Less than perfect quantum wavefunctions in momentum
Transcript of Less than perfect quantum wavefunctions in momentum
Less than perfect wave functions in momentum‐space:
How φ(p) senses disturbances in the force*,#φ(p)
Richard Robinett (Penn State)
M B ll i (D id C ll )M. Belloni (Davidson College)
May 25, 1977 Fall 2010
* To appear in Am. J. Phys #arxiv.org/abs/1010.4244
May 25, 1977
A pedagogical talk
Why a pedagogical talk?y p g g
• Eugene Golowich– “Most of us will make a much bigger contribution in education thanMost 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 ofAm. 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 y , p
silent until the end! (or until drinks tonight)• John Donoghue
– Focus on contact with experiments – maybe a nod to that?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 paperIt s what I have time for nowadays, and most recent paper
After all, role models are very importantAfter all, role models are very important
Richard Feynman (1918 – 1988) Nobel prize 1965
Richard Robinett (1953 ‐ )
No Nobel prize but not dead☺Nobel prize 1965 No Nobel prize but not dead☺
Connections between position‐ and t i QMmomentum‐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’
Si l δ( ) SW b i l– Single δ(x), ¶SW, quantum bouncer, etc. are singular– Finite wells are discontinuous V(X)– V(x) = F|x| has a discontinuous V’(x)V(x) F|x| has a discontinuous V (x)
• In such potentials, ψ(x) can be `kinky’ (discontinuous d i i d )derivative at some order)
• Does that `kink’ ha e a direct impact on φ(p)?• Does that `kink’ have a direct impact on φ(p)?– Yes!– It gives φ(p) a large‐|p| power‐law `tail’ which can beIt 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‐pStandard WKB like visualizations for x p
• Earliest picture I can find (Pauling and Wilson, 1935)
• Wigglier and smaller 0 ( i fnear x=0 (moving faster
there)
• Less wiggly and bigger near x = turning pointsnear x = turning points (moving slower there) Bumper sticker:
The wigglier ψ(x), the more momentum
Works for free particles tooWorks for free particles too
More wiggly in front(fast)
Less wiggly in back(slow)
Physics GRE problem
( )
Semi‐classical ‐‐|ψ(x)|2 versus |φ(p)|2Semi classical |ψ(x)| versus |φ(p)|
• SHO |ψ( )|2SHO |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 Wi Di t ib tiWigner Distribution
• Included in Physics Today review
June 2004
Included in Physics Today review article (on ‘revived classics’)
“…owe their renewed popularity topopularity to the upsurge of interest in quantum qinformation phenomena.”
How do YOU feel about h d bthe 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 kGaussian wave packet
Fast components outpace the slow ones
This is still very classical
The Wigner distribution is useful forThe 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 f h ‘ l h’after the ‘splash’
Right wall is here
+p0p0
‐p0
Smooth, classical, narrow, and going to
the right
Smooth, classical, wider, and going to
Full of wiggles, and very non‐positive when
the right the leftquantum interference effects are present.
BEFOREAFTER
DURING
Fractional quantum wave packet revivals(yielding Schrödinger cat‐type states)
• At Trev/4 you get a linearAt Trev/4, you get a linear combination of two ‘mini’‐packets two ‘bumps’ perpackets … two bumps per classical period.
• At Trev/3, you get even i i
Wigner distribution visualization
more interesting structures.
So, new stuff (?) from old examplesSo, new stuff (?) from old examples
• Many 1D textbook problems are based on `poorly y p p ybehaved’ 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)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) potentialSingle δ(x) potential
• Single attractive delta f i i l dfunction 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‐spaceSingle δ(x) potential in p space
Power‐law behavior of φ(p) for large |p| φ(p) g |p|Can rewrite in very suggestive way
Infinite square well (∞SW) example• Ψ(x) has a kink at eachkink at each wall
• Ψ’’(x) is singularsingular
• But <p2> is OKOK
ISW (cont’d)ISW (cont d)• Φ(p) has same power‐law type behaviorbehavior
• <p2> still well pbehaved
• Consistent with• Consistent with simple formula!
• Contributions from each wall
More complex example: The `half‐SHO’More complex example: The half SHO
• The `half oscillator’ is a familiar pedagogicalThe half oscillator is a familiar pedagogical example (see GRE examples below)
• Ψ(x) is easy to get (√2 ψ (x) for x ≥ 0 for n odd)• 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‐spaceHalf oscillator in p space• Re[ ] and Im[ ] parts give WKB type agreement to classical momentum gdistribution
• Looky here!• For large |p|, the Im[ ] part dies exponentially, while the Re[ ] gives the power‐law behavior we’ve seen.the power law behavior we ve seen.
p >> +Qn – deeply quantum limitclassical region
Lots more examples:i f h l l ?Can we infer the general result?
• Quantum bouncer (Airy function solutions)Quantum bouncer (Airy function solutions)– Another singular case
• Finite wells step potentials of various types• 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)General result (by example)• From all of these examples, we infer the
i l l l lsimple general result, namely
Quick proof – `hold your nose’ mathQuick 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≤ex
Proof (cont’d)( )
Do the resulting integrals exactly and then take some limitDo the resulting integrals exactly, and then take some limit.
Voilà
And the imaginary part gives you all of the other differences
Real‐life example (f ll h l )(finally, phenomenology)
• H‐atomato• Singular potential in 3D
• Semi‐classicalSemi classical WKB‐like limit
Smart people have done the H‐atom in momentum space
• Radial wave functionRadial wave function R(r) goes like rl
Th bi th l th• The bigger the l, the smoother it goes to zero
• So we’d expect power‐law behavior for φ(p)
• And φ(p) ~ 1/pl+4
H‐atom – ground state ‐ f(p) tailH atom ground state f(p) tail
• Ground state f(p)Ground state f(p)
• McCarthy and• McCarthy and Weigold data for φ|(p)|2 directlyφ|(p)| directly using (e,2e) method
• Large |p| power law tail clearly seentail clearly seen
Am. J. Phys. 51, 152‐152 (1983)A real “thought” experiment for the hydrogen atom