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For context, I'm designing a work sheet based around quantum tunneling. Currently I'm looking at the boundary conditions.

What I want to be able to do is to set the expression psi[1] equal to psi[2], but only for the value x = 0. Is this possible? I've tried using if statements, and I've considered converting these expressions into functions for this purpose, but I'm not having much luck. 

Thanks

Blanky

Hi, everyone.

I want to setup some conjugate rules for the quantum operator such as

conjugate(a) = a, conjugate(b) = -b.

And during the calculation progress, it works like

conjugate( k1*a + k2*conjugate(b) ) = conjugate(k1)*a + conjugate(k2)*b

#k1, k2 are complex numbers.

How can we build up this? I checked in the help system, but failed to find an answer.

I am trying to setup a general metric with the Physics package.  The metric is composed of the Minkowski metric plus the product of two null vectors. Here is the code:

retstart;

with(Physics);

Define(l[mu],eta[mu,nu]);

eta[mu,nu] := rhs(g_[]);

Setup(g[mu,nu]=eta[mu,nu]+l[mu]*l[nu]);

 

I get the following error:

Error, (in Physics:-Setup) wrong argument: g[mu, nu] = l[mu]*l[nu]+(Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 2) = -1, (2, 3) = 0, (2, 4) = 0, (3, 3) = -1, (3, 4) = 0, (4, 4) = 1}, storage = triangular[upper], shape = [symmetric]))

 

My plan is to apply the rule g[~mu,~nu]*l[mu]*l[nu] = 0 and calculate the Christoffel symbols using the metric.

I am trying to use the Physics package because the DifferentialGeometry package seemed focused on Newman-Penrose.  I will not be using NP for the calculations, only a strict calculation of the Einstein field equations from the given metric.

Thank you.

With the package VectorCalculus we can study the speed and acceleration to their respective components. Considering the visualizaccion and algebraic calculations and to check with their respective commands. Both 2D and 3D.

 

Velocidad-Aceleració.mw     (in spanish)

 

Lenin Araujo Castillo

Physics Pure

Computer Science

Hello, everyone. I faced some promblems on maple. Hope you can help me.

I want to set up the commutation rules like Pauli sigma matrix in Physics Packages,

but failed to get the correct one.

Here is my maple code:

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

Setup(quantumop = {Q}, algebrarule = {%AntiCommutator(Q[j], Q[k]) = 2*KroneckerDelta[j, k], %Commutator(Q[j], Q[k]) = 2*I*(Sum(LeviCivita[i, k, l]*Q[l], l = 1 .. 3))});

Commutator(Q[1], Q[2]);
                              
AntiCommutator(Q[1], Q[2]);
                              
AntiCommutator(Q[1], Q[1]);
                              
Commutator(Q[1], Q[1]);

In fact, Commutator(Q[1], Q[2]) give a incorrect result 0 while the correct answer is  2*I*Q[3].

Do anyone know how to resolve this?

The attached presentation is the last one of a sequence of three on Quantum Mechanics using Computer Algebra, covering the field equation for a quantum system of identical particles, its stationary solutions and the equations for small perturbations around them and, in this third presentation, the conditions for superfluidity of such a system of identical particles at low temperature. The novelty is again in how to tackle these problems in a computer algebra worksheet.

The Landau criterion for Superfluidity
  

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

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

(2) Maplesoft, Canada

 

A Bose-Einstein Condensate (BEC) is a medium constituted by identical bosonic particles at very low temperature that all share the same quantum wave function. Let's consider an impurity of mass M, moving inside a BEC, its interaction with the condensate being weak. At some point the impurity might create an excitation of energy `&hbar;`*omega[k] and momentum `&hbar;` `#mover(mi("k"),mo("&rarr;"))`. We assume that this excitation is well described by Bogoliubov's equations for small perturbations `&delta;&varphi;` around the stationary solutions `&varphi;```of the field equations for the system. In that case, the Landau criterion for superfluidity states that if the impurity velocityLinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) is lower than a critical velocity v[c] (equal to the BEC sound velocity), no excitation can be created (or destroyed) by the impurity. Otherwise, it would violate conservation of energy and momentum. So that, if LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) < v[c] the impurity will move within the condensate without dissipation or momentum exchange, the condensate is superfluid (Phys. Rev. Lett. 85, 483 (2000)). Note: low temperature liquid 4He is a well known example of superfluid that can, for instance, flow through narrow capillaries with no dissipation. However, for superfluid helium, the critical velocity is lower than the sound velocity. This is explained by the fact that liquid 4He is a strongly interacting medium. We are here rather considering the case of weakly interacting cold atomic gases.

