Sum rules in non-relativistic quantum mechanics: A pedagogical tutorial

R. W. Robinett (Penn State University)

(M. Belloni – Davidson College)

Foundations of Nonlinear Optics Tuesday, August 9th, 2016

Tufts University

Background/reference papers (all in the conference file)

• M. Belloni and RWR, ``Quantum mechanical sum rules for two (three) model systems’’, Am. J. Phys. 76, 798-806 (2008).

• M. Belloni and RWR, ``The infinite well and Dirac delta function potentials as pedagogical, mathematical, and physical models in quantum mechanics’’, Phys. Rep. 540, 25-122 (2014).

• M. Belloni and RWR, ``Less than perfect quantum wavefunctions in momentum-space: How φ(p) senses disturbances in the force”, Am. J. Phys. 79, 94-102 (2011).

• M. Belloni and RWR, ``Supersymmetric extensions of the infinite square well: Quantum wave functions and applications to energy-weighted sum rules”, (in preparation) – Thanks to the organizers for the push to finish this!

Overview/outline of topics • A different (non sum rule) pedagogical example

– φ(p) at large |p| in 1D • Derivations of sum rules • 1D examples of how they ‘work’ (checking them)

1. Harmonic oscillator 2. Infinite square well (ISW) 3. Single δ-function

• Sum rules in (almost) iso-spectral quantum systems – How energy-weighted sum rules ‘work’ in

supersymmetric extension(s) of the ISW

Momentum-space wave functions - ø(p)

• Pedagogical use, for semi-classical connections

ISW Harmonic oscillator Either ψn(x) or φn(p)

Measuring φ(p) (for large p) for reals! Power-law behavior for ‘spikey’ potentials

• Hydrogen atom

• Short-range (δ) interactions

D. Jin et al. group

E. Weigold, Am. J. Phys. 51, 152-152 (1983) A real “thought” experiment for the hydrogen atom

“Obi-Wan Kenobi Theorem”

• If…

• …then

• k = -1 – infinite potential (ISW/δ-function) - 1/p2

• k = 0 – discontinuous potential (finite well) - 1/p3

• k = +1 – potential with cusp (‘V’ potential) - 1/p4 • ... • Harmonic oscillator – perfectly behaved !

• Not power-law behavior – e-p^2

• Why do we even care about this for sum rules? – Convergence of the sums (steepest descent ideas)

Quantum mechanical sum rules • What do students need to know to derive sum rules?

– Complete set of states – OK to multiply by ① anywhere – Commutation relations

• Good references

– Bethe and Jackiw, Intermediate Quantum Mechanics – R. Jackiw, Phys. Rev. D 157, 1220-1225 (1967) – Belloni and RWR, AJP, 76, 798-806 (2008)

• Derivations most often relegated to ‘end-of-chapter’

problems

Just completeness first

• First of the Bethe-Jackiw dipole moment sum rules

Insert ①

Thomas-Reiche-Kuhn (TRK) sum rule: Start using commutators

Insert ① twice!

Historically important!

Lots of dipole moment sum rules with different powers of (Ek-En)

1. Closure

2. TRK

3. ‘Energy’

4. ‘Force times

momentum’

5. ‘Force squared’

And another energy-difference weighted quantity

• 2nd order perturbation theory result

• for Stark effect (or any linear potential, V(x) = Fx)

• …has the same format, but with energy differences

on the bottom – so same ‘math tricks’ should work

Even more sum rules

• Monopole

• Bethe-Bloch

• Wang* general result (TRK case using F(x) = x) * S. Wang, Phys. Rev. A 60, 262-266 (1999)

Pedagogical uses • Deriving sum rules (QM formalism)

• Checking/confirming identities for familiar model

cases (math formalism) 1. Harmonic oscillator 2. Infinite square well 3. Single δ-function

• Math methods (‘tricks’) for evaluating the sums (integrals) that appear

• How do the convergence properties of these sums relate to φ(p) and the ‘smoothness’ of the original ψ(x) and potential V(x)

Harmonic oscillator example • V(x) = mω2 x2/2 and En = (n+1/2)ħω and wave

functions well-known • Dipole (or any xp) matrix elements, <n|xp|k>,

can be obtained by using 1. identities for integrals over Hermite polynomials, 2. raising/lowering operators (much nicer!)

• Dipole matrix elements are easy to evaluate

• So the infinite sums are actually finite sums and

the series are ‘super-convergent’ • The TRK sum rule is trivial to verify for the SHO

SHO example (cont’d)

Φ(p) ~ e-p^2 !

Stark effect (linear potential) for the SHO

• 2nd order shift

• which is actually a classical result since

Infinite square well (ISW) (The workhorse, not showhorse, of QM)

• Convergence properties? Now it matters! • <n|x|k> ~ 1/k3 ,while ΔE ~ k2 for large k • The first three sum rules converge (as they must!)

but the rest do not

How to do the resulting sums • The TRK sum rule is given by • …with similar expressions for other (convergent)

sum rules • One ‘basic’ trick allows you to evaluate all such

sums and confirm the identities

Stark effect for the ISW

• Shift to ground state (n=1) energy is negative • Shifts to all other states (n>1) are positive! • Agrees with ‘Dalgarno-Lewis’ method calculations • Consistent with WKB ‘guess’ for power-law

potentials - Vk(x) = |x/a|k giving αn ~ n 2(2-k)/(2+k)

Same math tricks!

Single δ-function – the continuum matters!

Single δ-function (cont’d) • Matrix elements go like 1/k3 while ΔE goes like k2, so

again the first three sum rules converge • Same φ(p) behavior (1/p2, via the OWK theorem) as

well as matrix elements as ISW • The TRK (and all other convergent) sum rules require

integrals – use contour integration - math trick!

• The 2nd order Stark shift also works (Dalgarno-Lewis method or solving the problem with Airy functions)

Exploring energy weighted sum rules What role do the energies play?

• Each sum rule is an infinite set of constraints

• The energy differences are like the ‘rows’ in a tapestry

• The matrix elements (dynamics) are like the columns

• Can we have two systems with the same energies, but different dynamics?

Sum rules in isospectral systems • Supersymmetry allows you to generate pairs

of potentials with (almost) the same energy spectra

• Uses just raising and lowering operators The SUSY version of the SHO is the SHO!

Familiar 1D state -> Supersymmetric version V(-)(x) -> V(+)(x)

The SUSY version of the ISW

You can generate all of the new

ψn(x) and øn(p)

Super-ISW • For the Super-ISW, because of the smoother walls

– Ψn(x) → x2 as x → 0 – due to an “angular momentum barrier”

– Φn (p) → 1/p3 as p→∞ (Obi-Wan Kenobi theorem) – <n|x|k> → 1/k4 as k gets large, so all sums converge

faster and one additional sum rule is convergent

Blue – ISW Red = SUSY version

Super(symmetrize) me (again and again and again…..)

• You can repeatedly super-symmetrize the ISW*

• Note the “angular momentum” like S(S+1) factor • Increasingly ‘smooth’ wave functions at the

boundaries with ψn(x) → x(1+S)

• With φn(p) → 1/p(2+S) and matrix elements which converge faster and more sum rules converge

* A. Khare, AIP Conference Proceedings 744, 133 (2004)

Conclusions (and apologies!) • I talked way too long! • Sum rules are a great ‘quantum playground’

for many types of QM formalism • There are an infinite number of ‘parallel

universe’ playgrounds due to SUSY (ISWS!)

• Thanks again for the motivation to work on this new problem!