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ψψ
Quantum physics fromQuantum physics fromcoarse grained classical probabilitiescoarse grained classical probabilities
what is an atom ?what is an atom ?
quantum mechanics : isolated objectquantum mechanics : isolated object quantum field theory : excitation of quantum field theory : excitation of
complicated vacuumcomplicated vacuum classical statistics : sub-system of classical statistics : sub-system of
ensemble with infinitely many ensemble with infinitely many degrees of freedomdegrees of freedom
quantum mechanics quantum mechanics can be described by can be described by classical statistics !classical statistics !
quantum mechanics from quantum mechanics from classical statisticsclassical statistics
probability amplitudeprobability amplitude entanglemententanglement interferenceinterference superposition of statessuperposition of states fermions and bosonsfermions and bosons unitary time evolutionunitary time evolution transition amplitudetransition amplitude non-commuting operatorsnon-commuting operators violation of Bell’s inequalitiesviolation of Bell’s inequalities
essence of quantum essence of quantum mechanicsmechanics
description of appropriate description of appropriate subsystems of subsystems of
classical statistical ensemblesclassical statistical ensembles
1) equivalence classes of probabilistic 1) equivalence classes of probabilistic observablesobservables
2) incomplete statistics2) incomplete statistics 3) correlations between measurements based 3) correlations between measurements based
on conditional probabilitieson conditional probabilities 4) unitary time evolution for isolated 4) unitary time evolution for isolated
subsystemssubsystems
statistical picture of the statistical picture of the worldworld
basic theory is not deterministicbasic theory is not deterministic basic theory makes only statements basic theory makes only statements
about probabilities for sequences of about probabilities for sequences of events and establishes correlationsevents and establishes correlations
probabilism is fundamental , not probabilism is fundamental , not determinism !determinism !
quantum mechanics from classical statistics :quantum mechanics from classical statistics :not a deterministic hidden variable theorynot a deterministic hidden variable theory
Probabilistic realismProbabilistic realism
Physical theories and lawsPhysical theories and lawsonly describe probabilitiesonly describe probabilities
Physics only describes Physics only describes probabilitiesprobabilities
Gott würfeltGott würfelt
Physics only describes Physics only describes probabilitiesprobabilities
Gott würfeltGott würfelt Gott würfelt nichtGott würfelt nicht
Physics only describes Physics only describes probabilitiesprobabilities
Gott Gott würfelt würfelt
Gott würfelt nichtGott würfelt nicht
humans can only deal humans can only deal with probabilitieswith probabilities
probabilistic Physicsprobabilistic Physics
There is oThere is onene reality reality This can be described only by This can be described only by
probabilitiesprobabilities
one droplet of water …one droplet of water … 10102020 particles particles electromagnetic fieldelectromagnetic field exponential increase of distance exponential increase of distance
between two neighboring trajectoriesbetween two neighboring trajectories
probabilistic realismprobabilistic realism
The basis of Physics are The basis of Physics are probabilities probabilities
for predictions of real eventsfor predictions of real events
laws are based on laws are based on probabilitiesprobabilities
determinism as special case :determinism as special case :
probability for event = 1 or 0probability for event = 1 or 0
law of big numberslaw of big numbers unique ground state …unique ground state …
conditional probabilityconditional probability
sequences of sequences of events( measurements ) events( measurements )
are described by are described by
conditionalconditional probabilities probabilities
both in classical statisticsboth in classical statisticsand in quantum statisticsand in quantum statistics
w(tw(t11))
not very suitable not very suitable for statement , if here and nowfor statement , if here and nowa pointer falls downa pointer falls down
::
Schrödinger’s Schrödinger’s catcat
conditional probability :conditional probability :if nucleus decaysif nucleus decaysthen cat dead with wthen cat dead with wcc = 1 = 1 (reduction of wave function)(reduction of wave function)
one - particle wave one - particle wave functionfunction
from coarse graining from coarse graining of microphysical of microphysical
classical statistical classical statistical ensembleensemble
non – commutativity in classical statisticsnon – commutativity in classical statistics
microphysical ensemblemicrophysical ensemble
states states ττ labeled by sequences of labeled by sequences of
occupation numbers or bits noccupation numbers or bits ns s = 0 = 0 or 1or 1
ττ = [ n = [ nss ] = ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.[0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.
