Maple Questions and Posts

These are Posts and Questions associated with the product, Maple

Hi,

 I can’t use my "Filled" command to simultaneously explore my 3 graphs

Thanks for ideas

PétalesGraphes.mw

Hi, I am new to MAPLE and trying to solve the given BVP using shooting method but after defining the expression in u0 and u1, the code didn't compute the numerical value, and so that the whole code won't run exactly.
Please help me out to figure out the problem. I am looking for the answer. (NOTE: I copied this code from the web and it run so fine with the example but when I changed it according to me, it give me errors.)
The code is given below:
Thanks


restart;
with(DEtools);
with(plots);
dpdexi := 2;
K := 0.2;
epsilon := 0.0001;
                       epsilon := 0.0001
ode1 := (diff(phi(eta), eta) + epsilon*K*diff(phi(eta), eta)^3)/(1 + epsilon*diff(phi(eta), eta)^2) = dpdexi*eta;
bc1 := phi(0) = 0, phi(1) = 0;
a := 0;
b := 1;
ic1 := bc1[1], D(phi)(a) = alpha;
constraint := lhs(bc1[2]) - rhs(bc1[2]);
constraint = 0;
a0 := 0;
u0 := eval(constraint, phi(b) = PHIb(eval({ic1, ode1}, alpha = a0), phi(eta), b));
a1 := 0.01;
u1 := eval(constraint, phi(b) = PHIb(eval({ic1, ode1}, alpha = a1), phi(eta), b));
while fnormal(u1 - u0) <> 0 do
    z := solve(u1 = (u1 - u0)*(a1 - x)/(a1 - a0), x);
    a0, a1 := a1, z;
    u0, u1 := u1, eval(constraint, u(b) = Ub(eval({ic1, ode1}, alpha = z), u(x), b));
end do;
alpha_opt1 := z;
shoot_sol1 := evalf(dsolve(eval({ic1, ode1}, alpha = alpha_opt1), u(x), implicit));
infolevel[dsolve] := 3;
exact_sol1 := combine(dsolve({bc1, ode1}, u(x)));
infolevel[dsolve] := 0;
plot([rhs(exact_sol1), rhs(shoot_sol1)], x = 0 .. 1, color = [BLACK, RED], thickness = [5, 2], legend = ['exact', 'shooting'], title = "Figure 17.3");
evalf(eval(rhs(shoot_sol1), x = 1));

Hi..

OK, so my real-life example is I have a 3m gap between the house and the garage which I want to hang a sunshade. 
The hanging point on the house is 2.6m high and on the garage 2.1m (above ground). 

I have a choice between 2 sun sails, and I want to know -approximately- what sag I can expect from each.

I realise this is not exactly the same as the hanging wire problem i posted before.. but it my attempt anyway. Can anyone improve?

Sails.mw

there is this ref

https://www.mapleprimes.com/questions/147809-Real-Solution-For-Catenary-Differential-Equation

 

hello how do i define new units of multiple units. so i am able to choose them inside the Choose Unit in the rigth side

 

because as you can see there are not an option to define this as Unit -> [J/(K*mol)]

instead i have the option to use [m^2*kg/(s^2*K*mol)]

 

because i seen that you can defijne and install new units i just dont know how to impliment it so they alway remember my new units without i am suposed to initialize it with a command at top of my script

 

ps i also dont know how to define my new unit an just run that command at top of my script either.

but it would be better if it is permanent installed

 

i ask because i have several other units that is trublesome in maple (being in chemistry, we dont always use SI units) so molar [M] is also some times trublesome

 

as mapleprime often ask if we can include code i will do that

 

write this formula

3/(2)*8.3144621*J*K^(-1)*mol^(-1)

and use the simplify command in the right side and see that the unit is

thx

 

 

 

 

 

Hi. following on from

https://www.mapleprimes.com/questions/232817-Solving-Hanging-Cable-#comment281577

Second case: unequal poles.

I tried to work with vv's solution, but I got a problem... the required formula is

y=10.85378553130*cosh(0.0921337534371039*x) - 10.85378553130

Uneven.mw

Hi,

I have been trying to duplicate a solution to Schrodinger Eq from a utube video...the presenter use Wolfram software to graph and 

animate the plot...I have working on this all day..I a new user (several months)..any help would be appreciated.

