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Quantum Mechanics (II)...

ecterrab 14837
physics
December 06 2013
6 0

The attached presentation is the second one of a sequence of three that we want to do on Quantum Mechanics using Computer Algebra. The first presentation was about the equation for a quantum system of identical particles, the Gross-Pitaevskii equation (GPE). This second presentation is about the spectrum of its solutions. The level is that of an advanced undergraduate QM course. The novelty is again in the way these problems can be tackled nowadays in a computer algebra worksheet with Physics.

 

The Gross-Pitaevskii equation and Bogoliubov spectrum
  

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft, Canada

 

Departing from the equation for a quantum system of identical boson particles, i.e.the Gross-Pitaevskii equation (GPE), the dispersion relation for plane-wave solutions are derived, as well as the Bogoliubov equations and dispersion relations for small perturbations `δϕ` around the GPE stationary solutions.

Stationary and plane-wave solutions to the Gross-Pitaevskii equation
 


Problem: Given the Gross-Pitaevskii equation, NULL

I*`ℏ`*psi[t] = (G*abs(psi)^2+V)*psi-`ℏ`^2*%Laplacian(psi)/(2*m)

  

a) Derive a relationship between the chemical potential mu entering the phase of stationary, uniform solutions, the atom-atom interaction constant G and the particle density n = abs(psi)^2 in the absence of an external field (V = 0).
  

b) Derive the dispersion relation for plane-wave solutions as a function of G and n. 

  

 

Background: The Gross-Pitaevskii equation is particularly useful to describe Bose Einstein condensates (BEC) of cold atomic gases [3, 4, 5]. The temperature of these cold atomic gases is typically in the w100 nano-Kelvin range. The atom-atom interaction are repulsive for G > 0 and attractive for G < 0 , where G is the interaction constant. The GPE is also widely used in non-linear optics to model the propagation of light in optical fibers.

Solution
   

The Bogoliubov equations and dispersion relations
 

 

Problem: Given the Gross-Pitaevskii equation,

  

a) Derive the Bogoliubov equations, that is, equations for elementary excitations `&delta;&varphi;` and conjugate(`&delta;&varphi;`)around a GPE stationary solution `&varphi;`(x, y, z), NULL

 

"{[[i `&hbar;` (&PartialD;)/(&PartialD;t) `delta&varphi;`=-(`&hbar;`^2 (&nabla;)^2`delta&varphi;`)/(2 m)+(2 G |`&varphi;`|^2+V-mu) `delta&varphi;`+G `&varphi;`^2 (`delta&varphi;`),,],[i `&hbar;` (&PartialD;)/(&PartialD;t)( `delta&varphi;`)=+(`&hbar;`^2 (&nabla;)^2(`delta&varphi;`))/(2 m)-(2 G |`&varphi;`|^2+V-mu) (`delta&varphi;`)-G `delta&varphi;` ((`&varphi;`))^(2),,]]"

  


b) Show that the dispersion relations of these equations, known as the Bogoliubov spectrum, are given by
  

epsilon[k] = `&hbar;`*omega[k] and `&hbar;`*omega[k] = `&+-`(sqrt(`&hbar;`^4*k^4/(4*m^2)+`&hbar;`^2*k^2*G*n/m)),
  


where k is the wave number of the considered elementary excitation, epsilon[k] its energy or, equivalently, omega[k] its frequency.

Solution
   

NULL

References

NULL

[1] Gross-Pitaevskii equation (wiki)

[2] Continuity equation (wiki)
[3] Bose–Einstein condensate (wiki)

[4] Dispersion relations (wiki)

[5] Advances In Atomic Physics: An Overview, Claude Cohen-Tannoudji and David Guery-Odelin, World Scientific (2011), ISBN-10: 9812774963.

[6] Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN-13: 978-0123970237.