probabilities pprobabilities pττ > 0 > 0
function observablefunction observable
function observablefunction observable
ss
I(xI(x11)) I(xI(x44))I(xI(x22)) I(xI(x33))
normalized difference between occupied and empty normalized difference between occupied and empty bits in intervalbits in interval
generalized function generalized function observableobservable
normalizationnormalization
classicalclassicalexpectationexpectationvaluevalue
several species several species αα
positionposition
classical observable : classical observable : fixed value for every state fixed value for every state ττ
momentummomentum
derivative observablederivative observable
classical classical observable : observable : fixed value for every fixed value for every state state ττ
complex structurecomplex structure
classical product of classical product of position and momentum position and momentum
observablesobservables
commutes !commutes !
different products of different products of observablesobservables
differs from classical productdiffers from classical product
Which product describes Which product describes correlations of correlations of
measurements ?measurements ?
coarse graining of coarse graining of informationinformation
for subsystemsfor subsystems
density matrix from coarse density matrix from coarse graininggraining
• position and momentum observables position and momentum observables use only use only small part of the information contained small part of the information contained in in ppττ ,,
• relevant part can be described by relevant part can be described by density matrixdensity matrix
• subsystem described only by subsystem described only by information information which is contained in density matrix which is contained in density matrix • coarse graining of informationcoarse graining of information
quantum density matrixquantum density matrix
density matrix has the properties density matrix has the properties ofof
a quantum density matrixa quantum density matrix
quantum operatorsquantum operators
quantum product of quantum product of observablesobservables
the productthe product
is compatible with the coarse grainingis compatible with the coarse graining
and can be represented by operator productand can be represented by operator product
incomplete statisticsincomplete statistics
classical productclassical product
is not computable from information whichis not computable from information which
is available for subsystem !is available for subsystem ! cannot be used for measurements in the cannot be used for measurements in the
subsystem !subsystem !
classical and quantum classical and quantum dispersiondispersion
subsystem probabilitiessubsystem probabilities
in contrast :in contrast :
squared momentumsquared momentum
quantum product between classical observables :quantum product between classical observables :maps to product of quantum operatorsmaps to product of quantum operators
non – commutativity non – commutativity in classical statisticsin classical statistics
commutator depends on choice of product !commutator depends on choice of product !
measurement correlationmeasurement correlation
correlation between measurements correlation between measurements of positon and momentum is given of positon and momentum is given by quantum productby quantum product
this correlation is compatible with this correlation is compatible with information contained in subsysteminformation contained in subsystem
coarse coarse graininggraining
from fundamental from fundamental fermions fermions
at the Planck scaleat the Planck scaleto atoms at the Bohr to atoms at the Bohr
scalescale
p([np([nss])])
ρρ(x , x´)(x , x´)
quantumquantum particle from particle from classical probabilities in classical probabilities in
phase spacephase space
quantum particlequantum particleandand
classical particleclassical particle
quantum particle quantum particle classical particleclassical particle
particle-wave particle-wave dualityduality
uncertaintyuncertainty
no trajectoriesno trajectories
tunnelingtunneling
interference for interference for double slitdouble slit
particlesparticles sharp position and sharp position and
momentummomentum classical classical
trajectoriestrajectories
maximal energy maximal energy limits motionlimits motion
only through one only through one slitslit
double slit experimentdouble slit experiment
double slit experimentdouble slit experiment
one isolated particle ! one isolated particle ! no interaction between atoms passing through slitsno interaction between atoms passing through slits
probability –probability –distributiondistribution
double slit experimentdouble slit experiment
Is there a classical probability distribution in phase space ,Is there a classical probability distribution in phase space ,and a suitable time evolution , which can describe and a suitable time evolution , which can describe interference pattern ?interference pattern ?
quantum particle from quantum particle from classical probabilities in classical probabilities in
phase spacephase space probability distribution in phase space forprobability distribution in phase space for oonene particle particle
w(x,p)w(x,p) as for classical particle !as for classical particle ! observables different from classical observables different from classical
observablesobservables
time evolution of probability distribution time evolution of probability distribution different from the one for classical particledifferent from the one for classical particle
quantum mechanics quantum mechanics can be described by can be described by classical statistics !classical statistics !
