radiation it emits or adsorbs). Hence quantum experiments withlarge objects are notoriously difficult to perform. So far the largest molecule for whichquantum interference effects have been demonstrated is the buckyball: C60 (see below). Since the chiral states are given by:
  23. 23. 24 QUANTUM MECHANICS √ |1 > = ( |I > + |II >)/ 2 √ |2 > = (−|I > +|II >)/ 2 (2.17)if the molecule has left-handed chirality to begin with: |Ψ(0) >= |1 >, it will evolve as: 0 0 e−iE I t/ |I > + e−iE II t/ |II > |Ψ(t) >= √ 2The probability of finding the system with a left-handed chirality (state |1 >) after atime t is: 0 0 |e−iE I t/ + e−iE II t/ |2 P1 (t) = | < 1|Ψ(t) > |2 = = cos2 (At/ ) 4and the probability of finding the system with right-handed chirality is P2 (t) = sin2 (At/ ).When a given molecule is measured its chirality is well defined (either left or right), butmeasurements over many molecules yield the oscillating probability distribution justcomputed. Since cos2 (At/ ) = (1 + cos 2At/ )/2 the oscillation frequency is related tothe difference between the energy levels of the eigenstates ω0 = 2A/ . For NH3 thatfrequency ν0 = ω0 /2π = 24 Ghz is in the microwave range. For NHDT it is slighlylower due to the higher mass of Deuterium and Tritium. The essence of Bell’s inequalities may be grasped from this simple example. Imag-ine that a NHDT molecule prepared in a definite chiral state (|1 > or |2 >) is observeda time δt later so that the probability of finding it in the same state is 99%. If it ismeasured again a time δt later it will be observed in the same state as previously withprobability 99%. One may now ask: what is the probability of observing the systemin its initial state if we look at it a time not δt but 2δt later? If as Einstein believedthe system is at δt in a definite state that is only once in a hundred times differentfrom the initial state, then at 2δt the state of the system would be at worst twice ina hundred times different from the initial state, namely the probability of observingthe system in its initial state would be at worst 98%. However the QM prediction is:P(2δt) = cos2 (2Aδt/ ) = 1 − 4(Aδt/ )2 = 0.96 (since we assumed that P(δt) = 0.99,i.e. Aδt/ = 0.1). The QM mechanical prediction violates the lower bound (the Bellinequality) set by ”realistic” theories which assume that the system is in a definite statewhich we have simply no way of determining, not a ”meaningless” superposition ofleft and right-handed molecules. As mentioned earlier the experimental results (mea-sured not on chiral amonia molecules but some other two-state system) vindicate theQM prediction and rule out the ”realistic” theories.2.3.2 The amonia molecule in a constant electric fieldSince nitrogen is more electrophilic than hydrogen, it tends to be slightly more nega-tively charged than the hydrogen isotopes and the molecule ends up with a permanent
  24. 24. SIMPLE QM SYSTEMS 25electric dipole moment µ, as shown in Fig.2.6. In presence of an electric field E the en-ergy of a dipole is W = −µ · E, (Eq.??). If the electric field is along the x-axis in Fig.2.6,then we expect state |1 > to have higher energy than state |2 >. The Hamiltonian of themolecule in an external electric field is thus: E0 + W −A H= −A E0 − WNotice that in the eigenbasis representation (where the unperturbed Hamiltonian H0 isdiagonal), the perturbed Hamiltonian can be written as: E0 + A 0 0 W H = ΛT HΛ = H0 + δH = D + (2.18) 0 E0 − A W 0 √ √The eigenvalues of H (and H ) are: E I,II = E0 ± A2 + W 2 . Defining tan θ ≡ ( A2 + W 2 −W)/A the eigenvectors of H are: cos θ sin θ |I >= |II >= − sin θ cos θWhen W = 0 one recovers the previous result. When W A: |I > |1 > and |II >|2 >, in which case the enantiomers are also eigenstates. In practice however W =µE A and the energies vary as: E I,II = E I,II ± µ2 E 2 /2A. State |I > (with its dipole 0essentially opposite to the electric field) has higher energy than state |II >. Notice thatwe can write the energies E I,II in the following form (we will see later that this is ageneral result when the diagonal Hamiltonian is perturbed by an amount δH) δH12 δH21 EI = EI + 0 0 0 (2.19) E I − E II δH21 δH12 E II = E II + 0 0 0 (2.20) E II − E I These results suggest a way to separate the eigenstates by passing a beam of amoniamolecules through a strong electric field gradient. This field gradient generates a forceon the molecules: µ2 F I,II = − E I,II = E2 2Awhich separates them: state |I > is deflected to regions of small electric fields, whilestate |II > is deflected to regions of high electric fields. Thus can a sub-populationinversion be generated where high energy amonia molecules are separated from lowerenergy ones. This high energy sub-population can then be used to amplify microwaveradiaton by stimulated emission. The resulting device, known as a maser, was the firstimplementation of a stimulated radiation amplification device and served as the firstatomic clock.
