..ie where are you most likely to find the particle
Quantum mechanics a brief
Quantum MechanicsQuantum Mechanics
2The Quantum Mechanics View• All matter (particles) has wave-like properties– so-called particle-wave duality• Particle-waves are described in a probabilistic manner– electron doesn’t whiz around the nucleus, it has a probabilitydistribution describing where it might be found– allows for seemingly impossible “quantum tunneling”• Some properties come in dual packages: can’t know bothsimultaneously to arbitrary precision– called the Heisenberg Uncertainty Principle– not simply a matter of measurement precision– position/momentum and energy/time are example pairs• The act of “measurement” fundamentally alters the system– called entanglement: information exchange alters a particle’s state
Spring 2008 3Pre-quantum problems, cont.• Why was red light incapable of knocking electrons out of certainmaterials, no matter how bright– yet blue light could readily do so even at modest intensities– called the photoelectric effect– Einstein explained in terms of photons, and won Nobel Prize
4• Without Quantum Mechanics, we could never have designedand built:– semiconductor devices• computers, cell phones, etc.– lasers• CD/DVD players, bar-code scanners, surgical applications– MRI (magnetic resonance imaging) technology– nuclear reactors– atomic clocks (e.g., GPS navigation)• Physicists didn’t embrace quantum mechanics because it wasgnarly, novel, or weird– it’s simply that the #$!&@ thing worked so well
6Results• The pattern on the screen is an interferencepattern characteristic of waves• So light is a wave, not particulate• But repeat the experiment one photon at a time• Over time, the photons only land on theinterference peaks, not in the troughs– consider the fact that they also pile up in the middle!– pure ballistic particles would land in one of two spots
Classical mechanics is the mechanicsof everyday objects like tables andchairsSir Isaac Newton1. An object in motion tends to stay in motion.2. Force equals mass times acceleration3. For every action there is an equal andopposite reaction.
Classical mechanics reigned as the dominant theory ofmechanics for centuries1687 – Newton’s PhilosophiaeMathematica1788 – Lagrange’s MecaniqueAnalytique1834 – Hamiltonian mechanics1864 – Maxwell’s equations1900 – Boltzmann’sentropy equation
However, several experiments at the beginningof the 20th-century defied explanationThe UltravioletCatastropheThe HydrogenSpectrumThe Stern-GerlachExperimentNewtonian explanations forthese phenomena were wildlyinsufficient?
Classical Physics• Described by Newton’s Law of Motion (17thcentury)– Successful for explaining the motions ofobjects and planets– In the end of 19thcentury, experimentalevidences accumulated showing that classicalmechanics failed when applied to very smallparticles.),...,,(221 Ni iiUmpH rrr∑ +=Sir Isaac Newton
The failures of Classical Physics• Black-body radiation– A hot object emits light (consider hot metals)– At higher temperature, the radiation becomes shorterwavelength (red white blue)– Black body : an object capable of emitting andabsorbing all frequencies uniformly
The failures of classical physics• Experimental observation– As the temperature raised, the peak in theenergy output shifts to shorterwavelengths.– Wien displacement law– Stefan-Boltzmann lawWihelm Wien2max51cT =λ Kcm44.12 =c4/ aTVE ==Ε 4TM σ=
Rayleigh – Jeans law• First attempted to describe energydistribution• Used classical mechanics and equi-partition principle• Although successful at high wavelength, itfails badly at low wavelength.• Ultraviolet Catastrophe– Even cool object emits visible and UVregion– We all should have been fried !Lord RayleighλρddE = 48λπρkT=
Planck’s Distribution• Energies are limited to discrete value– Quantization of energy• Planck’s distribution• At high frequencies approaches the Rayleigh-Jeanslaw• The Planck’s distribution also follows Stefan-Boltzmann’s LasMax Planck,...2,1,0, == nnhE νλρddE =)1(8/5−= kThcehcλλπρkThckThce kThcλλλ≈−++=− 1....)1()1( /
15Let’s start with photon energy• Light is quantized into packets called photons• Photons have associated:– frequency, ν (nu)– wavelength, λ (λν = c)– speed, c (always)– energy: E = hν• higher frequency photons → higher energy → moredamaging– momentum: p = hν/c• The constant, h, is Planck’s constant– has tiny value of: h = 6.63×10-34J·s
Wave-Particle Duality-The particle character of wave• Particle character of electromagnetic radiation– Observation :• Energies of electromagnetic radiation of frequency vcan only have E = 0, h, v 2hv, …(corresponds to particles n= 0, 1, 2, … with energy = hv)– Particles of electromagnetic radiation : Photon– Discrete spectra from atoms and molecules can be explainedas generating a photon of energy hn .– ∆E = hv
Quantum mechanics was developed to explain these results anddeveloped into the most successful physical theory in history1900 – Planck’s constant1913 – Bohr’s modelof the atom1925 – Pauli exclusion principle1926 – Schrodinger equation1948 – Feynmann’s pathintegral formulationIncreasingweirdness1954 – Everett’s many-worldtheory
Although quantum mechanics applies to all objects, the effects of quantummechanics are most noticeable only for very small objectsHow small is very small?1 meter Looks classical1 micrometer Looks classical1 millimeter Looks classical1 nanometer Looks quantum!
