class 1. quantum mechanics

8/15/2019 Class 1. Quantum Mechanics

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Class 1. Quantum

mechanics: IADr. Marc Madou Chancellor’s Professor

UC Irvine 2012

Schrödinger's Cat could not cope

with a lifetime of uncertainty.


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In the IC and !M" #orld$ #e are movin% fast into the realmof &uantum mechanics. Moore’s la# mi%ht remain validuntil a'out 2020 'ut '( then the scale of electroniccom)onents #ill 'e at the molecular*atomic level$ andhence can no lon%er 'e descri'ed '( classical mechanics.

Quantum com)utin%$ nanotechnolo%($ includin%nanotu'es$ nano#ires$ 'iolo%ical nano+structures and&uantum dots$ all re&uire some %roundin% in &uantummechanics to 'e understood at all. Quantum mechanicsmust no# 'ecome a familiar tool not onl( to )h(sicists 'ut

also to materials scientists$ 'iolo%ists and electrical$mechanical and 'ioen%ineers.


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ContentsClassical ,heor( "tarts-alterin%

Quantum Mechanics tothe escue"chr/din%er’s !&uation


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Paul Drude(186 !1"#6$. Classical ,heor( "tarts

-alterin%

Modern condensedmatter physics reallystarted with the discoveryof the electron by J.J.Thompson in 1897.Drude describes electrons

in a metal as a freeelectron gas (FEG) toexplain electricalconductivity and Ohm’slaw with: ρ = m e

ne 2τ


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ρ = 1σ

=m ev thne 2λ

= m ene 2a

8k BTπ m e

Classical ,heor( "tarts

-alterin%Using the lattice constant, a,for the mean free path, and theMaxwell-Boltzmann-Equationat T=300K to calculate v th,values for the resistivity of a

metal are obtained that are sixtimes too large.Temperature dependence iswrong too. Expected was a √Tbut what we get

experimentally is:

ρ = ρ 0 + α T


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-alterin%Since solids contain a numberof atoms and electrons with

similar density, why the largeconductivity differences?More intriguing yet, based onthe above why would carbonallotropes come with such

highly varying electricalconductivities?


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-alterin%Drude using classical values forthe electron velocity v dx and

heat capacity C v,el , somehow gota number very close to theexperimental value. But howlucky that Drude dude was: by a

tremendous coincidence, theerror in each term he made wasabout two orders ofmagnitude…. in the oppositedirection %iscalculation


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For most metals the Hall constant is negative. But for Be andZn, for example, R H is positive. Of course energy bands

were not heard of yet and the results of a positive R H werebaffling at the time: how can we have q > 0 (even formetals!?).


-alterin%


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A blackbody of temperature T emits a continuousspectrum peaking at λ max . At very short and verylong wavelengths there is little light intensity, withmost energy radiated in some middle rangefrequencies. As the body gets hotter the peak of thespectrum shifts towards shorter wavelengths.Classical interpretation predicted somethingaltogether different; in the classical Rayleigh-Jeansmodel, instead of a peak in the blackbody radiationand a falling away to zero at zero wavelength themeasurements should go off scale at the shortwavelength end -towards “ultraviolet catastrophe”Max Planck took the revolutionary step that led toquantum mechanics :E = h ν


-alterin%

6.6&6 1#!

).s


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Planck, interestingly, never appreciated how farremoved from classical physics his work was. ThePlanck constant h, was a bit like an ‘uninvited guest’ ata dinner table; no one was comfortable with this newguest. Discontinuities in the nanoworld are meted out inunits based upon this constant. It is the underlyingreason for the perceived weirdness of the nanoworld;the existence of a least thing that can happen quantity-aquantum. The ubiquitous occurrence of discontinuitiesin the nanoworld, constantly upsets our common-senseunderstanding of the apparent continuity of themacroscopic world.


-alterin%

%a Planc* +pril & , 18-8 /cto0er , 1"


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Photoelectric effect; no electrons are ejected, regardless of theintensity of the light, unless the frequency exceeded a certainthreshold characteristic of the bombarded metal (red light did notcause the ejection of electrons, no matter what the intensity).The photoelectric phenomenon could not be understood without theconcept of a light particle, i.e., a quantum amount of light energy.

Einstein's 1905 paper explaining the photoelectric effect was one ofthe earliest applications of quantum theory and a major step in itsestablishment.The remarkable fact that the ejection energy was independent of thetotal energy of illumination showed that the interaction must be likethat of a particle which gave all of its energy to the electron!

Einstein reintroduced a modified form of the old corpuscular theoryof light, which had been supported by Newton but which was longabandoned.