  25. 25. 26 QUANTUM MECHANICSF. 2.8. Principle of operation of a maser. An amonia beam (here the chiral molecule NHDT) is sent trough a slit into a beam splitter that consists of a strong inhomoge- nous electric field. At high field one can separate the different enantiomers which are eigenstates of the energy. The high energy eigenstate (|I >) is sent into a cavity where it goes into the low energy state |II > by emitting stimulated radiation at the resonant frequency ω0 = (E I − E II )/2.3.3 The amonia maser and atomic clocksIn an amonia maser, Fig.2.8, the high energy state |I > selected as described, entersa resonant cavity (tuned to the transition frequency ω0 ). A population inversion of theamonia molecules is thus generated in the cavity and amplification of radiation can beexpected. The emitted radiation stimulated by the presence in the cavity of an externalsource generates a highly coherent microwave beam. To analyse the operation of a maser, let us assume that amonia molecules enter acavity in which they experience a time varying electric field: E = Ee−iωt x. This field ˆcouples with the dipole moment of the molecules to modulate their energy by W(t) =−µEe−iωt . The perturbed Hamitonian in the eigenbasis (|I > and |II >) is then, seeEq.2.18: EI 0 0 W(t) H = H0 + δH(t) = D + (2.21) 0 E II W ∗ (t) 0Looking for a general solution: Ψ(t) >= C I (t)|I > +C II (t)|II >, Eq.2.16 yields: i ∂t C I = E I C I + W(t)C II i ∂t C II = W ∗ (t)C I + E II C II (2.22)Looking for a solution C I,II = αI,II (t) exp(−iE I,II t/ ) yields the following equation forαI,II : i ∂t αI = W(t)e−i(E II −EI )t/ αII = −µEe−i(ω−ω0 )t αII i ∂t αII = W ∗ (t)e−i(E I −E II )t/ αI = −µEei(ω−ω0 )t αI (2.23)At the resonance: ω = ω0 , one obtains:
  26. 26. SIMPLE QM SYSTEMS 27 ∂2 αI,II + Ω2 αI,II = 0 t with Ω = µE/which solution is αI (t) = cos Ω(t − t0 ) and αII = sin Ω(t − t0 ). The probability of being instate |I > (α2 ) or II > (α2 ) oscillates with frequency 2Ω: the molecule go periodically I IIfrom stimulated emission in state |I > to adsorption in state II >. These oscillationsare different from the oscillations between the enantiomers |1 > and |2 > observed atfrequency ω0 in absence of electric field. This oscillation in the probability of observinga specific enantiomer is not associated to emission of any radiation, it results from thefact that the enantiomers are not eigenstates of the Hamiltonian: in absence of a timevarying electric field, if the molecule is in eigenstates |I > or |II > it remains there. If the frequency of the electric field is slightly off resonance ω − ω0 0 and if themolecule remains in the cavity for a short time t 1/Ω we may assume that αI 1and integrate the equation for αII to yield: ei(ω−ω0 )t − 1 αII (t) = iΩ i(ω − ω0 )The probability of transition from state |I > to state |II > is then: sin2 (ω − ω0 )t/2 PI→II (t) = |αII |2 = (Ωt)2 [(ω − ω0 )t/2]2The function sinc x ≡ sin x/x decays rapidly for values of |x| > π. Hence the transitionrate is significant only for frequencies which are very close to resonance: |ω − ω0 | <2π/t. If the molecule remains in the cavity for 1 sec, the relative possible detuning :|ν/ν0 − 1| = 1/ν0 t ∼ 4 10−11 . Only frequencies within that very narrow range can beamplified by stimulated emission. This allows for very high Q resonators, Eq.??, i.e.very precise frequency generators and clocks. Because of the sharpness of the functionsinc2 (ω − ω0 )t/2 one often rewrites the previous equation using the limit 2π lim sinc2 (ω − ω0 )t/2 = δ(ω − ω0 ) t→0 tto obtain the transition rate from state |I > to |II >: dPI→II (t) 2π|δH12 |2 BI,II = = 2πΩ2 δ(ω − ω0 ) = δ(E I − E II − ω) (2.24) dtNotice that if we had assumed the amonia beam to enter the cavity in state |II >, i.e.with αII = 1, then we could have similarly integrated the equation for αI to yield thetransition probability from state |II > to |I >, which comes to be: 2π|δH21 |2 BII,I = δ(E I − E II − ω) = BI,IIwhich was the intuitive assumption Einstein made: the rate of stimulated emission BI,IIis equal to the rate of adsorption BII,I . Since the energies of the excited states are never
