Quantum Mechanics Just Over the Horizon: Classroom Exercises Inform (New-ish)

Classic Experiments

Ricħard Robinett (Penn State University) (M. Belloni – Davidson College)

Physics Department Colloquium Thursday, Sept. 29th, 2016 West Virginia University

Eberly!

Academic history

• Asst/Assoc/Full Professor – 1986 - present - 2046

• Assoc Department Head – 1997 - present

• Director Ugrad Studies – 1997- present

• Director Grad Studies – 2003 - present

• Associate Dean (ECoS)

– 2013, 2015

Reference(s) for todays talk • The infinite well and Dirac delta function potentials as pedagogical,

mathematical and physical models in quantum mechanics, RWR/MB, Phys. Rep. 540, 25–122 (2014).

• Dimensional analysis as the other language of physics, RWR, Am. J. Phys. 83, 353-361 (2015)

• Less than perfect quantum wavefunctions in momentum-space: How ϕ(p) senses disturbances in the force, RRW/MB, Am. J. Phys. 79, 94-102 (2011).

• New identities from quantum-mechanical sum rules of parity-related potentials, RWR/MB et al. J. Phys. A: Math. Theor. 43 235202 (2010).

• Wigner quasi-probability distribution for the infinite square well: Energy eigenstates and time-dependent wave packets, RWR/MB, Am. J. Phys. 72, 1183-1192 (2004).

• Quantum wave packet revivals, RWR, Phys Rep. 392, 1–119 (2004). • Anatomy of a quantum bounce, M. A. Doncheski and RWR , Eur. J. Phys. 20

29–37 (1999) • The biharmonic oscillator and asymmetric linear potentials: from classical

trajectories to momentum-space probability densities in the extreme quantum limit, L J Ruckle, MB, and RWR, Eur. J. Phys. 33 ,1505–1525 (2012)

• Self-interference of a single Bose–Einstein condensate due to boundary effects, RWR, Phys. Scr. 73 681–684 (2006)

• Supersymmetric extensions of the infinite square well: Wave function properties and quantum sum rules, RWR/MB (in progress)

Topics for today: Pedagogy versus Experiment

1. Dimensional analysis • Gravitationally bound states of neutrons

2. Free-particle Gaussian wave packets • BECs

3. Wave packets in the infinite square well (ISW) • Quantum revivals and fractional revivals

4. |φ(p)| at large |p| - the ‘extreme’ quantum limit • H-atom and large-scattering length state systems

5. Supersymmetric version(s) of the ISW • New results • Curiosity • But outta time?

• ‘The Wiggles’ versus ‘The Spinners’

http://quantum.goetheanum.org/fileadmin/mas/Quantum/SternGerlach2.png

QM at the level of

PHYS 314 or

PHYS 451/452 level

Potential Energy

Kinetic Energy

Magnitude |ψ (x)| Wiggliness |dψ (x)/dx|

Big Small Big Small

Small Big Small Big

What sets the scales in E and L in terms of

m, g, and ħ?

L

E

Quantum bouncing ball

• E ≈ (mg2ħ2)1/3 ≈ 10-22 J • f ≈ (mg2/ħ)1/3 ≈ 1012 Hz (for m = 1 kg) • L ≈ (ħ2/m2g )1/3 ≈ 10-23 m

• E ≈ (mng2ħ2)1/3 ≈ 10-31 J ≈ 10-12 eV ≈ 1 peV • f ≈ (mng2/ħ)1/3 ≈ 103 Hz (for a neutron, mn) • L ≈ (ħ2/mn

2g )1/3 ≈ 10-5 m ≈ 10 μm

m, g, and ħ 1.

Where do neutrons, quantum mechanics, and gravity intersect?

• They’re spinors!

• They fall!

Where do neutrons, quantum mechanics, and gravity intersect?

• They have a phase in a gravitational field!

L ≈ (ħ2/m

n 2g ) 1/3 ≈ 10 μm

E ≈ (mn g

2ħ2) 1/3 ≈ 1 peV

f ≈ (mng2/ħ)1/3 ≈ 103 Hz E ≈ (mng2ħ2)1/3 ≈ 1 peV

2. Free particle Gaussian wave

packets

Spreading Gaussian wave packet Rojansky textbook (1938) – Fig. 19

Physics GRE: Good luck on 10/29!

Interlude: Ways to know you’re getting old(er)

And if you’re a physicist, you start knowing about the Wigner distribution!

Being bilingual in ‘x’ and ‘p’ at the same time (i.e. Spanglish)

• Ψ(x) and φ(p) but you can translate

• QM phase space? • Has the right

‘projections’!

• But it is NOT positive-definite, hence ‘quasi-probability density’

• But still useful!

• Wigner paper listed as No. 1 ‘Revived Classic’

“…have never knowingly seen it discussed….”

Rock on!!

x

p

Free p=0 Gaussian wave packets: Textbook versus research (BEC) versions

http://cua.mit.edu/ketterle_group/Projects_1996/Ballistic_expansion/TOF_figure.jpg

But how do you actually see the wiggles? • Two expanding p=0 Gaussians and interfere them

• Fringes get further apart in time • Higher |p| (shorter λ) components ‘hit’ first • Smaller |p| (longer λ) ones ‘hit’ later

And experimentally……

Phase-space (Wigner function) version

Interlude: Capturing a free-particle wave packet: One wall at a time – towards the ISW

Exact solution for ‘half-plane’, free-particle wave packets, for

any 1D solution, ψ(x,t)

The anatomy of a quantum ‘bounce’

‘’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.