quantum mechanics from quantum mechanics from classical statisticsclassical statistics
probability amplitudeprobability amplitude entanglemententanglement interferenceinterference superposition of statessuperposition of states fermions and bosonsfermions and bosons unitary time evolutionunitary time evolution transition amplitudetransition amplitude non-commuting operatorsnon-commuting operators violation of Bell’s inequalitiesviolation of Bell’s inequalities
quantum physicsquantum physics
wave functionwave function
probabilityprobability
phasephase
Can quantum physics be Can quantum physics be described by classical described by classical
probabilities ?probabilities ?“ “ No go “ theoremsNo go “ theorems
Bell , Clauser , Horne , Shimony , HoltBell , Clauser , Horne , Shimony , Holt
implicit assumption : use of classical correlation implicit assumption : use of classical correlation function for correlation between measurementsfunction for correlation between measurements
Kochen , SpeckerKochen , Specker
assumption : unique map from quantum assumption : unique map from quantum operators to classical observablesoperators to classical observables
zwitterszwitters
no different concepts for classical no different concepts for classical and quantum particles and quantum particles
continuous interpolation between continuous interpolation between quantum particles and classical quantum particles and classical particles is possible particles is possible
( not only in classical limit )( not only in classical limit )
classical particle classical particle without classical without classical
trajectorytrajectory
quantum particle quantum particle classical particleclassical particle
particle-wave particle-wave dualityduality
uncertaintyuncertainty
no trajectoriesno trajectories
tunnelingtunneling
interference for interference for double slitdouble slit
particle particle – wave – wave dualityduality
sharp position and sharp position and momentummomentum
classical trajectoriesclassical trajectories
maximal energy maximal energy limits motionlimits motion
only through one slitonly through one slit
no classical trajectoriesno classical trajectories
also for classical particles in microphysics :also for classical particles in microphysics :
trajectories with sharp position trajectories with sharp position
and momentum for each moment and momentum for each moment
in time are inadequate idealization !in time are inadequate idealization !
still possible formally as limiting casestill possible formally as limiting case
quantum particle classical quantum particle classical particleparticle
quantum - quantum - probability -probability -amplitude amplitude ψψ(x)(x)
Schrödinger - Schrödinger - equationequation
classical probability classical probability
in phase space w(x,p)in phase space w(x,p)
Liouville - equation for Liouville - equation for w(x,p)w(x,p)
( corresponds to Newton ( corresponds to Newton eq. eq.
for trajectories )for trajectories )
quantum formalism forquantum formalism forclassical particleclassical particle
probability distribution for probability distribution for oneone classical particle classical particle
classical probability distributionclassical probability distributionin phase spacein phase space
wave function for classical wave function for classical particleparticle
classical probability classical probability distribution in phase spacedistribution in phase space
wave function for wave function for classical classical particleparticle
depends on depends on positionposition and momentum ! and momentum !
CC
CC
wave function for wave function for oneoneclassicalclassical particle particle
realreal depends on position and momentumdepends on position and momentum square yields probabilitysquare yields probability
CC CC
similarity to Hilbert space for classical mechanicssimilarity to Hilbert space for classical mechanicsby Koopman and von Neumann by Koopman and von Neumann in our case : in our case : real real wave function permits computationwave function permits computationof wave function from probability distributionof wave function from probability distribution( up to some irrelevant signs )( up to some irrelevant signs )
quantum laws for quantum laws for observablesobservables
CC CC
xx
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ψψ
time evolution of time evolution of classical wave functionclassical wave function
Liouville - equationLiouville - equation
describes describes classicalclassical time evolution of time evolution of classical probability distributionclassical probability distributionfor one particle in potential V(x)for one particle in potential V(x)
time evolution of time evolution of classical wave functionclassical wave function
CC
CC CC
wave equationwave equation
modified Schrödinger - equationmodified Schrödinger - equation
CC CC
wave equationwave equation
CC CC
fundamenal equation for classical fundamenal equation for classical particle in potential V(x)particle in potential V(x)replaces Newton’s equationsreplaces Newton’s equations
particle - wave dualityparticle - wave duality
wave properties of particles :wave properties of particles :
continuous probability distributioncontinuous probability distribution
particle – wave dualityparticle – wave duality
experiment if particle at position x – yes or no :experiment if particle at position x – yes or no :discrete discrete alternativealternative
probability distribution probability distribution for findingfor findingparticle at position x :particle at position x :continuouscontinuous
11
11
00
particle – wave dualityparticle – wave duality
All statistical properties of All statistical properties of classicalclassical particles particles
can be described in quantum formalism !can be described in quantum formalism !
