1. Healthcare

The art of wavefunction visualisation
of amplitude and phase
Michael Bromley
Isaac Lenton
School of Maths/Physics
The University of Queensland
(Martin Kandes
Ricardo Carretero
San Diego State University)
y (osc. units)
from 5-D information
4
3
2
1
0
-1
-2
-3
-4
2π
π
0
-4 -3 -2 -1 0 1 2 3 4
x (osc. units)
3 vortices y’all
AIP 15min talk 2014
Bromley Group @ UQ
Computational AMO Physics
• Funding for my Future Fellowship comes from
http://www.smp.uq.edu.au/people/brom
• ultracold atom optics esp. BEC/atom interferometry
• atomic structure calculations, atomic clocks such as Al+
• non-linear optics of gases, eg. χ3 (I, ω)
• positron-atom interactions, eg. e+ Ca
Pre-school
• Mum bought me Commodore 64 (64KB RAM)
• Coded some BASIC games typed out from books.
• Amiga 500 (Motorola 68000 @ 7.09 MHz) 1024 KB RAM
• Learnt AmigaBASIC... DIY pictures of fractals etc
Movie 1 — Amiga raytracing software
• Inspired by, eg. The Amiga Juggler (1986)
https://www.youtube.com/watch?v=-yJNGwIcLtw
• First played with ray-tracing software
Single Particle Wavefunction
• The wavefunction, |Ψ(t)i, lies in Hilbert Space
• In position-space the wavefunction is complex-valued
(hx| ⊗ hy| ⊗ hz|) |Ψ(t)i = Ψ(x, y, z, t)
• represented through both an amplitude and (local) phase
Ψ (x, y, z, t) ≡ A (x, y, z, t) exp (i ϕ (x, y, z, t)) ,
• with frozen transverse, Ψ (x, y, z, t) = ψi (z)ψj (y)Ψ(x, t)
Ψ(x, t) = (hx| ⊗ hψi | ⊗ hψj |) |Ψ(t)i
Z Z
=
ψi∗ (z)ψj∗ (y)ψi (z)ψj (y)Ψ(x, t) dydz
Z
Z
= Ψ(x, t)
|ψi (z)|2 dz
|ψj (y)|2 dy = Ψ(x, t)
1-D Probability Density
• First computer visualisation of 1-D probability density
“Computer-Generated Motion Pictures of One-Dimensional
Quantum-Mechanical Transmission and Reflection Phenomena”,
Goldberg et al. Am. J. Phys. 35, 177 (1967)
3-D Wavefunctions I
• Consider the classic hydrogen (time-indep) eigenstates
|Ψnℓm i → Ψnℓm (~r, t) = exp(−iEn t/~)ψnℓ (r)Yℓm (θ, φ)
• Consider the three ℓ = 1 and m = −1, 0, 1
• visualise using contours based on probability density
http://commons.wikimedia.org/wiki/User:Geek3/hydrogen
• warning: two hours to generate each image!
Movie 2: 3-D Wavefunctions II
• Dauger atom-in-a-box (as seen in QM by Griffiths)
• i-thing apps http://daugerresearch.com/orbitals
start 2 min mark https://www.youtube.com/watch?v=OkRCnmpxZ4w
My computer science philosophy
• I’m a true believer in open-source (free-software)
• write my code in fortran using blend of OpenMP/MPI.
• At UQ we’ve built a (little supercomputer) with 12 Phi
• advantage is x86 architecture (also have GPUs/CUDA).
Intro to my atom optics research
• Consider the (single-particle) Rotating (BEC) NLSE
i
h
∂
ˆ z + g|ψ(r, t)|2 ψ(r, t)
ψ(r, t) = − 21 ∇2 + V (r, t) − ΩL
i ∂t
• Mean-field treatment with (g ∝ (N − 1) other atoms)
• Atoms are trapped eg. in all-optical ring-potential:
V(ρ,ϕ,z=0) (osc. units)
20
10
0
-20
-10
x (osc. units)
0
10
10
20 -20
-10
0
y (osc. units)
20
Example: The Sagnac effect
• Consider counter-propagating matter-waves in a ring
LΩ
L+
t>0
t=0
rotating
source
L−
t>0
L−
t= τ k/2
|Ω|
t= τ k/2
θ=|Ω|τ
L+
rotating
detector
t=0
k
t= τ k
first
collision
L−
http://arxiv.org/abs/1306.1308
L+
2-D Wavefunctions Ψ(x, y, t)
• Visualisation using fortran → gnuplot → ffmpeg
3-D Wavefunctions Ψ(x, y, z, t)
• Goal: HD-quality 3-D animations of 3-D wavefunctions
• using fortran → legacy VTK → paraview
• INSERT your cellophane NOW (h/t Tom Stace)
Thanks for your interest
• Free-software release of my holeynlse code + scripts
• Next I would like to visualise few-body systems such as