I am attaching a screenshot

Thanks 

Frank

 

I am attaching my Maple worksheet for reference 
 

NULL

BELOW IS ADRESSING PARTICLE MOTION AND HOW ELECTRONS MOVE AROUND AND HOW MAGNETIC FIELDS AFFECTION THE MOTION OF ELECTRONS

 

PARTICLE MOTION "IGNORING SPIN" WILL COME BACK TO THAT

restart

with(Physics); interface(imaginaryunit = i)

Setup(hermitianoperators = {H, O__H, O__S}, realobjects = {`&hbar;`, m, t, x_}, combinepowersofsamebase = true, mathematicalnotation = true)

[combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`&hbar;`, m, t, x_}]

(1)

with(Physics[Vectors])

WAVE FUNCTION OF PARITCLE. t =time , x= some position vector

x_ = `<,>`(x, y, z)

x_ = Vector[column](%id = 36893489722226370188)

(2)

psi(t, x_)

psi(t, x_)

(3)

psi(t, x, y, z)

psi(t, x, y, z)

(4)

p_

p_

(5)

ψ is xome complex number with a real and imaginary number, the absolute value of abs(psi(t, x_))^2
`and`((is*the*probabilty*density*of*finding*the)*particle*at*postion*vector*x, at*time*t), ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, int(abs(psi(t, x_))^2, V) = 1

abs(psi(t, x_))^2

 

is*the^2*probabilty*density*of*finding*particle*at*postion*vector*x and at*time*t, ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, abs(psi(t, x_))^2*V = 1

(6)

THE ABOVE EQUATION IS TRUE IF THE WAVE FUNCTION IS "PROPERLY NORMALIZED"

THE WAVE FUNCTION EVOLVES IN TIME ACCORDING TO SCHRODINGER EQ

"`i&hbar;`(&PartialD;Psi)/(&PartialD;t)= -(`&hbar;`^(2))/(2 m)(((&PartialD;)^2)/((&PartialD;)^( )x^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )y^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )z^2)(psi))",
diff(psi, x, x)+diff(psi, y, y)+((diff(psi, z, z))*IS*THE*KENETIC*ENERGY*OF*THE)*PARTICLE

restart

psi(t, x, y, z)

psi(t, x, y, z)

(7)

Loading PDEtools

I*`&hbar;`*(diff(Ket(psi, t), t))

I*`&hbar;`*(diff(Ket(psi, t), t))

(8)

SElns := I*`&hbar;`*(diff(Ket(psi, t), t)); 'SElns'

SElns

(9)

SErhs1 := `&hbar;`^2*(diff(psi, t, x, x)+diff(psi, t, y, y)+diff(psi, t, z, z))/(2*m)

0

(10)

``

 

``

Classical physics example

KE = p^2/(2*m)

KE = (1/2)*p^2/m

(11)

KE = (p[x]^2+p[y]^2+p[z]^2)/(2*m)

KE = (1/2)*(p[x]^2+p[y]^2+p[z]^2)/m

(12)

QUANTUM MECHANICS

``

p[x] = `&hbar;`*(Diff(p[x], x))/(I)

p[x] = -I*`&hbar;`*(Diff(p[x], x))

(13)

 

p[x]^2 = Diff(rhs(p[x] = -I*`&hbar;`*(Diff(p[x], x))))

p[x]^2 = Diff(-I*`&hbar;`*(Diff(p[x], x)))

(14)

substituting each value of p[x], p[y],p[z]  IN EQ 11 is the KE

ONE VERY IMPORTANT SOLUTUION TO SE IS LISTED BELOW

psi(t, `#mover(mi("x"),mo("&rarr;"))`)= "Pi^(-3/(4)) sigma^(-3/(2))C(t)^(-3/(2))Exp((-1/(2))(((x)^(2))/(sigma^(2 )C(t)))+ ((i p *x-(i( p)^(2)(t/(`&hbar;`^())))/(2 m))/(C(t)))"