Landau criterion for superfluidity

 

 

Background: For a BEC close to its ground state (at temperature T = 0 K), its excitations are well described by small perturbations around the stationary state of the BEC. The energy of an excitation is then given by the Bogoliubov dispersion relation (derived previously in Mapleprimes "Quantum Mechanics using computer algebra II").

 

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

 

where G is the atom-atom interaction constant, n is the density of particles, m is the mass of the condensed particles, k is the wave-vector of the excitations and omega[k] their pulsation (2*Pi time the frequency). Typically, there are two possible types of excitations, depending on the wave-vector k:

• 

In the limit: proc (k) options operator, arrow; 0 end proc, "epsilon[k]&sim;`&hbar;`*k*"v[c] with v[c] = sqrt(G*n/m), this relation is linear in k and is typical of a massless quasi-particle, i.e. a phonon excitation.

• 

In the limit: proc (k) options operator, arrow; infinity end proc, `&sim;`(epsilon[k], `&hbar;`^2*k^2/(2*m)) which is the dispersion relation of a free particle of mass "m,"i.e. one single atom of the BEC.

 

Problem: An impurity of mass M moves with velocity `#mover(mi("v"),mo("&rarr;"))` within such a condensate and creates an excitation with wave-vector `#mover(mi("k"),mo("&rarr;"))`. After the interaction process, the impurity is scattered with velocity `#mover(mi("w"),mo("&rarr;"))`.

 

a) Departing from Bogoliubov's dispersion relation, plus energy and momentum conservation, show that, in order to create an excitation, the impurity must move with an initial velocity

 

LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) >= v[c] and v[c] = sqrt(G*n/m)

 

  

When LinearAlgebra[Norm](`#mover(mi("v"),mo("&rarr;"))`) < v[c] , no excitation can be created and the impurity moves through the medium without dissipation, as if the viscosity is 0, characterizing a superfluid. This is the Landau criterion for superfluidity.

 

b) Show that when the atom-atom interaction constant G >= 0 (repulsive interactions), this value v[c] is equal to the group velocity of the excitation (speed of sound in a condensate).

Solution

   

 

References

NULL

[1] Suppression and enhancement of impurity scattering in a Bose-Einstein condensate

[2] Superfluidity versus Bose-Einstein condensation
[3] Bose–Einstein condensate (wiki)

[4] Dispersion relations (wiki)

 


Download QuantumMechanics3.mw   QuantumMechanics3.pdf

Edgardo S. Cheb-Terrab
Physics, Maplesoft

This is a 5-days mini-course I gave in Brazil last week, at the CBPF (Brazilian Center for Physics Research). The material will still receive polishment and improvements, towards evolving into a sort of manual, but it is also interesting to see it exactly as it was presented to people during the course. This material uses the update of Physics available at the Maplesoft Physics R&D webpage.

Mini-Course: Computer Algebra for Physicists

 

Edgardo S. Cheb-Terrab

Maplesoft

 

 

This course is organized as a guided experience, 2 hours per day during five days, on learning the basics of the Maple language, and on using it to formulate algebraic computations we do in physics with paper and pencil. It is oriented to people not familiar with computer algebra (sections 1-5), as well as to people who are familiar but want to learn more about how to use it in Physics.

 

Motivation

 

 

Among other things, with computer algebra:

 

• 

You can concentrate more on the ideas (the model and its formulation) instead of on the algebraic manipulations

• 

You can extend your results with ease

• 

You can explore the mathematics surrounding your problem

• 

You can share your results in a reproducible way - and with that exchange about a problem in more productive ways

• 

After you learn the basics, the speed at which algebraic results are obtained with the computer compensates with dramatic advantage the extra time invested to formulate the problem in the computer.

 

All this doesn't mean that we need computer algebra, at all, but does mean computer algebra can enrich our working experience in significant ways.

What is computer algebra - how do you learn to use it?

   

What is this mini-course about?

   

What can you expect from this mini-course?

   

 

Explore. Having success doesn't matter, using your curiosity as a compass does - things can be done in so many different ways. Have full permission to fail. Share your insights. All questions are valid even if to the side. Computer algebra can transform the algebraic computation part of physics into interesting discoveries and fun.

1. Arithmetic operations and elementary functions

   

2. Algebraic Expressions, Equations and Functions

   

3. Limits, Derivatives, Sums, Products, Integrals, Differential Equations

   

4. Algebraic manipulation: simplify, factor, expand, combine, collect and convert

   

5. Matrices (Linear Algebra)

   

6. Vector Analysis

   

7. Tensors and Special Relativity

   

8. Quantum Mechanics

   

9. General Relativity

   

10. Field Theory

   

BrasilComputacaoAlgebrica.mw.zip

 

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Hello everybody,

 

I want to use the physics package in the context of special relativity.  I am mostly interrested with the Lorenzts transformations, not having to unprotect gamma all the time, the 4-vectors for space, time and momentum in (ct,x,y).  For exemple, to be able to calculate the invariant (ct)^2-(r^2).  A small document (or worksheet) would be very nice of you.

