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Commutators,
eigenvalues,and
quantum mechanics
on surfaces.Evans Harrell
Georgia Tech
www.math.gatech.edu/~harrell
Tucson and Tours,
2004
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Dramatis personae
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Dramatis personae
• Commutator [A,B] = AB - BA
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Dramatis personae
• Commutator [A,B] = AB - BA
• The “nano” world
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Dramatis personae
• Commutator [A,B] = AB - BA
• The “nano” world
• Curvature
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Eigenvalues
• H uk = λ uk
• For simplicity, the spectrum will often be
assumed to be discrete. For example, the
operators might be defined on bounded
regions.
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Eigenvalues
• Laplacian - squares of frequencies of
normal modes of vibration
(acoustics/electromagnetics, etc.)
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Eigenvalues
• Laplacian - squares of frequencies of
normal modes of vibration
(acoustics/electromagnetics, etc.)
• Schrödinger Operator - energies of an atom
or quantum system.
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The spectral theorem for a general self-
adjoint operator
• The spectrum can be any closed subset of R.
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The spectral theorem for a general self-
adjoint operator
• For each u, there exists a measure µ, such that
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The spectral theorem for a general self-
adjoint operator
• Implication:
– If f(λ) ≥ g(λ) on the spectrum, then
– <u, f(H) u> ≥ <u, g(H) u>
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The spectrum of H
• For Laplace or Schrödinger not just any old
set of numbers can be the spectrum!
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“Universal” constraints on the spectrum
• H. Weyl, 1910, Laplace, λn ~ n2/d
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“Universal” constraints on the spectrum• H. Weyl, 1910, Laplace, λn ~ n2/d
• W. Kuhn, F. Reiche, W. Thomas, W. Heisenberg,
1925, “sum rules” for atomic energies.
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“Universal” constraints on the spectrum• H. Weyl, 1910, Laplace, λn ~ n2/d
• W. Kuhn, F. Reiche, W. Thomas, W. Heisenberg,1925, “sum rules” for atomic energies.
• L. Payne, G. Pólya, H. Weinberger, 1956: Thegap is controlled by the average of the smallereigenvalues:
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“Universal” constraints on the spectrum• H. Weyl, 1910, Laplace, λn ~ n2/d
• W. Kuhn, F. Reiche, W. Thomas, W. Heisenberg,1925, “sum rules” for atomic energies.
• L. Payne, G. Pólya, H. Weinberger, 1956: Thegap is controlled by the average of the smallereigenvalues:
• E. Lieb and W. Thirring, 1977, P. Li, S.T. Yau,
1983, sums of powers of eigenvalues, in terms of the phase-space volume.
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“Universal” constraints on the spectrum• H. Weyl, 1910, Laplace, λn ~ n2/d
• W. Kuhn, F. Reiche, W. Thomas, W. Heisenberg, 1925,“sum rules” for atomic energies.
• L. Payne, G. Pólya, H. Weinberger, 1956: The gap iscontrolled by the average of the smaller eigenvalues:
• E. Lieb and W. Thirring, 1977, P. Li, S.T. Yau, 1983,sums of powers of eigenvalues, in terms of the phase-spacevolume.
• Hile-Protter, 1980, Like PPW, but morecomplicated.
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PPW vs. HP
• L. Payne, G. Pólya, H. Weinberger, 1956:
• Hile-Protter, 1980:
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The universal industry after PPW
• Ashbaugh-Benguria 1991, proof of the
isoperimetric conjecture of PPW.
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“Universal” constraints on eigenvalues
• Ashbaugh-Benguria 1991, isoperimetric conjecture of
PPW proved.
• H. Yang 1991-5, unpublished, complicated formulae likePPW, respecting Weyl asymptotics.
• Harrell, Harrell-Michel, Harrell-Stubbe, 1993-present,
commutators.
• Hermi PhD thesis
• Levitin-Parnovsky, 2001?
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In this industry….
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In this industry….
