Download Quantum Angular Momentum Matrices

Quantum angular momentum
matrices – eigenvalues and SVs
Peter Loly
Department of Physics and Astronomy,
University of Manitoba, Winnipeg
WCLAM 14-15 May 2016
Abstract
• Replace “matrix norms” by “integer
measures”.
𝐽𝑥 for 𝑗=2, 𝑛 =2𝑗 + 1=5
0
1
0
0
0
1
0
𝟑
𝟐
0
0
0
𝟑
𝟐
0
𝟑
𝟐
0
0
0
𝟑
𝟐
0
1
0
0
0
1
0
𝐽𝑥 is non-negative
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Google Search
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matrix mechanics
-> IMAGES
Many of these matrices appear there.
Contemporary context includes “qubits”.
1925 Werner Heisenberg
91st anniversary!
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Niels Bohr, “Copenhagen” – 1912 electron orbits
June 1925 Heisenberg’s “matrix mechanics”, Uncertainty Principle.
Paul Dirac 1925 November 7; and 16th : Born, H and Jordan
Wolfgang Pauli 1926/27 – spin 1/2 matrices, Exclusion Principle.
1926 Erwin Schrӧdinger’s “wave mechanics”, his “cat”.
1972 B.L. van der Waerden, Notices of the AMS, Nov. 1997, 323-8.
Note remarks on C. Lanczos.
• Heisenberg just one of several students of Arnold Sommerfeld
(Munich) - to win Nobel Prize.
• Fritz London NOT! His student R.K.Eisenshitz, “Matrix Algebra for
Physicists”, c. 1972, supervised my undergraduate dissertation on
“The Theory of Measurement in Quantum Mechanics” in 1963.
Spin ½
1921 Stern-Gerlach experiment;
1927 Pauli matrices
• S=(ħ/2) σ; (ħ =h/2π); scale “half-Pauli” – set ħ=1
• USE “Pauli scale” [qubits] to avoid powers of ½
0 1
0 −𝑖
• σ𝑥 =
; σ𝑦 =
;
1 0
𝑖 0
1 0
1 0
2
• σ𝑧 =
; σ =3
;
0 −1
0 1
• Raising and lowering operators: σ± = σ𝑥 ±iσ𝑦 :
0 2
0 0
• σ+ =
; σ− =
;
0 0
2 0
𝑇 1 0
• Gramian: σ𝑖 (σ𝑖 ) =
, i = x, y, z
0 1
• N.B. by convention σ𝑖 is also used for singular values
Background
• Ph.D. – Interacting spin waves in Heisenberg’s
model of ferromagnetism using many-body
techniques (Green’s functions, quantum field
theory at finite temperatures) for boson and
spin models.
• Much later teaching “Introduction to
Theoretical Physics” with a chapter on
matrices … numerical examples of simple
matrices from MATLAB’s magic(n) function.
Overview
My interest in these fundamental square
matrices is in obtaining integer characteristic
polynomials from which integer sums of
eigenvalue and singular value powers follow.
They include imaginary matrices, and matrices
with no eigenvalues (j+, j-) where singular value
analysis becomes essential.
Previous work on integer matrices, especially
Latin squares and magic squares.
Matrix sums
• 1629 Albert Girard – powers of roots of
polynomials
• Later Newton’s identities
• G.A. Miller 1909, and 1916,1927
Girard’s identities 1629
• Characteristic polynomial:
for j=2,n=5: for 2𝑗𝑥,𝑦,𝑧 : cpe= -64 x+20 x3-x5
𝑛
1=1(𝑥
− 𝜆𝑖 )=𝑥 𝑛 − 𝑎1 𝑥 𝑛−1 + 𝑎2 𝑥 𝑛−2 − 𝑎3 𝑥 𝑛−3 +… = 0
• Sums of powers of roots:
• 𝐺𝑛 = 𝑛𝑖=1 𝜆𝑛𝑖
• 𝐺1 = 𝑎1 ; 𝐺2 = 𝑎12 − 2𝑎2 ; 𝐺3 = 𝑎13 − 3𝑎1 𝑎2 + 3𝑎3 ;
• Similar expressions follow for SVs with 𝜆𝑖 replaced by σ2𝑖
Vector Model
General formulae (ħ =h/2π)
(J.J.Sakurai, Modern Quantum Mechanics)
• J± = J𝑥 ±iJ𝑦 ⇒
•
1
3
5
j= ,1, ,2, ,…;
2
2
2
1
1
J𝑥 = (J+ +J− ); J𝑦 = (J+ − J− );
2
2𝑖
m=j,j-1,..,-j
• J𝑧 |j, m> =m ħ |j, m>
• J2 |j, m> = j(j+1) ħ2 |j, m>
• J± |j,m> = (𝑗 ∓ 𝑚)(𝑗 ± 𝑚 + 1) ħ|j, m ±1>
𝐽𝑥 for 𝑗=2, 𝑛 =2𝑗 + 1=5
0
1
0
0
0
1
0
𝟑
𝟐
0
0
0
𝟑
𝟐
0
𝟑
𝟐
0
0
0
𝟑
𝟐
0
1
0
0
0
1
0
𝐽𝑦 → 𝐽𝑥 𝑡𝑖𝑚𝑒𝑠 − 𝑖 𝑎𝑏𝑜𝑣𝑒 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙, 𝑖 𝑏𝑒𝑙𝑜𝑤;
𝑱𝒛 𝒐𝒏𝒍𝒚 𝒅𝒊𝒂𝒈𝒐𝒏𝒂𝒍: 𝟐, 𝟏, 𝟎, −𝟏, −𝟐;
𝒋𝟐 diagonal elements j(j+1)= 6, and all others vanish.