  27. 27. 28 QUANTUM MECHANICSperfectly sharp, due to the Heisenberg uncertainty principle that we shall see below anddue to thermal motion that leads to Doppler broadening of the emission lines (see sec-tion??), the δ-function is replaced by the density of states with the appropriate energy:ρ(E) (with E = E II + ω) and the normalisation ρ(E)dE = 1). The amonia maser was the first atomic (or molecular) clock. The present genera-tion of atomic clocks use a microwave transition in Cesium (Cs) as the reference fre-quency for the clock. The atoms are cooled to very low temperatures (µ◦ K) to reducethe Doppler broadening of their emission lines and keep them for as long as possiblein the cavity, usually a Penning trap (see ????). As a result the record for the frequencyprecision of atomic clock is: |ν/ν0 − 1| ∼ 10−16 .2.3.4 The energy spectrum of aromatic moleculesA simple generalization of the two-state system that we considered previously is then-state system consisting for example of a circular chain of n identical atoms such asbenzene C6 H6 (n = 6) around which electrons can hop. If an electron has energy E0when associated with a particular atom |n > and can hop only between nearest neighborsthe Hamiltonian for this system in the basis of the atom’s position |n > is:  E0 −A 0 . . . −A     −A E −A . . . 0       0  H= .   .      .  (2.25)  .    .  .       −A . . . 0 −A E0  The eigenstates of that system obey: H|Ψ >= E|Ψ >, with |Ψ >= n C n |n > Fromwhich we derive the equations: E − E0 C1 + C2 + Cn = 0 A E − E0 C1 + C2 + C3 = 0 A . . . E − E0 C1 + Cn−1 + Cn = 0 ALooking for a solution: Clm = exp i(2πlm/n) where l = 1, 2, ...., n we get: El − E0 = −(ei2πl/n + e−i2πl/n ) AThus the eigen-energies of the system are: El = E0 − 2A cos 2πl/n (2.26) For benzene (n = 6) we have: E6 = E0 − 2A; E1,5 = E0 − A; E3 = E0 + 2A and E2,4 =E0 + A. Energies E1,5,6 which are smaller that E0 are associated to so-called bonding
  28. 28. SIMPLE QM SYSTEMS 29F. 2.9. Band-gap theory of material. (a) If the gap between the valence and conduction band is large (a few electron-Volts) the material is an insulator. (b) Variation of the band-gap energy with wave-vector k. Top curve: energy of electrons Ee ; bottom curve: energy of holes Eh (increases towards the bottom). (c) A semi-conductor which due to thermal excitation or illumination has some electrons in the conduction band and some holes in the valence band. (d) a conductor is a material for which the valence and conduction band overlap or equivalently for which an energy band is not filled.orbitals or wave-functions, whereas energies E2,3,4 > E0 are associated to anti-bondingorbitals, on which we shall have more to say later. Notice that the eigen-state associated √to the maximally bonding orbital (l = 6) is fully symmetric: |Ψ6 >= (1, 1, 1, 1, 1, 1)/ 6,while that associated to the maximally anti-bonding orbital (l = 3)√ antisymmetric to isthe permutation of nearest neighbors: |Ψ3 >= (1, −1, 1, −1, 1, −1)/ Conduction bands in solidsAn interesting generalization of the previous analysis is the case of a long chain ofn atoms a distance a apart. In that case we have: Clm = exp[i(2πl/na)ma]. Since theposition