-alterin%


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The electron charge wasdetermined by Robert Millikanin 1909 and with that valueand the slope of the lines avalue for h of 6.626 x 10 -34 J.scan be calculated, identical tothe one derived from thehydrogen atom spectrum andblackbody radiation (seeabove)


-alterin%


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The young Einstein, in 1905, was thefirst scientist to interpret Planck’s

work as more than a mathematicaltrick and took the quantization oflight (E=h ν ) for physical reality. Hegave the uninvited dinner guest -thePlanck constant h-a place at thequantum mechanics dinner table.What Einstein proposed here wasmuch more audacious than themathematical derivations by Planckto explain the UV catastrophe away.


-alterin%


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-alterin%The three experiments thatmade the quantum

revolution, Black-bodyradiation, the photo-electriceffect and the Comptoneffect all indicate that lightconsists of particles.

+rthur 2arry Compton (18"&!1"6&$.


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Einstein’s specialrelativity allows one to

calculate the momentum,p, of a photon startingfrom:

De Broglie, while studying forhis PhD in Paris in 1924,

postulated that this lastequation also applied to amoving particle such as anelectron, in which case λ is the

wavelength of the waveassociated with the movingparticle, i.e., a “matter wave.”

Quantum Mechanics to

the escue

E 2 = (p c) 2 + m 0c2

( )2

E 2 = (p c) 2 + 0 or E = p c

p =Ec

=hν

c=

hλ

λ = hmv


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the escueIn his 1928 tests Thompson junior and Reidobserved interference patterns from electronsreflecting from a thin polycrystalline metal foilsurface.Clinton Davisson and Lester Germer, at Bell

Laboratories, in 1927 found the sameexperimental evidence: a beam of electronsscattered from a single crystal of nickel resultedin a diffraction pattern fitting the Braggdiffraction lawThis established the wave character of electrons,forming the basis of analytical techniques fordetermining the structures of molecules, solidsand surfaces, such as in LEED (low energyelectron diffraction) and SEM.

Da3isson!4ermer e periment (1"& $.


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the escueLouis-Victor, 7 th duc deBroglie (1892-1987)discovered thus that the secret

of Planck’s and Einstein’squanta lay in a general law ofnature i.e., the dual characterof waves and particles.Einstein commented about this

fantastic insight: “de Brogliehas lifted the great veil.”Even buckyballs are wavy


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the escueEinstein gave, what we had come tothink of as a wave (light) a particle

character and de Broglie gave whatwe thought of as a particle (electrons)a wave character. Radiation has wavecharacter and particle character andmatter has particle and wave

character or at the nanoscale, naturepresents itself with a wave-particleduality.

λ = hmv


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the escueThe wave-particle dualityintroduced in the previoussection forced physicists to

reconsider their description ofthe position and momentum ofvery small particles and is at thecore of the Heisenberg’suncertainty principle (HUP).In the nanoworld, theHeisenberg principle states thatthere are physical parameters inquantum physics whose valuescannot be known accuratelysimultaneously.


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the escue ∆px∆x≥2π =

∆ E∆ t ≥ h2π

= h


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The uncertainty about the energy of a particle depends on the time interval ∆tthat the system remains in a given energy state. Importantly this also meansthat conservation of energy can be violated if the time is short enough.

From the uncertainty principles it is possible that empty space locally does nothave zero energy but may actually have sufficient ∆E for a very short time ∆tto create particles and their antiparticles. This can be demonstrated through theCasimir effect.

This effect is also responsible for “lifetime broadening” of spectral lines.

Short-lived excited states (small ∆t) possess large uncertainty in the energy ofthe state (large ∆E). As a consequence, shorter laser pulses (e.g., femto andattosecond lasers) have broader energy (therefore wavelength) band widths.


the escue


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the escueBased on the idea that the 'vacuum' ofspace is actually a seething foam ofquantum fluctuations of differentfrequencies, Casimir proposed that iftwo electrically conducting, but

uncharged parallel plates were mounteda small distance apart in a vacuum, theywould tend to be drawn together. Animportant point is that the plates carryno electrical charge so that anyinteraction between the plates must

come from some other source.Using MEMS devices the Casimir forcehas been measured


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The existence of a Zero-Point–Energy: vibrational energycannot be zero even at T=0Kis also a consequence of theHeisenberg principle. If thevibration would cease atT=0K, then the position and

momentum would both be 0,violating the HUP.


the escue


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Schr ödinger, after attending a seminaron Einstein’s and de Broglie’s ideasthat wavelike entities can behave likeparticles and vice versa, thought thatthere must be a wave equation, Ψ (x,t),to describe particles.Schr ödinger’s picture of the atom hasthe electron standing waves vibrating intheir orbitals much like the vibrationson a string - but in three dimensionsinstead of one. In this figure a twodimensional representation ofSchr ödinger waves, like vibrations on adrum skin, is shown.