1. The arguments have varied, but always
essentially algebraic.
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In this industry….
1. The arguments have varied, but always
essentially algebraic.
2. Geometry often shows up - isoperimetric
theorems, etc.
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On a (hyper) surface,what object is most like
the Laplacian?
(Δ = the good old flat scalar Laplacian of Laplace)
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• Answer #1 (Beltrami’s answer): Consider
only tangential variations.
• At a fixed point, orient Cartesian x0 with the
normal, then calculate
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Difficulty:
• The Laplace-Beltrami operator is an
intrinsic object, and as such is unaware
that the surface is immersed!
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Answer #2
The nanophysicists’ answer• E.g., Da Costa, Phys. Rev. A 1981
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Answer #2:
- ΔLB + q,
Where the effective potential q responds to
how the surface is immersed in space.
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Nanoelectronics
• Nanoscale = 10-1000 X width of atom
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Nanoelectronics
• Nanoscale = 10-1000 X width of atom
• Foreseen by Feynman in 1960s
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Nanoelectronics
• Nanoscale = 10-1000 X width of atom
• Foreseen by Feynman in 1960s
• Laboratories by 1990.
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Nanoelectronics• Quantum wires - etched semiconductors,
wires of gold in carbon nanotubes.
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Nanoelectronics• Quantum wires
• Quantum waveguides - macroscopic in two
dimensions, nanoscale in the width
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Nanoelectronics• Quantum wires
• Quantum waveguides
• Designer potentials - STM places individual
atoms on a surface
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• Answer #2 (The nanoanswer):
- ΔLB + q
• Since Da Costa, PRA, 1981: Perform a
singular limit and renormalization toattain the surface as the limit of a thindomain.
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Thin domain of fixed width
variable r= distance from edge
Energy form in separated variables:
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Energy form in separated variables:
First term is the energy form of Laplace-Beltrami.
Conjugate second term so as to replace it by a potential.
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Some subtleties• The limit is singular - change of dimension.
• If the particle is confined e.g. by Dirichlet
boundary conditions, the energies all
diverge to +infinity
• “Renormalization” is performed to separate
the divergent part of the operator.
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The result:
- ΔLB + q,
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Principal curvatures
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The result:
- ΔLB + q,
d=1, q = -κ2/4 ≤ 0 d=2, q = - (κ1-κ2)2/4 ≤ 0
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Consequence of q(x) ≤ 0:
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Consequence of q(x) ≤ 0:• If there is any bending at all, and the wire or
waveguide is large and asymptotically flat,
then there is always a bound state below theconduction level.
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Consequence of q(x) ≤ 0:• If there is any bending at all, and the wire or
waveguide is large and asymptotically flat,
then there is always a bound state below theconduction level.
• By bending or straightening the wire,
current can be switched off or on.
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Difficulty:
• Tied to a particular physical model -
other effective potentials arise from other
physical models or limits.
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Some other answers• In other physical situations, such as
reaction-diffusion, q(x) may be other
quadratic expressions in the curvature,usually q(x) ≤ 0.
• The conformal answer: q(x) is a
multiple of the scalar curvature.
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Heisenberg's Answer(if he had thought about it)
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Heisenberg's Answer(if he had thought about it)
Note: q(x) ≥ 0 !
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Commutators: [A,B] := AB-BA
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Commutators: [A,B] := AB-BA
1. Quantum mechanics is the effect thatobservables do not commute:
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• Canonical commutation:
[Q, P] = i
• Equations of motion, “Heisenberg picture”
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Canonical commutation
[Q, P] = i
= 1
Represented by Q = x, P = - i d/dx
Canonical commutation is then just
the product rule:
Set
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Commutators: [A,B] := AB-BA2. Eigenvalue gaps are connected to commutators:
H uk =λ
k uk , H self-adjoint
Elementary gap formula:
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Commutators: [A,B] := AB-BA
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What do you get whenyou put canonical
commutation togetherwith the gap formula?