Nonetheless, quantum mechanics isstill very important.How important is very important?Without quantum mechanics:All atoms would be unstable.UniverseexplodesChemical bonding would beimpossible.All moleculesdisintegrateMany biological reactionswould not occur.Life doesnot existNeil Shenvi’s dissertation title:Vanity of Vanities, All is VanityMinimalconsequences
When you start going subatomic or even smaller, things getstrange. That strangeness, however, can lead to some pretty coolinventions.
All information about a system is provided by thesystem’s wavefunction.( )xΨxPr( )xxInteresting facts about the wavefunction:1. The wavefunction can be positive, negative, or complex-valued.2. The squared amplitude of the wavefunction at position x isequal to the probability of observing the particle at position x.3. The wave function can change with time.4. The existence of a wavefunction implies particle-wave duality.
At a given instant in time, the position and momentum of aparticle cannot both be known with absolute certaintyThis consequence is known as Heisenberg’s uncertainty principleClassical particle Quantum particleWavefunction = ψ(x)Hello, my name is:Classical particlemy position is 11.2392…Angmy momentum is -23.1322… m/s“I can tell you my exact position, but then Ican’t tell you my momentum. I can tellyou my exact momentum, but then I can’ttell you my position. I can give you apretty good estimate of my position, butthen I have to give you a bad estimate ofmy momentum. I can…”????
a particle can be put into a superposition ofmultiple states at onceClassical elephant:Valid states:Quantum elephant:GrayMulticoloredGray MulticoloredValid states:+Gray AND Multicolored
Properties are actions to be performed, notlabels to be readClassical Elephant: Quantum Elephant:The ‘position’ of an object existsindependently of measurement and issimply ‘read’ by the observerPosition = hereColor = greySize = large‘Position’ is an action performed on anobject which produces some particularresultPosition:In other words, properties like position or momentum do not exist independent ofmeasurement! (*unless you’re a neorealist…)
WAVESA wave is nothing but disturbance which is occurredin a medium and it is specified by its frequency,wavelength, phase, amplitude and intensity.PARTICLESA particle or matter has mass and it is located at asome definite point and it is specified by its mass,velocity, momentum and energy.
• The physical values or motion of a macroscopicparticles can be observed directly. Classicalmechanics can be applied to explain that motion.• But when we consider the motion of Microscopicparticles such as electrons, protons……etc.,classical mechanics fails to explain that motion.• Quantum mechanics deals with motion ofmicroscopic particles or quantum particles.
de Broglie hypothesis• In 1924 the scientist named de Broglieintroduced electromagnetic waves behaveslike particles, and the particles likeelectrons behave like waves called matterwaves.• He derived an expression for thewavelength of matter waves on the analogyof radiation.
• According to Planck’s radiation law• Where ‘c’ is a velocity of light and ‘λ‘is a wavelength.• According to Einstein mass-energy relationFrom 1 & 2)1..(..........λϑchhE==phmchchmc===λλλ2)2......(2mcE =
de Broglie wavelength associated withelectronsLet us consider the case of an electron of restmass m0 and charge ‘ e ‘ being accelerated by apotential V volts.If ‘v ‘ is the velocity attained by the electron dueto accelerationThe de Broglie wavelength 020221meVveVvm==AVmeVmhvmh000026.122==⇒=λλλ
Characteristics of Matter waves• Lighter the particle, greater is the wavelengthassociated with it.• Lesser the velocity of the particle, longer thewavelength associated with it.• For V= 0, λ=∞ . This means that only with movingparticle matter wave is associated.• Whether the particle is charged or not, matterwave is associated with it. This reveals that thesewaves are not electromagnetic but a new kind ofwaves .