I understande that someone could say that I want to use a gun to kill a fly.  But this is a process that will lead me in using it in general relativity and using tensors in my calculations.  It would be very interresting to have a Student,Physics package.  Don't you think so?

Thank you in advance for your trouble and comprehension.

 

--------------------------------------
Mario Lemelin
Maple 18 Ubuntu 13.10 - 64 bits
Maple 18 Win 7 - 64 bits messagerie : mario.lemelin@cgocable.ca téléphone :  (819) 376-0987

can we load part of a package not the package whole !? for example from physics package i only need diff tool not all of the package tools,can i do sth for that !? 

hi,i want to take differential with respect to another differential using physics package,but using D instead of diff,could anyone help me do that ? for example :

restart; with(Physics):
A1 := -(1/24)*1*rho*((diff(phi[1](x, t), t))^2)*(h^3)-(1/2)*1*rho*((diff(u[ref](x, t), t))^2)*h-(1/2)*rho*((diff(w(x, t), t))^2)*h+(1/24)*1*1*((diff(phi[1](x, t), x))^2)*(h^3)+(1/2)*1*(1*((diff(u[ref](x, t), x)+(1/2)*(diff(w(x, t), x))^2)^2)+K*1*((diff(w(x, t), x)+phi[1](x, t))^2))*h-1*q*w(x, t):

A2:=-diff(diff(A1,diff(u[ref](x,t),x)),x);

here i want to compute A2 using D command,not diff and i do not want use convert command ! i just need to calculate A2 directly using D command. tnx for your help.

 

I wanted to define a 4th order tensor in the following way,

    restart;
    with(Physics);
    Setup(mathematicalnotation = true);
    delta := KroneckerDelta;
    L[i, j, k, l] := lambda*delta[i, j]*delta[k, l]+mu*(delta[i, k]*delta[j, l]+delta[i, l]*delta[j, k]);

but when I type L[1, 1, 1, 1] it doesn't replace i, j, j and l by 1. i
do i need to define it in a different way?

which method i input a series of random number it output a periodic waveform

how to express bosonic and fermonic with physics package in maple?

what are the difference between upperscript and underscript?

Vector using package Physics, LinearAlgebra.

Vectores.mw     (in spanish)

I was recently asked about performing some General Relativity computations from a paper by Plamen Fiziev, posted in the arXiv in 2013. It crossed my mind that this question is also instrumental to illustrate how these General Relativity algebraic computations can be performed using the Physics package. The pdf and mw links at the end show the same contents but with the Sections expanded.

General Relativity using Computer Algebra

 

Problem: for the spacetime metric,

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -exp(lambda(r)), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = exp(nu(r))}))

 

a) Compute the trace of

 

"Z[alpha]^(beta)=Phi R[alpha]^(beta)+`&Dscr;`[alpha]`&Dscr;`[]^(beta) Phi+T[alpha]^(beta)"

 

where `&equiv;`(Phi, Phi(r)) is some function of the radial coordinate, R[alpha, `~beta`] is the Ricci tensor, `&Dscr;`[alpha] is the covariant derivative operator and T[alpha, `~beta`] is the stress-energy tensor

 

T[alpha, beta] = (Matrix(4, 4, {(1, 1) = 8*exp(lambda(r))*Pi, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 8*r^2*Pi, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 8*r^2*sin(theta)^2*Pi, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 8*exp(nu(r))*Pi*epsilon}))

b) Compute the components of "W[alpha]^(beta)"" &equiv;"the traceless part of  "Z[alpha]^(beta)" of item a)

 

c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)" obtained in b)

 

Background: The equations of items a) and b) appear in a paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by Plamen Fiziev, a Maple user.  These equations model a problem in the context of a Branse-Dicke theory with vanishing parameter "omega." The Brans–Dicke theory is in many respects similar to Einstein's theory, but the gravitational "constant" is not actually presumed to be constant - it can vary from place to place and with time - and the gravitational interaction is mediated by a scalar field. Both Brans–Dicke's and Einstein's theory of general relativity are generally held to be in agreement with observation.

 

The computations below aim at illustrating how this type of computation can be performed using computer algebra, and so they focus only on the algebraic aspects, not the physical interpretation of the results.

a) The trace of "  Z[alpha]^(beta)=Phi R[alpha]^(beta)+`&Dscr;`[alpha]`&Dscr;`[]^(beta) Phi+T[alpha]^(beta)"

   

b) The components of "W[alpha]^(beta)"" &equiv;"the traceless part of " Z[alpha]^(beta)"

   

c) An exact solution for the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)"

   

 

GeneralRelativit.pdf   GeneralRelativity.mw

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

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