of atom m is x = ma we may write Ck (x) = exp ikx with k = 2πl/na. Like theoscillation modes on a string of length na, the electron’s eigenstates are 1D transversewaves of wavelength λ = 2π/k = na/l. The energy of such a mode is: E(k) = E0 − 2A cos ka (2.27)The energy of the electron is bounded: E0 − 2A < E(k) < E0 + 2A. If each atomcontributes two electrons, these 2n electrons will occuy all the n-energy states (we shallsee later that each state can accomodate 2 electrons) and electron hoping in this energy-band will be impossible. If however the on-site energy E0 can possess discrete values(El as it does indeed, for example in Bohr’s model), the coupling of the n−atoms willgenerate energy bands around each energy value El into which electron may hop. Thisforms the basis of the band-theory of conduction in materials, see Fig.2.9: a materialwill conduct if there are empty states into which electrons can hop. If the low energy
  29. 29. 30 QUANTUM MECHANICSF. 2.10. A p-type semiconductor consists of a material (usually Silicon, Si) doped with an element (such as Aluminium) which having less electrons in its outer shell than Si tend to trap electrons from the lattice, leaving a electron vacancy instead: a hole. This depletes the valence bands of electrons allowing conduction through motion of holes. A n-type semiconductor consists of a material doped with an elec- tron donor, an element (such as Phosphate) which has more electrons in its outer shell than Si. This injects electrons into the conduction band allowing the material to conduct electricity. A pn-junction is formed when p-type and n-type semiconduc- tors are brought in contact. Such a junction can be used as a Light Emitting Diode (LED) when electron from the n-side of the junction recombine with holes from the p-side. The wavelength of emitted light is determined by the energy-gap between the conduction and valence and bands.(valence) band is filled with electrons and the next (conduction) band is empty butmany electron-volts (eV) above it, then the electrons have no states in which to go andthe material is an insulator. If the conduction band overlaps with the valence band thenthe electrons have empty states to hop to and current can flow in the material. Theinteresting and technologically important case is the situation where the band-gap Egbetween the conduction and valence bands is small: Eg 1eV (the gap for Silicon(Si) is: Eg = 1.1eV; for Germanium (Ge): Eg = 0.72eV). In that case electrons can betransfered from the valence to the conduction band by thermal agitation or via the photo-electric effect resulting in a material that can behave either as a metal or an insulatordepending on the external conditions (temperature, voltage, illumination wavelengthand intensity, etc.). In particular the introduction of atomic impurities (doping) that donate electrons to(or accept electrons from) a semiconductor lattice creates a situation where few elec-trons occupy an almost empty conduction band (or few holes (electron vacancies) oc-cupy an almost full valence band). As we shall see later the electrons in an atom oc-cupy certain orbitals or shells around the nucleus. Atoms (such as Phosphate or Ar-senic) that have more electrons in their outer shell than the bulk semiconductor (theso-called majority carrier, usually Silicon) will usually donate an electron to the lat-tice, whereas atoms that have less electrons in their outer shell (such as Boron or Alu-minium) will accept an electron from the lattice. The doping creates so called n- and
