Less than perfect wave functions in momentum-space:

How φ(p) senses disturbances in the force*,#

Richard Robinett (Penn State)M. Belloni (Davidson College)

* To appear in Am. J. Phys #arxiv.org/abs/1010.4244

May 25, 1977

A pedagogical talk

Fall 2010

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

Physics GRE problem

More wiggly in front(fast)

Less wiggly in back(slow)

Semi-classical --|ψ(x)|2 versus |φ(p)|2

• SHO

• ∞SW

• V(x) = F|x|

|ψ(x)|2|φ(p)|2

|ψ(x)|2 |φ(p)|2

|ψ(x)|2 |φ(p)|2 |ψ(x)|2 |φ(p)|2

Revived interest in the Wigner Distribution

• Included in Physics Today review article (on ‘revived classics’)

“…owe their renewed popularity to the upsurge of interest in quantum information phenomena.”

June 2004

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’

Smooth, classical, narrow, and going to

the right

Smooth, classical, wider, and going to

the left

Full of wiggles, and very non-positive when

quantum interference effects are present.

BEFOREAFTER

DURING

+p0

-p0

Right wall is here

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.

• At Trev/3, you get even more interesting structures.

Wigner distribution visualization

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

ISW (cont’d)• Φ(p) has same

power-law type behavior

• <p2> still well behaved

• Consistent with simple formula!

• Contributions from each wall

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.

p >> +Qn – deeply quantum limitclassical region

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

Assume the kink is at x = 0, split it there, and add convergence factors

ex

Look at I1,2(p) separately

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 power-law 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!