It can be proved that the matter waves travel faster than light.We know thatThe wave velocity (ω) is given byAs the particle velocity v cannotexceed velocity of light c,ω is greater than velocity of light.hmcmchmcEhE222=→===ϑϑϑvcwmvhhmcwheremvhhmcww222&))((=====λϑϑλ
Experimental evidence for matter waves1.Davisson and Germer ’s Experiment.
DAVISSON & GERMER’S EXPERMENT• Davison and Germer first detected electron wavesin 1927.• They have also measured de Broglie wave lengthsof slow electrons by using diffraction methods.Principle:• Based on the concept of wave nature of matter fastmoving electrons behave like waves. Henceaccelerated electron beam can be used fordiffraction studies in crystals.
Experimental arrangement• The electron gun G produces a fine beam ofelectrons.• It consists of a heated filament F, which emitselectrons due to thermo ionic emission• The accelerated electron beam of electrons areincident on a nickel plate, called target T. The targetcrystal can be rotated about an axis perpendicular tothe direction of incident electron beam.• The distribution of electrons is measured by using adetector called faraday cylinder c and which ismoving along a graduatedcircular scale S.• A sensitive galvanometer connected to the detector.
ResultsResults• When an electron beam accelerated by 54 volts wasdirected to strike the nickel crystal, a sharp maximum inthe electron distribution occurred at scattered angle of 500with the incident beam.• For that scattered beam of electrons the diffracted anglebecomes 650.• For a nickel crystal the inter planer separation isd = 0.091nm.
250250650Incident electron beamDiffracted beam650Iθ0V = 54v500
• According to Bragg’s law• For a 54 volts , the de Broglie wavelength associated with the electron isgiven by• This is in excellent agreement with theexperimental value.• The Davison - Germer experimentprovides a direct verification of deBroglie hypothesis of the wave natureof moving particle.nmnmnd165.0165sin091.02sin20=×=××=λλλθnmAAV166.05426.1226.1200===λλλ
Heisenberg realised that• In the world of very small particles, one cannot measure anyproperty of a particle without interacting with it in some way• This introduces an unavoidable uncertainty into the result• One can never measure all theproperties exactlyWerner Heisenberg (1901-1976)
Measuring the position andmomentum of an electron• Shine light on electron and detect reflectedlight using a microscope• Minimum uncertainty in positionis given by the wavelength of thelight• So to determine the positionaccurately, it is necessary to uselight with a short wavelength
Measuring the position and momentumof an electron (cont’d)• By Planck’s law E = hc/λ, a photon with a short wavelength has alarge energy• Thus, it would impart a large ‘kick’ to the electron• But to determine its momentum accurately,electron must only be given a small kick• This means using light of long wavelength!
Fundamental Trade Off …• Use light with short wavelength:– accurate measurement of position but not momentum• Use light with long wavelength– accurate measurement of momentum but not position
Heisenberg’s Uncertainty PrincipleThe more accurately you know the position (i.e.,the smaller ∆x is) , the less accurately you know themomentum (i.e., the larger ∆p is); and vice versa
Heisenberg uncertainty principlestatement• This principle states that the product ofuncertainties in determining the both positionand momentum of particle is approximatelyequal to h / 4Π.Where Δx is the uncertainty in measuringpositiondetermine the position and Δp is theuncertainty in determining momentum.• This relation shows that it is impossible toπ4hpx ≥∆∆
• This relation is universal and holds for all canonicallyconjugate physical quantities like1. Angular momentum & angle2. Time & energyππθ44hEthj≥∆∆≥∆∆Consequences of uncertainty principle• Explanation for absence of electrons in the nucleus• Diffraction of electrons through single slit.• Existence of protons and neutrons inside nucleus.• Uncertainty in the frequency of light emitted by anatom.• Energy of an electron in an atom• .
Physical significance of the wave function• The wave function ‘Ψ’ has no direct physicalmeaning. It is a complex quantity representingthe variation of a Matter wave.• The wave function Ψ( r, t ) describes the positionof a particle with respect to time.• It can be considered as ‘probability amplitude’since it is used to find the location of theparticle.