  30. 30. MOMENTUM AND SPACE OPERATORS 31p-type semiconductors that have electrons in their conduction-band (n-type) or holes intheir valence-band (p-type), see Fig.2.10. The energy of the electrons moving near thebottom Emin = E0,c − 2Ac of the conduction band (i.e. when ka << 1 in Eq.2.27) is: E(k) = Emin + k /2mc 2 2where the effective mass of the electron in the conduction band is defined as: mc = /∂k E(k) = 2 /2a2 Ac which can be quite different from the mass of a free electron. 2 2Notice that if, as proposed by de Broglie, we identify the momentum of the electronas p = k , then Eq.2.3.5 represents the kinetic energy of an electron with mass mc atthe bottom of the conduction band. Similarly the energy of the hole near the top of thevalence band is: Eh (k) = Emax − 2 k2 /2mvThe movement of holes in the valence band is similar to that of air bubbles in water: asthe moving water displaces the bubble up, the electrons moving in the opposite directionto the hole minimize their energy by displacing it to the top of the valence band. As aresult holes have minimal energy at the top of the valence band: their energy increasesas k increases. Doped semiconductors are the basic ingredients of all the semi-conductor industry(transistors, diodes, integrated circuits, etc.). For example the coupling of p-type andn-type semiconductors, generate a pn-junction which acts like a diode (current flows inonly one direction). It can also be used as a powerful and efficient light source. Electronsin the n-type part of the junction can recombine with holes in the p-type (i.e. transit tothe valence band) with emission of light (just as in the atomic or molecular transitionsdiscussed earlier in the context of stimulated emission and the laser). The advantageof these Light Emitting Diodes (LED) is that by appropriate tuning of the energy gap(appropriate choice of the semi-conducting material) one can tune the wavelength atwhich the LED will emit light. For example in the red and infrared part of the spectrumGalium-Arsenide (GaAs) is the material of choice for LEDs. The intensity of light iscontrolled by the current flowing through the junction. These LED are used in all kindof electronic displays, in new high efficiency spot-lamps and traffic lights, in the remotecontrol of various electronic device, in the laser diode of DVD players, etc.2.4 Momentum and space operatorsIn the example above we considered the case of a QM system that could occupy onlydiscrete states |n >. It is easy to generalize that to free particles that can be found atcontinuous positions |x >. In that case the general wave-function in position (or real)space can be formally written as: |Ψ >= Ψ(x)|x > x
  31. 31. 32 QUANTUM MECHANICSWhere Ψ(x) is the probability amplitude of finding the particle at position |x >, so that: P(x) = Ψ∗ (x)Ψ(x) (2.28)Of course the probability of finding the particle somewhere is one, so that the wave-function Ψ(x) is normalized: dxΨ∗ (x)Ψ(x) = 1 (2.29)The mean position of the particle is: < x >= dxP(x)x = dxΨ∗ (x)xΨ(x) (2.30)For example, the wavefunction can be: 1 e−(x−x0 ) /4σ 2 2 Ψ(x) = (2πσ2 )1/4for which the probability distribution: 1 e−(x−x0 ) /2σ 2 2 P(x) = √ 2πσcorresponds to a gaussian distribution with mean < x >= x0 and standard deviation < (x− < x >)2 > = σ. In Fourier-space (see Appendix) we can write: 1 1 Ψ(x) = √ dkΨ(k)eikx = √ d pΨ(p)eipx/ (2.31) 2π 2πwhere we used de Broglie’s relations: p = k. Conversely: 1 Ψ(p) = √ dxΨ(x)e−ipx/ (2.32) 2πΨ(p) is the wavefunction in the momentum (or Fourier) space: |Ψ >= p Ψ(p)|p >. Forthe example chosen above the momentum wavefunction is: e−ipx0 / ∞ dxe−(x−x0 ) /4σ e−ip(x−x0 )/ 2 2 Ψ(p) = √ 2π (2πσ2 )1/4 −∞ −ipx0 / −p2 σ2 / 2 ∞ e e = √ dxe−(x−x0 +2ipσ/ )2 /4σ2 2π (2πσ2 )1/4 −∞ √ 2σ/ −ipx0 / −p2 σ2 / 2 = e e (2π)1/4Being the Fourier transform of Ψ(x), Ψ(p) satisfies the normalization condition:
