quantum mechanics and probability possible research topics ... · classical versus quantum...

Quantum Mechanics and ProbabilityPossible Research Topics for PhD Students

Volker Betz

Warwick, 25 January 2011

Quantum Mechanics...

Volker Betz Quantum Mechanics and Probability

Classical versus Quantum MechanicsClassical Mechanics

I Particles represented by their position x ∈ R3 and theirmomentum p ∈ R3.

I A particle moving in a potential V is modelled by the equation

x(t) =1

mp(t), p(t) = −∇V (x(t)).

Quantum MechanicsI Particles represented by normalized wavefunction ψ ∈ L2(R3).I No definite position or momentum (uncertainty principle).

But we can determine the probability of finding particles withcertain position or momentum:

Prob(x ∈ A) =

∫A|ψ(x)|2 dx, Prob(p ∈ A) =

∫A|ψ(k)|2 dk.

I Dynamics are governed by the Schrodinger equation:

i~∂tψ(x, t) = Hψ(x, t), H = − ~2

2m∆+V. H ’Hamiltonian’


Schrodinger equation for a molecule

i~∂tψ(x,y, t) = Hψ(x,y, t),

with

H = −Nnuc∑i=1

~2

2mnuc,i∆xi−

Nel∑i=j

~2

2mel∆yj+Vnuc(x)+Vel(y)+Vn,e(x,y)

I Vnuc = Coulomb repulsion between nuclei.

I Vel = Coulomb repulsion between electrons.

I Ve,n = Coulomb attraction between electrons and nuclei.

I Atomic units: ~ = mel = 1.

I mnuc � mel, for simplicity mnuc,i = ε−2 for all i.

I Time change: τ = εt, so ∂tψ = ε∂τψ.

I Define electronic Hamiltonian Hel(x) acting in L2(dy).



i∂tψ(x,y, t) = Hψ(x,y, t),

with

H = −Nnuc∑i=1

1

2mnuc,i∆xi−

Nel∑i=j

1

2∆yj +Vnuc(x)+Vel(y)+Vn,e(x,y)










i∂tψ(x,y, t) = Hψ(x,y, t),

with

H = −Nnuc∑i=1

ε2

2∆xi −

Nel∑i=j

1

2∆yj + Vnuc(x) + Vel(y) + Vn,e(x,y)










iε∂tψ(x,y, t) = Hψ(x,y, t),

with

H = −Nnuc∑i=1

ε2

2∆xi −

Nel∑i=j

1











iε∂tψ(x,y, t) =

(−ε

2

2∆x +Hel(x)

)ψ(x,y, t),

with

H = −Nnuc∑i=1

ε2

2∆xi −

Nel∑i=j

1










The Born-Oppenheimer approximation

iε∂tψ(x,y, t) =

(−ε

2

2∆x +Hel(x)

)ψ(x,y, t),

I Hel(x) acts on L2(dy) for all x.

I Idea: The nuclei move slowly, so the electrons stay in theireigenstates, which depend on the position of the nuclei.

I Diagonalisation: Let U(x) be the unitary operator on L2(dy)that diagonalises Hel(x). Thus

U(x)Hel(x)U∗(x) =

E1(x) 0 · · ·0 E2(x) 0

0 0. . .

I Ej(x) are the electronic energy bands.

I U(x) and ε2∆x do not commute. This couples the energybands. But the coupling is small.

I Born-Oppenheimer approximation: ignore the coupling!


The importance of coupling: Photo-Dissociation ofNaI

I A short laser pulse excites theelectrons from the ground state.

I The nuclei now feel a force due tothe changed electronic configurationand travel according to theBorn-Oppeneimer approximation.

I At the avoided crossing, the BOapproximation is not very good.Transitions occur, i.e. some part ofthe electronic wave functionswitches back to the ground state.

I The part that makes the transitionsis typically very small.

First, we could show experimentally that the wave packetwas highly localized in space, ∼0.1 Å, thus establishing theconcept of dynamics at atomic-scale resolution. Second, thespreading of the wave packet was minimal up to a fewpicoseconds, thus establishing the concept of single-molecule

trajectory; i.e., the ensemble coherence is induced effectively,as if the molecules are glued together, even though we startwith a random and noncoherent ensemblesdynamics, notkinetics. Third, vibrational (rotational) coherence was observedduring the entire course of the reaction (detecting products or

Figure 7. Femtochemistry of the NaI reaction, the paradigm case. The experimental results show the resonance motion between the covalent andionic structures of the bond, and the time scales for the reaction and for the spreading of the wave packet. Two transients are shown for theactivated complexes in transition states and for final fragments. Note the “quantized” behavior of the signal, not simply an exponential rise or decayof the ensemble. The classical motion is simulated as trajectories in space and time. Reference 42.

Feature Article J. Phys. Chem. A, Vol. 104, No. 24, 2000 5671


Superadiabatic transitions: results and projects

I Determining non-adiabatic transitions quantitatively is animportant open problem in computational chemistry.

I Many approaches are very ad-hoc and justified only by thefact they work more or less.

I With Benjamin Goddard: we found an expicit formula thatworks very well for one-atomic molecules.

I Big challenge: extend this to several nuclear degrees offreedom.

Possible projectsI Extend the theory of superadiabatic transitions (high risk /

high impact).I Use the techniques developed in some previous work to give

rigorous proofs in many areas of exponential asymptotics(some progress should be guaranteed).

