physical chemistry ii: quantum chemistry lecture 10
TRANSCRIPT
Time-dependentSchrodingerEquation
• Time-evolutionofaquantumstateisfullydescribedbytheSchrodingerequation:
• Solvingtheenergyeigenstatesgivestheideal“basis”todescribetime-evolution:Thisisnecessaryforanalyticalsolutions.
i! ∂∂t
Ψ(t) = H Ψ(t)
H φn = En φn
Ψ(t) = cn (0)e−iEnt! φn
n∑
Time-evolutionofWaveFunctions• Time-evolutionintheenergyeigenbasis:Easytosolvewhentheenergyeigenstatesareknownçnotaneasytask
• Wavefunctionsare“COMPLEX”• Eacheigenstateoscillatesinadifferentfrequency• Interferencesbetweendifferentoscillatingwavesconstitutesthefullquantumdynamics
Ψ(t) = cn (0)e−iEnt! φn
n∑
Time-dependentSchrodingerEquation
• Actually,anyorthonormalbasissetsareequallyvalidforsolvingtheSchrodingerequation
• NumericallysolvetheSchrodingerequation:1. Findanumericallyconvenient&efficientbasis2. ConstructtheHamiltonianmatrixandtheinitial
wavefunctionket(columnvector)inthebasis3. Findeigenvalues&eigenvectorsoftheHamiltonian
tosolvethetime-independentproblem4. Propagatethewavefunctionstepbystepintime
usinganefficient&robustalgorithm
Poorman’sDiscreteVariableRepresentation
• Thesimplestbasischoiceistousediscretegridpointsinspace
• Wavefunctionthenrepresentsbyavectorof“pointvalues”
Ψ(x)
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Ψ( 1)
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Ψ( 8)
Ψ =
Ψ(x1)
Ψ(x2 )
Ψ(x3)
Ψ(x4 )
!
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟
Poorman’sDiscreteVariableRepresentation
• Hamiltonian:• Thepotentialoperatorisdiagonalinpositionandeasytoconstructinthediscretebasis
V =
V (x1) 0 0 0 !
0 V (x2 ) 0 0 !
0 0 V (x3) 0 !
0 0 0 V (x4 ) !
" " " " #
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟
H = T +V
• Thekineticenergyoperator:
Poorman’sDiscreteVariableRepresentation
T =
2k −k 0 0 !−k 2k −k 0 !0 −k 2k −k !0 0 −k 2k !" " ! ! !
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
T = −!2
2md 2
dx2
−!2
2md 2
dx2Ψ(x)
x=xn
≈−!2
2m1Δ
Ψn+1 −Ψn
Δ−Ψn −Ψn−1
Δ
⎛
⎝⎜
⎞
⎠⎟
=!2
2mΔ22Ψn −Ψn−1 −Ψn+1( )
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Δ
k = !2
2mΔ2
TimePropagation• Schrodingerequation:
• Iterationwithasmalltimestepδt:
• Normally,analgorithmthatincludeshigher-ordercontributions(e.g.δt2)isusedtoensureefficiencyandaccuracy.Forexample,theCrank-Nicolsonmethodprovidedsignificantimprovementinaccuracy.
i! ∂∂t
Ψ(t) = H Ψ(t)
Ψ(t +δt) ≈ Ψ(t) − iδt!H Ψ(t)
MatlabExamples
Matlab: http://www.mathworks.com/
DiscreteVariableRepresentation• dvr_eigen.m• Poorman’sDVR:801
pointsfromx=-4tox=4• KineticOp:
T(n,n)=2*kT(n,n+1)=T(n+1,n)=-k
• PotentialOp:V(x)onthediagonal
• DVReigenstatesyieldswavefunctionsdirectly– Harmonic– Quarticdoublewell– Biasedquarticdoublewell– Anyboundpotential!!
T=zeros(Nx);foridxi=1:(Nx-1)T(idxi,idxi)=2*k;T(idxi,idxi+1)=-1*k;T(idxi+1,idxi)=-1*k;endT(Nx,Nx)=2*k;V=diag(potential(xvec));%HamiltonianH=T+V;%eigenvaluesandvectors[U,E]=eig(H);
DynamicsviaDVR
• Poorman’sDVR+Crank-Nicolson
• Additionaltricks:focusonlowenergyeigenstates
• Gaussianwavepacketdynamics– Harmonicpotential– Quarticdoublewellpotential
– Biasedquarticpotential
%eigenvaluesandvectors[U,E]=eig(H);%numofeigenstatestouseNred=50%truncatedHamiltonian(diagform)Hred=E(1:Nred,1:Nred);%eigenvectors,usedforbasischangeUred=U(:,1:Nred);%transformPsi_0totheeigenspacePsi_0red=Ured'*Psi_0;
GaussianWavepacketDynamicsviaDVR
• Gaussianwavepacket:superpositionofeigenstates,normallynon-stationary
• Groundstate?(dvr_propagate1.m)
• LowenergyGaussianwavepacketinaharmonicpotential(dvr_propagate2.m)
– Oscillationsinpositionandwidth(p)– Centerofthewavepacketbehavesclassically
• Displacedground-stateGaussian(dvr_propagate3.m)
– Minimumuncertaintywavepacket– Width&shapeinvariant:coherentstate
GaussianWavepacketDynamicsviaDVR
• HighenergyGaussianwavepacketinaharmonicpotential(dvr_propagate4.m)
– Superpositionofmanymoreeigenstatesbecauseofenergeticaccessibility
– Morecomplexdynamicsduetointerferencesofmodeswithabroaddistributionoffrequencies
– Recurrencesoccurduetosymmetry
• Quarticdoublewell(dvr_propagate[5,6].m)– Lowenergy:tunneling– Highenergy:scattering
LimitationsofDVR+CN
• Infeasiblefortreatinghighdimensionalsystems:N=(Nx)d
• Unfitforextensivesystems&highlyexcitedstates
• Difficult&computationallyexpensiveforaccuratelongtimedynamics
• Inefficientforsystemsexhibitingseparationoftemporalorspatialscales
• Inefficientforcalculatingrates,yields,…etc.
InternetResources
• MatlabscriptsshowntodayalreadyonCEIBA• JavaAppletdemoof1DQMsystems:http://www.falstad.com/qm1d
• JavaAppletdemoof2DQMsystems:http://www.falstad.com/qm2dbox
• PhETSimulationsofQuantumPhenomena:http://phet.colorado.edu/simulations/index.php?cat=Quantum_Phenomena
Remarks• Intuitionsfromclassicalmechanicsstillexplainsmanyquantumphenomena,however,adjustmentsarerequiredforsuperpositions&interferences(coherence),tunneling,measurement…etc.
• Energyeigenstatesformthebasisfordescribingtimeevolutionofquantumstates
• Solutionstothetime-independentSchrodingerequationarefundamentaltounderstandquantumphenomenainphysics&chemistry
SchemeforTimePropagation
• tls_prog_exactdiag.m:exactpropagatorthroughdiagonalizingH
• tls_prog_euler.m:Eulermethod
• tls_prog_cranknicolson.m:Crank-Nicolsonscheme
• TLSdynamicsasanexample
• Implementations• Stability,accuracy,&efficiency– Sizeofeachtimestep– Accuracyatlongertimes– Efficiencyofthealgorithm
• Bytheway,TLSdynamics:fixenergygap,varyJ