no quantum particles yet !no quantum particles yet !
modification of Liouville modification of Liouville equationequation
evolution equationevolution equation
time evolution of probability has to time evolution of probability has to be specified as fundamental lawbe specified as fundamental law
not known a priori not known a priori Newton’s equations with trajectories Newton’s equations with trajectories
should follow only in should follow only in limiting caselimiting case
zwitterszwitters
same formalism for quantum and same formalism for quantum and classical particlesclassical particles
different time evolution of probability different time evolution of probability distributiondistribution
zwitters : zwitters :
between quantum and classical particle –between quantum and classical particle –
continuous interpolation of time continuous interpolation of time evolution equationevolution equation
quantum time evolutionquantum time evolution
modification of evolution modification of evolution forfor
classical probability classical probability distributiondistribution
HHWW
HHWW
CC CC
quantum particlequantum particle
evolution equationevolution equation
fundamental equation for fundamental equation for quantum particle in potential Vquantum particle in potential V
replaces Newton’s equationsreplaces Newton’s equations
CCCC CC
modified observablesmodified observables
Which observables should Which observables should be chosen ?be chosen ?
momentum: p or ?momentum: p or ?
position : x or ?position : x or ?
Different possibilities , should be Different possibilities , should be adapted to specific setting of adapted to specific setting of measurementsmeasurements
position - observableposition - observable
different observables according to different observables according to experimental situationexperimental situation
suitable observable for microphysis has suitable observable for microphysis has to be foundto be found
classical position observable : idealization classical position observable : idealization of infinitely precise resolutionof infinitely precise resolution
quantum – observable : remains quantum – observable : remains computable with coarse grained computable with coarse grained informationinformation
1515
quantum - observablesquantum - observables
observables for classical observables for classical position and momentumposition and momentum
observables for observables for quantum - quantum - position and momentumposition and momentum
… … do not commute ! do not commute !
uncertaintyuncertainty
quantum – observables containquantum – observables contain statistical partstatistical part ( similar to entropy , temperature )( similar to entropy , temperature )
Heisenberg’sHeisenberg’suncertainty relationuncertainty relation
Use quantum observables Use quantum observables for description of for description of measurements of measurements of
position and momentum of position and momentum of particles !particles !
quantum particlequantum particle
with evolution equationwith evolution equation
all expectation values and correlations forall expectation values and correlations forquantum – observables , as computed fromquantum – observables , as computed fromclassical probability distribution ,classical probability distribution ,coincide for all times precisely with coincide for all times precisely with
predictions predictions of quantum mechanics for particle in of quantum mechanics for particle in
potential Vpotential V
CCCC CC
classical probabilities – classical probabilities – not a deterministic classical not a deterministic classical
theorytheory
quantum paquantum particle from rticle from classical probabilities in phase space !classical probabilities in phase space !
quantum formalismquantum formalismfrom classical from classical probabilitiesprobabilities
pure statepure state
described by quantum wave functiondescribed by quantum wave function
realized by classicalrealized by classicalprobabilities in theprobabilities in theformform
time evolution described bytime evolution described bySchrödinger – equationSchrödinger – equation
density matrix anddensity matrix andWigner-transformWigner-transform
Wigner – transformed densityWigner – transformed densitymatrix in quantum mechanicsmatrix in quantum mechanics
permits simple computationpermits simple computationof expectation values ofof expectation values ofquantum mechanicalquantum mechanicalobservablesobservables
can be constructed from classical wave function !can be constructed from classical wave function !