  32. 32. MOMENTUM AND SPACE OPERATORS 33 Ψ∗ (p )Ψ(p) 1= dxΨ∗ (x)Ψ(x) = d pd p dxei(p−p )x/ 2π = d pΨ∗ (p)Ψ(p) = d pP(p) (2.33)where we used the identity (see Appendix on Fourier transforms): dxeipx/ = 2π δ(p) (2.34)Eq.2.33 is known in the theory of Fourier transforms as Parseval’s theorem. For a Gaus-sian wavefunction Ψ(x) (see above example) P(p) corresponds to a Gaussian centeredon p = 0 with standard deviation < (p− < p >)2 > = /2σ. The mean value of themomentum satisfies: < p >= d pΨ∗ (p)pΨ(p) = d p d pΨ∗ (p )δ(p − p)pΨ(p)Using the identity Eq.2.34 and Eq.2.31 allow us to express the momentum operator inreal space as: 1 <p>= dxd p d pΨ∗ (p )ei(p−p )x/ pΨ(p) 2π 1 ∂ = dx d p Ψ∗ (p )e−ip x/ dp Ψ(p)eipx/ 2π i ∂x ∂ = dxΨ∗ (x) Ψ(x) ≡ dxΨ∗ (x) pΨ(x) ˆ i ∂xwhere we identify the momentum operator in real space as: ∂ p= ˆ ≡ −i ∂ x (2.35) i ∂xSimilarly by writing the mean position < x > in momentum space we come to identifythe position operator in momentum space as: ∂ x=i ˆ ≡ i ∂p (2.36) ∂pNotice that momentum and space-operators do not commute. In real space: < x|[x, p]|x >= dxΨ∗ (x)(x p − px)Ψ(x) = ˆ ˆ dxΨ∗ (x)[x∂ x Ψ − ∂ x (xΨ)] = i iWe would have obtained the same result by computing the commutator in momentumspace: < p|[x, p]|p >= d pΨ∗ (p)( x p − p x)Ψ(p) = i . Since two observable cannot ˆ ˆshare the same eigenstates if they do not commute, the position and momentum of aparticle cannot be simultaneously determined with absolute precision. The same resultholds also for the commutator of time and energy: < x|[H, t]|x >=< x|Ht − tH|x >= dxΨ∗ (x)(i ∂t t − ti ∂t )Ψ(x) = i
  33. 33. 34 QUANTUM MECHANICS2.4.1 Heisenberg uncertainty principleFrom the result that momentum and space do not commute we can derive the Heisenberguncertainty principle. Consider the action on state |ψ > of the operator xo +iλpo where λis a number and the operators xo and po are the deviation of the position and momentumoperators from their mean: xo = x− < x > and po = p− < p >: |φ >= (xo + iλpo )|ψ >Since [xo , po ] = [x, p] = i the positiveness of the probability implies that: 0 ≤ < φ|φ > = < ψ|(xo − iλpo )(xo + iλpo )|ψ > = < ψ|xo + iλ[xo , po ] + λ2 p2 |ψ > 2 o = < ψ|xo |ψ > − λ+ < ψ|p2 |ψ > λ2 = P2 (λ) 2 oFor the quadratic polynomial P2 (λ) to be non-negative for any real λ, its determinanthas to satisfy: 2 − 4 < ψ|xo |ψ >< ψ|p2 )|ψ >≤ 0 2 oOr in terms of the position and momentum variables: 2 < ∆x2 >< ∆p2 > ≥ (2.37) 4This is Heisenberg’s principle: it sets a limit on the precision with which one can mea- √sure both the position δx = < ∆x2 > and the momentum δp = < ∆p2 > of a physi-cal system. The smallest uncertainty (the equality in Eq.2.37) is obtained for a Gaussianprobability distribution as can be verified from the example worked out above. Sincethe Hamiltonian and time operators do not commute similar uncertainty relation can beobtained for the energy and time uncertainties: 2 < ∆E 2 >< ∆t2 > ≥ (2.38) 4 Notice that Heisenberg principle is a direct mathematical consequence of the QMdescription of physical systems by a complex wave-function Ψ(x) (Eq.2.13) and of deBroglie’s relation between wavelength and momentum. Heisenberg uncertainty princi-ple is a tautology: a consequence of the definition of Fourier transforms (see Appendix).In the context of communication it has been known for a long time: a very short timesignal is spread over a very large frequency spectrum. In the context of optics we havealso encountered it in the diffraction pattern from a hole which is larger the smaller thehole is.