𝐽𝑥 , 𝐽𝑦 NOT Latin (only 3 symbols);
Not magic (4 different row, column, diagonal sums); Not anti-magic;
Not affine, nor doubly affine, … not stochastic…
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Sums
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EV2= 𝑖 λ𝑖 2
SV2= 𝑖 σ𝑖 2
And because it was useful for magic squares:
SV4= 𝑖 σ𝑖 4
Table 1: 2𝐽𝑥 [x,y,z] “Pauli scale”
<N.B. the factor “2” to avoid powers of ½: i.e. ¼, ⅛,… >
n=2j+1
j
r
EVs
SVs
EV2 or
𝑬𝑽𝟐
and SV2
SV4 or
𝑺𝑽𝟒
2
½
2
±1
1,1
2
2
3
1
2
±2,0
2,2,0
8
32
4
3/2
4
±3,±1
3,3,1,1
20
164
5
2
4
±4,±2,0
4,4,2,2,0
40
544
6
5/2
6
±5,±3,±1
5,5, 3,3,1,1
70
1414
7
3
6
±6, ±4,±2,0
6,6, 4,4,2,2,0
112
3136
EV2 and SV2 sums
• For 2𝐽𝑧 the even order (half-integral spin, 1/2,3/2,5/2,..) EV2,
namely 2,20,70,168,330,.. interlace perfectly with the odd order
(integral spin 1,2,3,,...) values 8,40,112,240,440,.., for the combined
integer series: 2,8,20,40,70,112,168,240,330,440,...
• This is Sloane's OEIS A007290 2*binomial(n,3) (Formerly M1831)
0,0,0,2,8,20,40,70,112,168,240,330,440,572,728,910,
• "This is the sequence for nuclear magic numbers in an idealized
spherical nucleus under the harmonic oscillator model.” - Jess
Tauber, May 20 2013"
• O. Haxel, JHD Jensen (NL 1963 with Maria Goeppert-Mayer) and HE
Suess, 1949 Phys. Rev. 75, 1766 – just a couple of pages referencing
three longer ones to appear in German journals.
𝑗=2, 𝐽+ (and 𝐽− )
0
2
0
0
0
0
0
𝟔
0
0
0
𝟎
0
𝟔
0
0
0
0
0
2
0
0
0
0
0
𝐽− ℎ𝑎𝑠 𝑠𝑎𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑏𝑒𝑙𝑜𝑤 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙
Both nilpotent – thanks Rachel Quinlan!
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Table 2: 𝐽+− , EVs vanish, SVs do not!
n=2j+1
j
r
EVs
SVs
2
½
1
0
1,0
3
1
2
0
4
3/2
3
0
5
2
4
0
6
5/2
5
0
7
3
6
0
𝑺𝑽𝟒
𝑺𝑽𝟐
1
1
𝟐, 𝟐, 0
4
8
2, 𝟑, 𝟑, 0
10
34
20
104
3, 2 𝟐, 2 𝟐, 𝟓, 𝟓, 0
35
259
2 𝟑,2 𝟑, 𝟏𝟎, 𝟏𝟎, 𝟔, 𝟔, 0
56
560
𝟔, 𝟔, 2, 2, 0
OEIS
𝑺𝑽𝟐 :
A000292 Tetrahedral (or triangular pyramidal)
numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.
