Download Less than perfect wave functions in momentum-space

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!