CC CC
quantum observables quantum observables andand
classical observablesclassical observables
zwitterzwitter
difference between quantum and difference between quantum and classical particles only through classical particles only through different time evolutiondifferent time evolution
continuous continuous interpolatiointerpolationn
CLCL
QMQMHHWW
zwitter - Hamiltonianzwitter - Hamiltonian
γγ=0 : quantum – particle=0 : quantum – particle γγ==ππ/2 : classical particle/2 : classical particle
other interpolating Hamiltonians possible !other interpolating Hamiltonians possible !
HowHow good is quantum good is quantum mechanics ?mechanics ?
small parameter small parameter γγ can be tested can be tested experimentallyexperimentally
zwitter : no conserved microscopic zwitter : no conserved microscopic energyenergy
static state : orstatic state : or
CC
ground state for zwitterground state for zwitter
static state with loweststatic state with lowest
eigenstate for quantum energyeigenstate for quantum energy
zwitter –ground state has admixture of zwitter –ground state has admixture of excited excited
levels of quantum energylevels of quantum energy
quantum - energyquantum - energy
energy uncertainty ofenergy uncertainty ofzwitter ground statezwitter ground state
also small shift of energyalso small shift of energy
experiments for experiments for determination ordetermination orlimits on zwitter – limits on zwitter –
parameter parameter γγ ??
ΔΔE≠0E≠0
almost degenerate almost degenerate energy levels…?energy levels…?
lifetime of nuclear spin states > 60 hlifetime of nuclear spin states > 60 hγγ < 10 < 10-14 -14 ( Heil et al.) ( Heil et al.)
experiments for experiments for determination ordetermination orlimits on zwitter – limits on zwitter –
parameter parameter γγ ??
lifetime of nuclear spin states > 60 h ( Heil et al.) : lifetime of nuclear spin states > 60 h ( Heil et al.) : γγ < 10 < 10-14-14
sharpened observables –sharpened observables –between quantum and between quantum and
classicalclassical
ß=0 : quantum observables, ß=1 :classical ß=0 : quantum observables, ß=1 :classical observablesobservables
weakening of uncertainty weakening of uncertainty relationrelation
exp
eri
men
exp
eri
men
t ?
t ?
quantum particles and quantum particles and classical statisticsclassical statistics
common concepts and formalism for quantum common concepts and formalism for quantum and classical particles :and classical particles :
classical probability distribution and wave classical probability distribution and wave functionfunction
different time evolution , different Hamiltoniandifferent time evolution , different Hamiltonian quantum particle from “ coarse grained “ quantum particle from “ coarse grained “
classical probabilities ( only information classical probabilities ( only information contained in Wigner function is needed )contained in Wigner function is needed )
continuous interpolation between quantum continuous interpolation between quantum and classical particle : zwitterand classical particle : zwitter
conclusionconclusion
quantum statistics emerges from classical quantum statistics emerges from classical statisticsstatistics
quantum state, superposition, quantum state, superposition, interference, entanglement, probability interference, entanglement, probability amplitudeamplitude
unitary time evolution of quantum unitary time evolution of quantum mechanics can be described by suitable mechanics can be described by suitable time evolution of classical probabilitiestime evolution of classical probabilities
conditional correlations for measurements conditional correlations for measurements both in quantum and classical statisticsboth in quantum and classical statistics
experimental challengeexperimental challenge
quantitative tests , how accurate the quantitative tests , how accurate the predictions of quantum mechanics are predictions of quantum mechanics are obeyedobeyed
zwitterzwitter sharpened observables sharpened observables small parameter : small parameter : “ “ almost quantum mechanics “almost quantum mechanics “
endend
Quantenmechanik aus Quantenmechanik aus klassischen klassischen
WahrscheinlichkeitenWahrscheinlichkeitenklassische Wahrscheinlichkeitsverteilung kann klassische Wahrscheinlichkeitsverteilung kann
explizit angegeben werden für :explizit angegeben werden für :
quantenmechanisches Zwei-Zustands-quantenmechanisches Zwei-Zustands-System Quantencomputer : Hadamard System Quantencomputer : Hadamard gategate