  34. 34. SCHROEDINGER’S EQUATION 352.5 Schroedinger’s equationWe have determined the representation of the position operator in momentum space andof the momentum operator in real space. Note that in its eigenspace the momentumspace p is diagonal, i.e. it is a number. We can now write the representation of the ˆHamiltonian in any of these Hilbert spaces. It is often easier to work in real space, inwhich case the Hamiltonian which is the sum of kinetic and potential energy is writtenas: p2 ˆ 2 2 H= + V(x) = − (∂2 + ∂2 + ∂2 ) + V(x) = − x y z 2 + V(x) (2.39) 2m 2m 2mand Schroedinger’s equation, Eq.2.16 can be recast as: ∂ 2 i Ψ(x, t) = − 2 Ψ(x, t) + V(x)Ψ(x, t) (2.40) ∂t 2mMultiplying Eq.2.40 by Ψ∗ , its complex conjugate by Ψ and subtracting the two yieldsa conservation law for the probability distribution P(x) = |Ψ(x)|2 : 2 i ∂t P = − (Ψ∗ 2 Ψ−Ψ 2 Ψ∗ ) 2m 2 = · (Ψ∗ Ψ − Ψ Ψ∗ ) 2mwhich can be recast in the usual form (see Eq.?? for the charge distribution in EM): Ψ∗ vΨ + c.c. ˆ ∂t P + ·J=0 with J = Ψ∗ Ψ + c.c. = (2.41) 2im 2where c.c stands for the complex conjugate and v = ˆ /im is the velocity operator.Eq.2.41 is a very important self-consistency check of Quantum Mechanics, since forP(x) to be interpreted as a probability distribution it must satisfy a conservation law.This equation expresses the intuitive expectation that the change in the probability offinding a particle at a given position is equal to the particle flux gradient.2.5.1 Diffraction of free particlesIf the potential is null V(x) = 0 then the eigen-solutions of Schroedinger’s equation,Eq.2.40: 2 dΨ i =− 2 Ψ (2.42) dt 2mare plane waves: Ψ(x, t) = eik·x−iEk t/with p = k and Ek = k /2m. These plane-waves are eigenmodes of the momentum 2 2operator: pΨ(x, t) = −i ˆ Ψ(x, t) = p Ψ(x, t)
  35. 35. 36 QUANTUM MECHANICSF. 2.11. The double-slit or Young’s experiment. (a) a wave passing through two slits in a screen generates two wave-sources which interference creates on a far-away screen a pattern of interference consisting of alternating minima and maxima of intensity. (b) a particle in state |O > impinging on a double slit generates two states |I > and |II > corresponding to its passage through slit 1 or 2. The phase of these states evolves as exp ikl. If their coherence is maintained they can interfere on a screen a large distance z from the slits, generating an oscillating pattern related to their phase difference: φint = k(l2 − l1 ) = kd sin θ. (c) Observation of the interference pattern of electron passing through a double slit and impinging on a camera. Each electron is observed as a particle (white dot) with a well defined position on the camera. The QM interference pattern is only visible when a sufficiently large number of particles has been observed (A.Tonomura, Proc.Natl.Acad.Sci. 102, 14952 (2005). One of the most striking confirmations of the QM mechanics picture is the obser-vation of a diffraction pattern like the one seen with electro-magnetic radiation whena free particle is passed through one or a few slits (see ??? and Fig.2.11(a)). Let theparticle be in an eigenstate |0 >= Ψ(x, t) = eikz−iEk t/as it impinges on a screen that is absoring except for two apertures of size a a distanced apart. In the far-field, i.e. at distances z d2 /λ the wave amplitude is given byHuygens’s principle, Eq.??: eik(z+(x +y )/2z 2 2 Ψdi f f (x, y, z) = dx dy e−i(kx x +ky y ) (2.43) iλzWhere k x = kx/z and ky = ky/z. Thus the probability of detecting a particle on a screena distance z from the slits is: 4a2 |Ψdi f f |2 = sinc2 k x a cos2 k x d/2 (2.44) (λz)2where sinc x ≡ sin x/x. If the distance between the diffraction slits is much larger thantheir width (d a), the probability oscillates with a period: δx = λz/d. While each