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

Transcription

1 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 9 th, 2016 Tufts University

2 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, (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, (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, (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!

3 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

4 Momentum-space wave functions - ø(p) Pedagogical use, for semi-classical connections ISW Harmonic oscillator Either ψ n (x) or φ n (p)

5 Measuring φ(p) (for large p) for reals! Power-law behavior for spikey potentials Hydrogen atom E. Weigold, Am. J. Phys. 51, (1983) A real thought experiment for the hydrogen atom Short-range (δ) interactions D. Jin et al. group

6 Obi-Wan Kenobi Theorem If then

7 k = -1 infinite potential (ISW/δ-function) - 1/p 2 k = 0 discontinuous potential (finite well) - 1/p 3 k = +1 potential with cusp ( V potential) - 1/p 4... 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)

8 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, (1967) Belloni and RWR, AJP, 76, (2008) Derivations most often relegated to end-of-chapter problems

9 Just completeness first Insert 1 First of the Bethe-Jackiw dipole moment sum rules

10 Thomas-Reiche-Kuhn (TRK) sum rule: Start using commutators Insert 1 twice!

11 Historically important!

12 Lots of dipole moment sum rules with 1. Closure 2. TRK 3. Energy 4. Force times momentum 5. Force squared different powers of (E k -E n )

13 And another energy-difference weighted quantity 2 nd 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

14 Monopole Even more sum rules Bethe-Bloch Wang* general result (TRK case using F(x) = x) * S. Wang, Phys. Rev. A 60, (1999)

15 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)

16 Harmonic oscillator example V(x) = mω 2 x 2 /2 and E n = (n+1/2)ħω and wave functions well-known Dipole (or any x p ) matrix elements, <n x p k>, can be obtained by using 1. identities for integrals over Hermite polynomials, 2. raising/lowering operators (much nicer!)

17 SHO example (cont d) Dipole matrix elements are easy to evaluate So the infinite sums are actually finite sums and the series are super-convergent Φ(p) ~ e -p^2! The TRK sum rule is trivial to verify for the SHO

18 Stark effect (linear potential) for the SHO 2 nd order shift which is actually a classical result since

19 Infinite square well (ISW) (The workhorse, not showhorse, of QM) Convergence properties? Now it matters! <n x k> ~ 1/k 3,while ΔE ~ k 2 for large k The first three sum rules converge (as they must!) but the rest do not

20 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

21 Stark effect for the ISW Same math tricks! 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 - V k (x) = x/a k giving α n ~ n 2(2-k)/(2+k)

22 Single δ-function the continuum matters!

23 Single δ-function (cont d) Matrix elements go like 1/k 3 while ΔE goes like k 2, so again the first three sum rules converge Same φ(p) behavior (1/p 2, 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 2 nd order Stark shift also works (Dalgarno-Lewis method or solving the problem with Airy functions)

24 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?

25 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)

26 The SUSY version of the ISW You can generate all of the new ψ n (x) and ø n (p)

27 Super-ISW For the Super-ISW, because of the smoother walls Ψ n (x) x 2 as x 0 due to an angular momentum barrier Blue ISW Red = SUSY version Φ n (p) 1/p 3 as p (Obi-Wan Kenobi theorem) <n x k> 1/k 4 as k gets large, so all sums converge faster and one additional sum rule is convergent

28 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 ψ n (x) x (1+S) With φ n (p) 1/p (2+S) and matrix elements which converge faster and more sum rules converge * A. Khare, AIP Conference Proceedings 744, 133 (2004)

29 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 (ISW S!) Thanks again for the motivation to work on this new problem!