I Many unsolved intriguing side-issues.I Mathematical techniques: Asymptotics, complex variables,

PDE, functional analysis.


Superadiabatic transitions: results and projects

I Determining non-adiabatic transitions quantitatively is animportant open problem in computational chemistry.

I Many approaches are very ad-hoc and justified only by thefact they work more or less.

I With Benjamin Goddard: we found an expicit formula thatworks very well for one-atomic molecules.

I Big challenge: extend this to several nuclear degrees offreedom.

Possible projectsI Extend the theory of superadiabatic transitions (high risk /

high impact).I Use the techniques developed in some previous work to give

rigorous proofs in many areas of exponential asymptotics(some progress should be guaranteed).

I Many unsolved intriguing side-issues.I Mathematical techniques: Asymptotics, complex variables,

PDE, functional analysis.Volker Betz Quantum Mechanics and Probability

... and Probablity


Feynman Path integrals and Feynman-Kac formula

Recall the Schrodinger equation:

i∂tψ(x, t) = Hψ(x, t), H = −1

2∆ + V. H ’Hamiltonian’

It is solved by

e−itH ψ0 = limn→∞

(e

i2n

∆ einV)n

= limn→∞

1

(2πit/n)n/2

∫ n−1∏j=1

dxj ei∑n−1

j=01

2t/n|xj+1−xj |2−V (xj+1) t

n

’=’

∫exp

(i

∫ t

0

1

2|X(s)|2 − V (X(s)) ds

)dX

Classical action integral in the exponent!

Unfortunately, this cannot be made into mathematics. But the’imaginary time version’ can, and is the Feynman-Kac formula:

e−tH ψ0(x) =

∫exp

(−∫ t

0V (Xs) ds

)dWx(X)


Feynman Path integrals and Feynman-Kac formula

Recall the Schrodinger equation:

i∂tψ(x, t) = Hψ(x, t), H = −1

2∆ + V. H ’Hamiltonian’

It is solved by

e−itH ψ0 = limn→∞

(e

i2n

∆ einV)n

= limn→∞

1

(2πit/n)n/2

∫ n−1∏j=1

dxj ei∑n−1

j=01

2t/n|xj+1−xj |2−V (xj+1) t

n

’=’

∫exp

(i

∫ t

0

1

2|X(s)|2 − V (X(s)) ds

)dX

Classical action integral in the exponent!Unfortunately, this cannot be made into mathematics. But the’imaginary time version’ can, and is the Feynman-Kac formula:

e−tH ψ0(x) =

∫exp

(−∫ t

0V (Xs) ds

)dWx(X)


Bose-Einstein condensation

Wave function of N indistinguishable particles

Ψ(x1, . . . , xN ) ∈ L2symm(R3N ),

i.e.Ψ(x1, . . . , xN ) = Ψ(xπ(1), . . . , xπ(N)).

Hamiltonian:

H = − ~2

2m

N∑i=1

∆i +∑

1 6 i<j 6 N

U(xi − xj) on L2symm(R3N ).

U is a repulsive pair potential.

One is interested in Tr e−βH , which can be expressed through theFeynman-Kac formula. But we have to take symmetrisation intoaccount, which introduces permutations.


Spatial random permutations: finite volume

I Λ ⊂ Rd , finite volmue V .x = {x1, . . . , xN} ⊂ Λ ⊂ Rd

I SN = set of permutations onπ : {1, . . . , N} → {1, . . . , N}.

I Typical example for a measureon SN :

Px({π}) =1

Y (x)exp

(−

N∑i=1

|xi − xπ(i)|2).

I Aim: Study the infinite volume limit:

V,N →∞, N

V= ρ

ρ is the density of points in Λ.

I Question: Existence and distribution of infinite cycles.


Spatial random permutations: finite volume

I Λ ⊂ Rd , finite volmue V .x = {x1, . . . , xN} ⊂ Λ ⊂ Rd

I SN = set of permutations onπ : {1, . . . , N} → {1, . . . , N}.

I Typical example for a measureon SN :

Px({π}) =1

Y (x)exp

(−

N∑i=1

ξ(xi − xπ(i))

).

I Aim: Study the infinite volume limit:

V,N →∞, N

V= ρ

ρ is the density of points in Λ.

I Question: Existence and distribution of infinite cycles.


Macroscopic cycles: heuristics

Px({π}) =1

Y (x)exp

(−

N∑i=1

ξ(xi − xπ(i))

).

I The spatial structure suppresses long jumps.

I For low density ρ long cycles are therefore unlikely.

I Conjecture: there is a phase transition, i.e. infinite cycles onlyappear above a critical density ρc.

I Important extension: Pair interaction between jumps. E.g.jumps are discouraged to cross, or they want to be parallel,etc...


Spatial random permutations: results and projects

I (with Daniel Ueltschi) In a spatially averaged model we knowthat a phase transition exists, also for some form ofinteraction.

I We argue non-rigorously that this should have someimplications for BEC.

Projects

I Understand the non-spatially averaged model: Existence ofthe infinite volume limit, infinite cycles, decay of correlations...

I Understand the point process associated to the spatiallyaveraged model: Conjecture: points repel each other.

I Incorporate ’real’ pair interactions into any model.

Methods: Classical probability theory, statistical mechanics.

All of these projects have a (relatively) easy entry point, but arecompletely open ended and may get very hard, but also veryrewarding!


quantum mechanics and probability possible research topics ... · classical versus quantum...

Documents