"chr/din%er’s !&uation


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Example 2: harmonic oscillators,Hooke’s law (F=-kx)As the amplitude (A) can takeany value, this means that the

energy (E) can also take anyvalue – i.e., energy is continuous.Any energy value is allowed bysimply changing the force

constant k.

d 2xdt 2

= − k m

x

2 mcost snmm =2m 2+



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"chr/din%er’s !&uationIn the late 18th century the mathematician Pierre Simon de Laplace (1749-1827)encapsulated classical determinism as follows: “…if at one time we knew the positionsand motion of all the particles in the Universe, then we could calculate their behavior atany other time, in the past or the future.” In classical physics, particles and trajectoriesare real entities and it is assumed that the universe exists independently from theobserver, that it is predictable and that for every effect there is a cause so experimentsare reproducible.

Heisenberg’s uncertainty principle destroyed all this. In quantum physics the measuredand unmeasured particle are described differently. The measured particle has definiteattributes such as position and momentum, but the unmeasured particle does not haveone but all possible attribute values, as Nick Herbert in his book Quantum Realitywrites …somewhat like a broken TV that displays all its channels at the same time.


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Erwin Schr ödinger (1887-1961), in1926, encouraged by Debye whoremarked that there should be a waveequation to describe the de Brogliewaves, proposed a wave equation thatcan be applied to any physical systemin which it is possible to describe theenergy mathematically. In onedimension he postulated:

∂ 2Ψ (x,t)∂

x2 +

8π 2m

h2 E - V(x,t)[ ] Ψ (x,t) = 0


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"chr/din%er’s !&uationThe first term is the rate of change of the rate of change of the wave function withdistance x. The energy of the particle is E and the potential energy function todescribe the forces acting upon the particle is represented by V(x, t). The Schr ödinger equation has the same central role in quantum mechanics thatNewton’s laws have in mechanics and Maxwell’s equations in electromagnetism.

Solutions to Newton’s equations are of the form v=f(x , t), while solutions to thewave are called wave functions Ψ (x, t).Like Newton’s equation, it describes the relation between kinetic energy, potentialenergy, and total energy. If one knows the forces involved, one can calculate thepotential energy V and solve the equation to find Ψ .Solving the Schr ödinger equation specifies Ψ (x,t) completely, except for a constant;if Ψ (x,t) is a solution then A Ψ (x,t) is a solution as well.

∂ 2Ψ (x,t)∂

x2 +

8π 2m

h2 E - V(x,t)[ ] Ψ (x,t) = 0


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The so-called “Copenhagen Interpretation” of Schr ödinger’s equation is thatthe Ψ (x,t) function is not some physical representation of a physical substanceas in classical physics (e.g., the amplitude of a water wave) but a “probabilityamplitude” of the particle which, when squared, gives the probability offinding the particle at a given place at a given time: | Ψ (x, t)| 2dx = probabilitythe particle will be found between x and x + dx at time t and the wavefunction

itself has no physical meaning.Since the probability that the particle is somewhere must equal one, it holdsthat one can normalize this probability function as:

The Copenhagen interpretation also holds that an unmeasured particle in acertain sense is not real: its attributes are created or realized by the measuringact. Another way of saying this is that a wavefunction collapses uponmeasurement; before measurement a particle is described by a wave functiondescribed by the Schr ödinger equation but upon measuring that particle’swave suddenly and discontinuously collapses .


| Ψ (x,t) | 2−∞

+∞

∫ dx = 1

∂ 2 Ψ (x,t)∂ x 2 +

8π 2 mh 2 E - V(x,t)[ ] Ψ (x,t) = 0


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"chr/din%er’s !&uationBecause (x,y,z,t) is complex and can be positive or negative, it cannot beΨthe probability directly. The Born interpretation of Ψ places restrictions onthe form of the wavefunction:

(a) Ψ must be continuous (no breaks);(b) The gradient of Ψ (d Ψ /dx) must be continuous (no kinks);

(c) Ψ must have a single value at any point in space;(d) Ψ must be finite everywhere;(e) Ψ cannot be zero everywhere.

∂2Ψ (x,t)∂x 2

+8π 2m

h 2E - V(x,t)[ ] Ψ (x,t) = 0


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