(Formerly M3382 N1363)
𝑺𝑽𝟒 :
A033455 Convolution of nonzero squares
A000290 with themselves.
DISCLAIMER!
• https://en.wikipedia.org/wiki/Magic_number
_(physics)
• The seven most widely recognized magic
numbers as of 2007 are 2, 8, 20, 28, 50, 82,
and 126 (sequence A018226 in OEIS).
End
• Thanks to Andrew Senchuk, Adam Rogers
• 12May - Conjugate transpose used for jy…
• Question: Are there other physics topics
where singular values, sums of EVs and SVs, …
might be useful?
LINSTAT-2012, IWMS-21 at Bedlewo, Poland
http://home.cc.umanitoba.ca/~loly/
-> NEWS 2013 in LATEST NEWS…
• "Signatura of magic and Latin integer squares:
isentropic clans and indexing.” by Ian Cameron, Adam
Rogers and Peter D. Loly, from 2012 conference
published in Discussiones Mathematicae Probability
and Statistics, 33(1-2) (2013) 121-149, from
http://www.discuss.wmie.uz.zgora.pl/ps"
• “plain version of 2012 conference paper published in
Discussiones Mathematicae Probability and Statistics,
33(1-2) (2013) 121-149. [DMPS]
• “Data Appendix for Signatura”: Ian Cameron, Adam
Rogers and Peter D. Loly
Some talks and papers on
“magical squares”
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LOLY, The Invariance of the Moment of Inertia of Magic Squares , The Mathematical Gazette, 88,
March 2004, 151-153.
Daniel Schindel, Matthew Rempel and Peter LOLY, Enumerating the bent diagonal squares
of Dr Benjamin Franklin FRS, Proceedings of the Royal Society A: Physical, Mathematical and
Engineering, 462 (2006) 2271-2279.
LOLY, Franklin Squares: A Chapter in the Scientific Studies of Magical Squares, Complex Systems
17 (2007) 143-161 from NKS2006 (Wolfram-Mathematica).
Eigenvalues in the Universe of Matrix Elements 1..n-squared. IWMS-16 2007 – Introductory
Keynote Talk (with Ian Cameron and Adam Rogers) – SVDs:
Peter LOLY, Ian Cameron, Walter Trump and Daniel Schindel, Magic square spectra, Linear Algebra
and its Applications, 430 (2009)
WCLAM2008 Winnipeg – Compound MS - Adam Rogers with Loly and Styan – Discussion - Loly:
some are matrices with only the linesum EV of doubly stochastics.
Cameron, Adam Rogers & Peter Loly, Signatura of magic and Latin integer squares: isentropic
clans and indexing., IWMS 2012 Bedlewo, Poland – VIDEO
http://www.physics.umanitoba.ca/~icamern/Poland2012/
Discussiones Mathematicae Probability and Statistics, 33(1-2) (2013) 121-149.
CMS2014 Winnipeg – Multimagic (Knut Vik) Latin Squares with Cameron and Rogers
WCLAM2016 - now
Shannon Entropy and Magical Squares
• Newton, P.K. & DeSalvo, S.A. (2010) [“NDS”]
“The Shannon entropy of Sudoku matrices”, Proc. R.
Soc. A 466:1957-1975 [Online Feb. 2010.]
• Immediately clear to us that we could extend NDS
using our studies of the singular values of magic
squares at IWMS16 in 2007, published in Lin. Alg.
Appl. 430 (10) 2659-2680, 2009.
• DMPS
• http://home.cc.umanitoba.ca/~loly/
Shannon Entropy
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NDS formula: h= (-)∑ni=1 σi ln(σi); < σi are normalized σi >
hmax = ln(n)
NDS compression measure:
C=1-h/ln(n)*100%
[100% - uniform; 0% - random]
Effective rank: erank=exp[h]
N.B. h,C and erank are related.
An index, R, of the sum of the 4th powers of the SVs
(omitting the first linesum SV) is also a useful metric.
Magic squares, Morse—Hedlund
sequence, Hilbert-Kamke and Waring
problems
• Sergeyev, Y.D., “Generation of symmetric
exponential sums”, arXiv:1103.2043v1
[math.NT] 10 Mar 2011
“Almost All Integer Matrices Have No
Integer Eigenvalues”, Greg Martin and
Erick B. Wong, UBC:
• www.math.ubc.ca/~gerg/papers/downloads/
AAIMHNIE.pdf