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Commutators and gaps
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The fundamental eigenvalue gap
• In quantum mechanics, an excitation energy
• In “spectral geometry” a geometric quantity
small gaps indicate decoupling (dumbbells)
(Cheeger, Yang-Yau, etc.)
large gaps indicate convexity/isoperimetric(Ashbaugh-Benguria)
Γ := λ2 - λ1
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Gap Lemma
H = your favorite self-adjoint operator, u1 thefundamental eigenfunction, and G is whatever you
want.
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Gap Lemma
H = your favorite self-adjoint operator, u1 thefundamental eigenfunction, and G is whatever you
want. CHOOSE IT WELL!
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Commutators and gaps
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Commutators and gaps
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Commutators and gaps
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But on the spectrum,
(λ - λ1) (λ2 - λ1) ≤ (λ - λ1)2
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Universal Bounds using Commutators
– Play off canonical commutation relations
against the specific form of the operator:H = p2 + V(x)
– Insert projections, take traces.
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Universal Bounds using Commutators
• A “sum rule” identity (Harrell-Stubbe, 1997):
Here, H is any Schrödinger operator, p is the gradient
(times -i if you are a physicist and you use atomic units)
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Universal Bounds with Commutators
• Compare with Hile-Protter:
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Universal Bounds with Commutators
• No sum on j - multiply by f(λ j), sum and
symmetrize
• Numerator only kinetic energy - no
potential.
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Among the consequences:
• All gaps: [λn,λn+1] ⊆ [λ-(n),λ+(n)], where
• The constant σ is a bound for the kinetic
energy/total energy. (σ
=1 for Laplace, but1/2 for the harmonic oscillator)
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Among the consequences:
• All gaps: [λn,λn+1] ⊆ [λ-(n),λ+(n)], where
• Dn is a statistical quantity calculated from
the lower eigenvalues.
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Among the consequences:
• All gaps: [λn,λn+1] ⊆ [λ-(n),λ+(n)], where
• Sharp for the harmonic oscillator for all n!
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And now for some completely
different commutators….
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Commutators: [A,B] := AB-BA3. Curvature is the effect that motions do not commute:
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Commutators: [A,B] := AB-BA
• More formally (from, e.g., Chavel, Riemannian Geometry, A Modern
Introduction: Given vector fields X,Y,Zand a connection ∇, the curvature tensor isgiven by:
R(X,Y) = [∇Y ,∇X ] - ∇[Y,X]
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Commutators: [A,B] := AB-BA3a. The equations of space curves are commutators:
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Commutators: [A,B] := AB-BA3a. The equations of space curves are commutators:
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Commutators: [A,B] := AB-BA3a. The equations of space curves are commutators:
Note: curvature is defined by a second commutator
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The Serret-Frenet equations as
commutator relations:
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Sum on m and integrate. QED
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Sum on m and integrate. QED
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Interpretation:
Algebraically, for quantum mechanics on a
wire, the natural H0
is not
p2,
but rather
H1/4 := p2 + κ2/4.
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That is, the gap for any H is
controlled by an expectation value
of H1/4.
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Bound is sharp for the circle:
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Gap bounds for (hyper) surfaces
Here h is the sum of the principal curvatures.
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where
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Bound is sharp for the sphere:
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Spinorial Canonical
Commutation
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Spinorial Canonical
Commutation
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Sum Rules
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Corollaries of sum rules
• Sharp universal bounds for all gaps
• Some estimates of partition function
Z(t) = ∑ exp(-t λk)
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Speculations and open problems
• Can one obtain/improve Lieb-Thirring bounds as a
consequence of sum rules?
• Full understanding of spectrum of Hg.
What spectral data needed to determine the curve?
What is the bifurcation value for the minimizer of λ1?
• Physical understanding of Hg and of the spinorial operators
it is related to.
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Sharp universal bound
for all gaps
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Partition function
Z(t) := tr(exp(-tH)).
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Partition function
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which implies