 

 

QuantumMechanics.pdf     Download QuantumMechanics2.mw

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Physics updates and multi-index summatio...

ecterrab 14837
physics summation
December 05 2013
4 0

Hi,
Relevant developments in Physics happened during the last month and a 1/2, some of them of interest beyond the use of this package. Among the most exciting things I can mention:

  1. The redefinition of the derivative rule for the complex components (abs, argument, conjugate, Im, Re, signum) together with the introduction of Wirtinger calculus, as an user-option to work with complex variables. In other words: it is now possible to compute taking z and its conjugate as independent variables and in equal footing.
  2. Introduction of textbook mathematical display for all the inert functions of the mathematical language, also for unknown functions f(x).
  3. New options in Physics:-Setup to indicate that some mathematical objects are real (different from assume(x, real), while integrated with `is` and `coulditbe`).
  4. A rather large number of micro developments advancing the integration of Physics with simplify, expand and combine.
  5. Another large number of micro developments for quantum mechanics.
  6. New options in Physics:-Setup to redefine sum as Physics:-Library:-Add, and with that have access to multiindex summation directly from sum, for instance as in sum(f(i, j), i + j <= n), including related typesetting copy & paste.

As usual the latest version of the package is available for download in the Maplesoft Physics: Research & Development webpage  and in the zip there is a worksheet illustrating all these developments. Below I'm copying the section related to the new redefinesum option of Physics:-Setup and multiindex summation.

Thanks to everyone who provided feedback, it has been of great value and at the root of this new round of developments.

December 4
 
• 

New option in Setup: redefinesum, so that the sum command is redefined in such a way that
    a) the sum arguments are processed in a way avoiding premature evaluation and related unexpected results or error interruptions
    b) the sum command includes new functionality present in Physics:-Library:-Add to perform sum over integer values of many indices, as in

"(&sum;)S(i,j)"     or  "(&sum;)S(i,j)" 

> 

restart; with(Physics); Setup(notation = true)

`* Partial match of 'notation' against keyword 'mathematicalnotation'`

  

[mathematicalnotation = true]

 (1.1)

New option: redefine sum so that its arguments are processed by the more modern Physics:-Library:-Add and so that it can perform multiindice summation.

 

Example:

By default, the sum command is not redefined, so the value of redefinesum is

> 

Setup(redefinesum)

[redefinesum = false]

 (1.2)

Consider this multiindex summation functionality of the Physics:-Library:-Add command

> 

Library:-Add(f[i, j], 1 <= i+j and i+j <= n)

Physics:-Library:-Add(f[i, j], i+j <= n, lowerbound = 1)

 (1.3)

For instance, for n = 2,

> 

eval(Physics[Library]:-Add(f[i, j], i+j <= n, lowerbound = 1), n = 2)

f[0, 1]+f[1, 0]+f[0, 2]+f[1, 1]+f[2, 0]

 (1.4)

This functionality can be plugged directly into the sum command. For that purpose, set redefinesum to true

> 

Setup(redefinesum = true)

[redefinesum = true]

 (1.5)

You can now compute directly with sum. The left-hand side is inert while the right-hand side is computed

> 

(%sum = sum)(f[i, j], i+j <= 2)

%sum(f[i, j], i+j <= 2) = f[0, 0]+f[0, 1]+f[1, 0]+f[0, 2]+f[1, 1]+f[2, 0]

 (1.6)
> 

(%sum = sum)(f[i, j], 1 <= i+j and i+j <= 2)

%sum(f[i, j], 1 <= i+j and i+j <= 2) = f[0, 1]+f[1, 0]+f[0, 2]+f[1, 1]+f[2, 0]

 (1.7)
> 

value(%sum(f[i, j], 1 <= i+j and i+j <= 2) = f[0, 1]+f[1, 0]+f[0, 2]+f[1, 1]+f[2, 0])

f[0, 1]+f[1, 0]+f[0, 2]+f[1, 1]+f[2, 0] = f[0, 1]+f[1, 0]+f[0, 2]+f[1, 1]+f[2, 0]

 (1.8)

