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Less than perfect wave functions in momentum-space: How φ(p) senses disturbances in the force*,# Richard Robinett (Penn State) M. Belloni (Davidson College) May 25, 1977 * To appear in Am. J. Phys A pedagogical talk Fall 2010 #arxiv.org/abs/1010.4244 Why a pedagogical talk? • Eugene Golowich – “Most of us will make a much bigger contribution in education than in research” – maybe a pedagogical talk? • Barry Holstein – Am. J. Phys ‘guru’ for years and encyclopedic knowledge of everything - maybe something with some history? – Explaining complex ideas at the ugrad level – If Barry knows that this has all been done before, please let him be silent until the end! (or until drinks tonight) • John Donoghue – Focus on contact with experiments – maybe a nod to that? – Systematic expansions in everything • It’s what I have time for nowadays, and most recent paper After all, role models are very important Richard Feynman (1918 – 1988) Nobel prize 1965 Richard Robinett (1953 - ) No Nobel prize but not dead☺ Connections between position- and momentum-space in QM • Review of some pedagogical aspects of x-p in QM • Wiggles in ψ(x) depend on V(x) and show connections to p-space – Bound state problems and free particles • Momentum-space φ(p) also shows semi-classical behavior • Wigner distribution illustrates x-p correlations • Are there other connections? One we hadn’t seen before! New connections? (today’s talk) • Many of the most familiar 1D QM problems are based on potentials which are `less than perfect’ – Single δ(x), SW, quantum bouncer, etc. are singular – Finite wells are discontinuous V(X) – V(x) = F|x| has a discontinuous V’(x) • In such potentials, ψ(x) can be `kinky’ (discontinuous derivative at some order) • Does that `kink’ have a direct impact on φ(p)? – Yes! – It gives φ(p) a large-|p| power-law `tail’ which can be written down knowing only ψ(x) at the `kink’ Standard WKB-like visualizations for x-p • Earliest picture I can find (Pauling and Wilson, 1935) • Wigglier and smaller near x=0 (moving faster there) • Less wiggly and bigger near x = turning points (moving slower there) Bumper sticker: The wigglier ψ(x), the more momentum Works for free particles too Less wiggly in back (slow) Physics GRE problem More wiggly in front (fast) Semi-classical --|ψ(x)|2 versus |φ(p)|2 • SHO |ψ(x)|2 |φ(p)|2 • ∞SW |ψ(x)|2 |φ(p)|2 |ψ(x)|2 |φ(p)|2 • V(x) = F|x| |ψ(x)|2 |φ(p)|2 Revived interest in the Wigner Distribution June 2004 • Included in Physics Today review article (on ‘revived classics’) “…owe their renewed popularity to the upsurge of interest in quantum information phenomena.” How do YOU feel about the Wigner distribution • Referee report describing his/her experience with the Wigner distribution… “...never knowingly seen it…” (like the House Un-American Activities Committee?) Wigner distribution for free-particle Gaussian wave packet Fast components outpace the slow ones This is still very classical The Wigner distribution is useful for non-classical things, like wave packet revivals Look at wave packet motion in the infinite well! ‘’Wigner’s eye view’’, before, during, and after the ‘splash’ Right wall is here +p0 -p0 Smooth, classical, narrow, and going to the right BEFORE Full of wiggles, and very non-positive when quantum interference effects are present. DURING Smooth, classical, wider, and going to the left AFTER Fractional quantum wave packet revivals (yielding Schrödinger cat-type states) • At Trev/4, you get a linear combination of two ‘mini’packets … two ‘bumps’ per classical period. Wigner distribution visualization • At Trev/3, you get even more interesting structures. So, new stuff (?) from old examples • Many 1D textbook problems are based on `poorly behaved’ potentials • Resulting ψ(x) `less than perfect’ in some derivative • Wiggliness of ψ(x) has connections to p • What effect does a ‘generalized kink’ in ψ(x) have on φ(p) – Big kinks φ(p) at large |p| • Consider three simple cases to `experiment’ – Single δ(x), ∞SW, and `half oscillator’ Single δ(x) potential • Single attractive delta function potential and discontinuity • Normalized wave function • Poorly behaved ψ’’(x) • But <p2> is OK Both give the same result Single δ(x) potential in p-space Power-law behavior of φ(p) for large |p| Can rewrite in very suggestive way Infinite square well (∞SW) example • Ψ(x) has a kink at each wall • Ψ’’(x) is singular • But <p2> is OK • Φ(p) has same power-law type behavior • <p2> still well behaved • Consistent with simple formula! • Contributions from each wall ISW (cont’d) More complex example: The `half-SHO’ • The `half oscillator’ is a familiar pedagogical example (see GRE examples below) • Ψ(x) is easy to get (√2 ψn(x) for x ≥ 0, for n odd) • Φ(p) can be obtained numerically `Half-oscillator’ in p-space • Re[ ] and Im[ ] parts give WKB type agreement to classical momentum distribution • Looky here! • For large |p|, the Im[ ] part dies exponentially, while the Re[ ] gives the power-law behavior we’ve seen. classical region p >> +Qn – deeply quantum limit Lots more examples: Can we infer the general result? • Quantum bouncer (Airy function solutions) – Another singular case • Finite wells, step potentials of various types – V(x) just discontinuous • V(x) = F|x| (Airy function solutions) – V’(x) discontinuous • `Biharmonic oscillator’ – V’’(x) discontinuous General result (by example) • From all of these examples, we infer the simple general result, namely Quick proof – `hold your nose’ math Do the real and imaginary parts separately – nothing new here Look at I1,2(p) separately Assume the kink is at x = 0, split it there, and add convergence factors e x Proof (cont’d) Do the resulting integrals exactly, and then take some limit. Voilà And the imaginary part gives you all of the other differences Real-life example (finally, phenomenology) • H-atom • Singular potential in 3D • Semi-classical WKB-like limit Smart people have done the H-atom in momentum space • Radial wave function R(r) goes like rl • The bigger the l, the smoother it goes to zero • So we’d expect powerlaw behavior for φ(p) • And φ(p) ~ 1/pl+4 More smart people… H-atom – ground state - (p) tail • Ground state (p) • McCarthy and Weigold data for φ|(p)|2 directly using (e,2e) method • Large |p| power law tail clearly seen Am. J. Phys. 51, 152-152 (1983) A real “thought” experiment for the hydrogen atom Conclusions • It’s still fun to do physics… • …even pedagogical stuff • Thanks to the UMass group for everything!