ΨΨ*or ׀Ψ ׀2is theprobability density function.ΨΨ*dx dy dz gives the probability of finding the electronin the region of space between x and x + dx, y and y + dy,z and z + dz.The above relation shows that’s a ‘normalization condition’of particle.112--*==∫∫∞+∞+∞∞dxdydzdxdydzψψψ
Schrödinger time independent waveequation• Schrödinger wave equation is a basic principle of afundamental Quantum mechanics.• Consider a particle of mass ‘m’ ,moving withvelocity ‘v’ and wavelength ‘λ’. According to deBroglie,)1.(..........mvhph==λλDirac, Heisenberg, and Schrödinger (L to R) atthe Stockholm train station on their way to theNobel Prize ceremony, December 1933.
• According to classical physics, the displacement for amoving wave along X-direction is given by• Where ‘A’ is a amplitude ‘x’ is a position co-ordinate and‘λ’ is a wave length.• The displacement of de Broglie wave associated with amoving particle along X-direction is given by)2sin( xAS ×=λπ)2sin(),( xAtr ×=λπψ
If ‘E’ is total energy of thesystem)2......(222)(2..222222VEmhmhVEmhVmpVEEKEPE−=+=+→+=+=λλλ
Periodic changes in ‘Ψ’ are responsible for thewave nature of a moving particle)3(4114).2sin(4).2sin(]2[).2cos(2)(].2sin[)(222222222222222→−=−=−=−===dxddxdxAdxdxAdxdxAdxdxAdxddxdψψπλψλπψλπλπψλπλπψλπλπψλπψ
0]41[22222222222222222222=−+−−=−=−−=−−=ψπψψπψψψπψψπλVEhmdxdVEhmdxdVEdxdmhVEdxdmhVEmhThis is Schrödinger time independent wave equation in one dimension.From equation 3……0][822222222=−+∂∂+∂∂+∂∂ψπψψψVEhmzyxIn three dimensional way it becomes…..
Particle in a one dimensional potentialbox• Consider an electron of mass ‘m’ in an infinitely deep one-dimensional potential box with a width of a ‘ L’ units inwhich potential is constant and zero.LxxxvLxxv≥≤∞=〈〈=&0,)(0,0)(X=0 X=LV=0
The motion of the electron in one dimensionalbox can be described by the Schrödingersequation.0][2222=−+ ψψVEmdxdInside the box the potential V =0EmkwherekdxdEmdxd222222222,,00][2=→=+=+ψψψψkxBkxAx cossin)( +=ψThe solution to above equation can be written as
Where A,B and K are unknown constants and tocalculate them, it is necessary to applyboundary conditions.• When X = 0 then Ψ = 0 i.e. |Ψ|2= 0 ……. aX = L Ψ = 0 i.e. |Ψ|2= 0 …… b• Applying boundary condition ( a ) to equation ( 1 )A Sin K(0) + B Cos K(0) = 0 B = 0• Substitute B value equation (1)Ψ(x) = A Sin Kx
Applying second boundary condition for equation(1)Substitute B & K value in equation (1)To calculate unknown constant A, considernormalization condition.LnknkLkLkLAkLkLAππ====+=0sin0sincos)0(sin0LxnAx)(sin)(πψ =
The particle Wave functions & their energyEigen values in a one dimensional square wellpotential are shown in figure.LznLynLxnLnπππψ 3213sinsinsin)/2(=Normalized Wave function in three dimensions is given by
Probability density01-1 -0.5 0 0.5 101-1 -0.5 0 0.5 101-1 -0.5 0 0.5 101-1 -0.5 0 0.5 1|u1(x)|2 |u2(x)|2|u3(x)|2 |u4(x)|2x/a x/ax/a x/aP x x t( ) | ( , ) |= ψ 2For first foureigenfunctions forparticle in a box
Conclusions1.The three integers n1,n2and n3 called Quantum numbersare required to specify completely each energy state.2.The energy ‘ E ’ depends on the sum of the squares of thequantum numbers n1,n2and n3 but not on their individualvalues.3.Several combinations of the three quantum numbers maygive different wave functions, but not of the same energyvalue. Such states and energy levels are said to bedegenerate.
Finally, quantum mechanics challenges our assumption thatultimate reality will accord with our natural intuition about whatis reasonable and normalClassical physics Quantum physicsI think it is safe to say that no one understands quantum mechanics. Do not keepsaying to yourself, if you can possibly avoid it, But how can it possibly be likethat? … Nobody knows how it can be like that. – Richard Feynman