The formula for the integer power of a sum

> 

(a+b+c)^n = sum(factorial(n)*a^p*b^q*c^r/(factorial(p)*factorial(q)*factorial(r)), p+q+r = n)

(a+b+c)^n = sum(Physics:-`*`(Physics:-`*`(Physics:-`*`(Physics:-`*`(factorial(n), Physics:-`*`(1, Physics:-`^`(Physics:-`*`(Physics:-`*`(factorial(p), factorial(q)), factorial(r)), -1))), Physics:-`^`(a, p)), Physics:-`^`(b, q)), Physics:-`^`(c, r)), p+q+r = n)

 (1.9)
> 

eval((a+b+c)^n = sum(Physics[`*`](Physics[`*`](Physics[`*`](Physics[`*`](factorial(n), Physics[`*`](1, Physics[`^`](Physics[`*`](Physics[`*`](factorial(p), factorial(q)), factorial(r)), -1))), Physics[`^`](a, p)), Physics[`^`](b, q)), Physics[`^`](c, r)), p+q+r = n), n = 2)

(a+b+c)^2 = a^2+2*a*b+2*a*c+b^2+2*b*c+c^2

 (1.10)
> 

eval((a+b+c)^n = sum(Physics[`*`](Physics[`*`](Physics[`*`](Physics[`*`](factorial(n), Physics[`*`](1, Physics[`^`](Physics[`*`](Physics[`*`](factorial(p), factorial(q)), factorial(r)), -1))), Physics[`^`](a, p)), Physics[`^`](b, q)), Physics[`^`](c, r)), p+q+r = n), n = 3)

(a+b+c)^3 = a^3+3*a^2*b+3*a^2*c+3*a*b^2+6*a*b*c+3*a*c^2+b^3+3*b^2*c+3*b*c^2+c^3

 (1.11)

Verify whether this equation is true

> 

(`@`(evalb, expand))((a+b+c)^3 = a^3+3*a^2*b+3*a^2*c+3*a*b^2+6*a*b*c+3*a*c^2+b^3+3*b^2*c+3*b*c^2+c^3)

true

 (1.12)

Besides this new functionality, the redefined sum does a more modern handling of its arguments, consider a typical problem posted in Maple primes

> 

a := 1; b := 2; j := 3

1

  

2

  

3

 (1.13)

In the following summation, j is a dummy summation index, so the value just assigned, j := 3, is not expected to interfer with the summation. This is the case with the redefined sum

> 

sum(f(j), j = a .. b)

f(1)+f(2)

 (1.14)

while without redefining sum the input above is interrupted with an error message. Likely, in this other case also reported in Mapleprimes

> 

g := proc (j) options operator, arrow; if j::odd then G[j] else 0 end if end proc

proc (j) options operator, arrow; if j::odd then G[j] else 0 end if end proc

 (1.15)

the following two summations can be performed after having redefining sum:

> 

sum(g(i), i = 1 .. f)

sum(g(i), i = 1 .. f)

 (1.16)

For the summation above, without redefining sum, it returns 0 instead of unevaluated, because of a premature evaluation of the function g(i) with an unassigned index i before performing the summation. Returning unevaluated as (1.16) permits evaluate the sum at a latter moment, for instance attributing a value to f

> 

eval(sum(g(i), i = 1 .. f), f = 3)

G[1]+G[3]

 (1.17)

And this other sum where f is given from the begining also returns 0 without redefining sum

> 

sum(g(i), i = 1 .. 3)

G[1]+G[3]

 (1.18)

Problems like this other one reported in Mapleprimes here also get resolved with this redefinition of sum.

 

 

Download sum_in_physics.mw

Edgardo S. Cheb-Terrab
Physics, Maplesoft

 

Quantum Mechanics using Computer Algebra...

ecterrab 14837
physics
November 27 2013
12 0

The attached presentation is the first one of a sequence of three that we wanted to do on Quantum Mechanics using Computer Algebra. The level is that of an advanced undergraduate QM course. Tackling this topic within a computer algebra worksheet in the way it's done below, however, is an entire novelty, and illustrates well the kind of computations that can be done today with Maple & Physics.