BEFORE AFTER DURING

+p0

-p0

Add a second wall (but in 2D)

3. The infinite square well (ISW)

ψ∞(x,t) = Σn {ψ(x+2Ln,t) - ψ (-x+2Ln,t)}

4. But you don’t really do it that way…. ISW wave packet construction

• Put a Gaussian in the ISW… • …much narrower than the

well, so you pretend the walls don’t matter

• Calculate the expansion

coefficients via an = ∫ψ(G)(x,0) un(x) dx and ψ(x,t) = Σ an un(x) e-iE(n)t/ħ

• And then let ‘er go!

Can you make real wave packets?

Can you see their classical periodicity?

Do try this at home!

Time-dependence of the ISW wave packet

• Time-development determined by e-iE(n)t/ħ factors, where E(n) = E0 n2 and E0 = ħ2π2/2mL2

• At a time given by Trev = 2πħ/E0, all of the exp(-i2πn2) phases will return to unity,

• …and the wave packet will return exactly to its initial state at Trev…a quantum revival

• Exact for the ISW – approximate otherwise 2

• NHRA

• NASCAR

• But exact revivals also happen at Trev/2

• So we get two ‘mini-packets’!

But there are lots of integers!

General results from number theory: Gauss sums

Fractional revivals in popular culture

https://www.youtube.com/watch?v=5H44R6TpWG8

http://sciencedemonstrations.fas.harvard.edu/presentations/pendulum-waves

Or search ‘harvard pendulum waves YouTube Phillip Glass’

Or search ‘Army of Darkness Tiny Ash scene ‘

“Half psychotic, sick, hypnotic Got my blueprint, it's symphonic”

Optics version • The Talbot effect is a near-field diffraction effect first observed in 1836 by Henry Fox Talbot.[1] When a plane wave is incident upon a periodic diffraction grating, the image of the grating is repeated at regular distances away from the grating plane. The regular distance is called the Talbot length, and the repeated images are called self images or Talbot images. Furthermore, at half the Talbot length, a self-image also occurs, but phase-shifted by half a period (the physical meaning of this is that it is laterally shifted by half the width of the grating period).

https://en.wikipedia.org/wiki/Talbot_effect

One of the earliest mentions of ‘revivals’ is in a pedagogical journal

Ugrad at the time!

Now a

prof at IIT

4. Classical versus quantum momentum distributions:

Bouncing ball

|ψn(x)|2 versus x

|φn(p)|2 versus p

Classically disallowed

region

Classical and quantum momentum distributions:

Infinite square well

Infinite square well (ISW): It’s JUST BARELY a member of the QM family!

• Ψ(x) has a kink at each wall

• Ψ’’(x) is

singular

• But <p2> is OK

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

How <p2> work in momentum space?

• ISW φ(p) has power-law type behavior – 1/p2

• <p2> still well

behaved for ISW

• Single-δ has simple φ(p) – 1/p2 and <p2> also converges!

• So |φ(p)| for large |p| is useful

Not here!

But here!!

Are there real singular potentials with power-law ø(p) ‘tails’?

• H-atom V(r) ≈ 1/r

• For the H

ground state

• Another power law!

Gebenbaur

But how about the δ-function potential?

Shamelessly copied from http://www.int.washington.edu/PROGRAMS/int20/talks/Deborah.Jin.pdf

• If V(x) is discontinuous in the ‘k’-th derivative, V(k)(x)? • k = -1 – infinite potential (ISW/δ-function) – ø(p) ≈

1/p2

• k = 0 – discontinuous potential (finite well) - ø(p) ≈ 1/p3

• k = +1 – potential with cusp (‘V’ potential) - ø(p) ≈ 1/p4

• ... • Harmonic oscillator – perfectly behaved !

• Not power-law behavior, but pne-p^2

Potentials!

Working on this since 1977!

Bucket list - □

The Obi-Wan Kenobi theorem

• Potential? • k = -1 (singular)

• k = 0

(discontinuous)

• k=1 (cusp)

• ……..

The Belloni-Robinett theorm

5. Isospectral quantum systems • Supersymmetry allows you to generate pairs

of potentials with (almost) the same energy spectra

• Generalized raising and lowering operators

Familiar 1D state -> Supersymmetric version V(-)(x) -> V(+)(x)

• The SUSY version of the 1D SHO is the 1D SHO! • The SUSY version of the 3D Coulomb problem is the

3D Coulomb problem (different angular momentum values)

• For most other systems, the SUSY version is different

1D SUSY QM

The SUSY version of the ISW

You can generate all of the new (S=1) ψn(x) and øn(p)

Super-ISW • For the Super-ISW, because of the smoother walls

– Ψn(x) → x2 as x → 0 – due to an “angular momentum barrier”

– Φn (p) → 1/p3 as p→∞ (Obi-Wan Kenobi theorem!)

Blue – ISW Red = SUSY

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 ψn(x) → x(1+S)

• With φn(p) → 1/p(2+S) and matrix elements which converge faster and more sum rules converge!

* A. Khare, AIP Conference Proceedings 744, 133 (2004)

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?

Conclusions • There are many modern(ish) experimental

realizations of quantum mechanics that are ‘just over the horizon’ from standard problems

• To mix another metaphor – ‘old chestnuts’ can still have neat things inside

• Be curious!

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 9th, 2016

Tufts University