Vier-Zustands-System ( CNOT gate )Vier-Zustands-System ( CNOT gate ) verschränkte Quantenzuständeverschränkte Quantenzustände InterferenzInterferenz
Bell’sche UngleichungenBell’sche Ungleichungen
werden verletzt durch werden verletzt durch bedingtebedingte Korrelationen Korrelationen
Bedingte Korrelationen für zwei EreignisseBedingte Korrelationen für zwei Ereignisse
oder Messungen reflektieren bedingte oder Messungen reflektieren bedingte WahrscheinlichkeitenWahrscheinlichkeiten
Unterschied zu klassischen Korrelationen Unterschied zu klassischen Korrelationen
( Klassische Korrelationen werden implizit zur ( Klassische Korrelationen werden implizit zur Herleitung der Bell’schen Ungleichungen Herleitung der Bell’schen Ungleichungen verwandt. )verwandt. )
Bedingte Dreipunkt- Korrelation nicht kommutativBedingte Dreipunkt- Korrelation nicht kommutativ
RealitätRealität
KorrelationenKorrelationen sind physikalische Realität , nicht sind physikalische Realität , nicht nur Erwartungswerte oder Messwerte einzelner nur Erwartungswerte oder Messwerte einzelner ObservablenObservablen
Korrelationen können Korrelationen können nicht-lokal nicht-lokal sein ( auch in klassischer Statistik ) ; sein ( auch in klassischer Statistik ) ; kausale Prozesse zur Herstellung kausale Prozesse zur Herstellung nicht-lokaler Korrelationen nicht-lokaler Korrelationen erforderlicherforderlich
Korrelierte Untersysteme sind nicht separabel in Korrelierte Untersysteme sind nicht separabel in unabhängige Teilsysteme – unabhängige Teilsysteme – Ganzes mehr als Ganzes mehr als Summe der TeileSumme der Teile
EPR - ParadoxonEPR - Paradoxon
Korrelation zwischen zwei Spins wird Korrelation zwischen zwei Spins wird bei Teilchenzerfall hergestelltbei Teilchenzerfall hergestellt
Kein Widerspruch zu Kausalität oder Kein Widerspruch zu Kausalität oder Realismus wenn Korrelationen als Teil der Realismus wenn Korrelationen als Teil der Realität verstanden werdenRealität verstanden werden
hat mal nicht Recht )hat mal nicht Recht )((
Untersystem und Untersystem und Umgebung:Umgebung:
unvollständige Statistikunvollständige Statistik
typische Quantensysteme sind typische Quantensysteme sind UntersystemeUntersysteme
von klassischen Ensembles mit unendlich vielenvon klassischen Ensembles mit unendlich vielen
Freiheitsgraden ( Umgebung )Freiheitsgraden ( Umgebung )
probabilistischeprobabilistische Observablen für Untersysteme : Observablen für Untersysteme :
Wahrscheinlichkeitsverteilung für Messwerte Wahrscheinlichkeitsverteilung für Messwerte
in Quantenzustandin Quantenzustand
conditional correlationsconditional correlations
example :example :two - level observables , two - level observables , A , B can take values ± 1A , B can take values ± 1
conditional probabilityconditional probability
probability to find value +1 for probability to find value +1 for product product
of measurements of A and Bof measurements of A and B
… … can be expressed in can be expressed in terms of expectation valueterms of expectation valueof A in eigenstate of Bof A in eigenstate of B
probability to find A=1 probability to find A=1 after measurement of B=1after measurement of B=1
measurement correlationmeasurement correlation
After measurement A=+1 the system After measurement A=+1 the system must be in eigenstate with this eigenvalue.must be in eigenstate with this eigenvalue.Otherwise repetition of measurement couldOtherwise repetition of measurement couldgive a different result !give a different result !
measurement changes statemeasurement changes statein all statistical systems !in all statistical systems !
quantum and classicalquantum and classical
eliminates possibilities that eliminates possibilities that are not realizedare not realized
physics makes statements physics makes statements about possibleabout possible
sequences of events sequences of events and their probabilitiesand their probabilities
M = 2 :M = 2 :
unique eigenstates for unique eigenstates for M=2M=2
eigenstates with A = 1eigenstates with A = 1
measurement preserves pure states if projectionmeasurement preserves pure states if projection
measurement correlation measurement correlation equals quantum correlationequals quantum correlation
probability to probability to measure measure A=1 and B=1 :A=1 and B=1 :
sequence of three sequence of three measurements measurements
and quantum commutatorand quantum commutator
two measurements commute , not threetwo measurements commute , not three