Ground state of a quantum system of identical boson particles
  

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

 

Departing from the Energy of a quantum system of identical boson particles, the field equation is derived. This is the Gross-Pitaevskii equation (GPE). A continuity equation for this system is also derived, showing that the velocity flow satisfies `&x`(VectorCalculus[Nabla], `#mover(mi("v"),mo("&rarr;"))`) = 0, i.e.: is irrotational.  

The Gross-Pitaevskii equation
 

NULL


Problem: derive the field equation describing the ground state of a quantum system of identical particles (bosons), that is, the Gross-Pitaevskii equation (GPE).

 

Background: The Gross-Pitaevskii equation is particularly useful to describe Bose Einstein condensates (BEC) of cold atomic gases [3, 4, 5], that is, an ensemble of identical quantum boson particles that interact with each other with an interaction constant G. The temperature of these cold atomic gases is typically in the w100 nano-Kelvin range. The atom-atom interactions are repulsive for G > 0 and attractive for G < 0  (which could lead to some instabilities). The GPE is also widely used in non-linear optics to model the propagation of light in optical fibers. In this area, GPE is known as "non-linear Schrödinger equation", and the non-linearity comes from the Kerr effect [6].

Solution
   

Continuity equation for a quantum system of identical particles
   

References

``

[1] Gross-Pitaevskii equation (wiki)

[2] Continuity equation (wiki)
[3] Bose–Einstein condensate (wiki)

[4] Bose-Einstein Condensation in Dilute Gases, C. J. Pethick and H. Smith, Second Edition, Cambridge (2008), ISBN-13: 978-0521846516.

[5] Advances In Atomic Physics: An Overview, Claude Cohen-Tannoudji and David Guery-Odelin, World Scientific (2011), ISBN-10: 9812774963.

[6] Nonlinear Fiber Optics, Fifth Edition (Optics and Photonics), Govind Agrawal, Academic Press (2012), ISBN-13: 978-0123970237.

 


Downlioad: QuantumMechanics1.mw,    QuantumMechanics1.pdf

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Inert Mathematical Functions...

ecterrab 14837
November 19 2013
10 0

Hi
In connection with recent developments in the Physics package, we now have mathematical typesetting for all the inert functions of the mathematical language. Hey! This is within the Physics update available on the Maplesoft Physics: Research & Development webpage. 

I think this is an interesting development that will concretely change the computational experience with these functions: it is not the same to compute with something you see displayed as %exp(x) instead of the same computation but flowing with it nicely displayed as an exponential function with the e in grey, reflecting that Maple understands this object as the exponential inert function, with known properties (all those of the active exp function), and so Maple can compute with the inert one taking these properties into account while not executing the function itself - and this is the essence of the inert function behaviour.

Introducing mathematical display, copy and paste for all these inert functions of the mathematical language concretely increases the mathematical expressiveness of the system, for teaching, working and also for presenting ideas.

Attached is a brief illustration.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

InertMathematicalFun.mw  InertMathematicalFun.pdf

Maple 17 and Mathematica 9 in solving Ka...

ecterrab 14837
October 19 2013
21 10

Hi
Just finished updating the comparison between Maple 17.02 and Mathematica 9.01 in solving the 1390 Ordinary Differential Equations (ODEs) of Kamke's book:

  • Mathematica solved 80% in 7 hours and 8 minutes
     
  • Maple solved 97.5% in 43 minutes

While trying to solve the whole set, Mathematica hanged with 90 of these ODEs while Maple hanged with 6 ODEs. A pdf with a summarizing table and all the details is linked below

It is also relevant here that Maple's dsolve has close to half of its code implementing more modern methods, not found in Kamke, illustrated in the Maple 'what's new in DEs' help pages of the last 10 releases; for these other kinds of equations the difference is more impressive. I'll see to prepare another post about that.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Comparison_Kamke.pdf

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