C(t) = 1 +  "(`i&hbar;` t))/((m sigma^(2))), sigma= initial position uncertainty of the particle,m,mass of particle"

```#mover(mi("p"),mo("&rarr;"))`=<p[x],p[y],p[z]>...just 3 Real no...this is NOT THE SAME AS EQ 14

There are 3 parmeters in the above eq σ (NOT PAULI), m ,`#mover(mi("p"),mo("&rarr;"))`

abs(psi(t, `#mover(mi("x"),mo("&rarr;"))`))^2 = 1/Pi^(3/2)Exp(-(`#mover(mi("x"),mo("&rarr;"))`-i*`#mover(mi("p"),mo("&rarr;"))`(t/m))^2/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2))/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)^(3/2)

int(abs(psi(t, `#mover(mi("x",fontweight = "bold"),mo("&rarr;",fontweight = "bold"))`))^2, V) = 1

THE ABOVE FUNCTION WILL BE A MAXIMUM WHEN `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t/m

distribution,

n. 1. (Statistics) the set of possible values of a random variable, or points in a sample space , considered in terms of their theoretical or observed frequency . 2. also called generalized function. a generalization of the concept of a function, defined as continuous linear functionals  over spaces of infinitely differentiable functions, introduced so that all continuous functions possess partial distributional derivatives (also called Schwartzian derivatives) that are again distributions. This leads to so-called weak solutions of differential equations and is of importance in the theory of partial differential equations ...

THE EQ YIEDS A GAUSSIAN "BELL SHAPE IN 2 D ONLY, CANNOT BE DRAWN 3D, 
abs(psi)^2*y*axis, `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t*x*axis/m, width = sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t and sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t = ((0*this)*will*equal*sigma*initial*uncertainity*of*the*position*of)*the*particle

t→∞ width = `#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m)

``

restart

with(Student[VectorCalculus])

with(VectorCalculus)

with(plots)

 

ln(1) := ts

ts

(15)

ts := 2

2

(16)

sigma := 4

4

(17)

h := 1

1

(18)

M := 1

1

(19)

P := 1

1

(20)

x

x

(21)

psi(x, t)

psi(x, t)

(22)

I = sqrt(-1)

I = I

(23)

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

h, I, ts, M

1, I, 2, 1

(24)

sigma^2

16

(25)

I*h*ts/M

2*I

(26)

 

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

2+2*I

2+2*I

(27)

(-1/2)*(x^2)

-(1/2)*x^2

(28)

"((-1/2)((x^2)/((sigma^2+(I*h*(t/M))))"

NULL

-(1/2)*x^2/(2+2*I)

(-1/8+(1/8)*I)*x^2

(29)

``

``

"Exp(((-1/2)((x^2)/((sigma^2+(I*h*(t/M)))))+(((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

NULL

I*P*x/h

I*x

(30)

 

I*P^2*t/((2*M)*h)

((1/2)*I)*t

(31)

I*x-((1/2)*I)*t

I*x-((1/2)*I)*t

(32)

"((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

``

NULL

I*h*t

I*t

(33)

sigma^2+M

17

(34)

1+I*t*(1/17)

1+((1/17)*I)*t

(35)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(36)

NULL

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(37)

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(38)

simplify(psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)))

psi(x, t) = exp(-I*(-2*x+t)*(I*t+34)/((2*I)*t+34))

 

Explore(psi(x, t) = exp(VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(I, `-`(2*x)+t), VectorCalculus:-`+`(VectorCalculus:-`*`(I, t), 34)), 1/VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`*`(2, I), t), 34)))), parameters = [[t = 0 .. 40, controller = slider], [x = -20 .. 80, controller = slider]], loop = never, size = NoUserValue, numeric = false, echoexpression = true)

(39)

with(plots)

implicitplot(abs(psi(x, ts*t)), Re(psi(x, ts*t)), Im(psi(x, ts*t)), x = -20 .. 80, scaling = constrained)

Error, (in plots/implicitplot) invalid input: lhs received Im(psi(x, 2*t)), which is not valid for its 1st argument, expr

 

NULL


 

Download Lec7QuantumWaveSE.mw
 

NULL

BELOW IS ADRESSING PARTICLE MOTION AND HOW ELECTRONS MOVE AROUND AND HOW MAGNETIC FIELDS AFFECTION THE MOTION OF ELECTRONS

 

PARTICLE MOTION "IGNORING SPIN" WILL COME BACK TO THAT

restart

with(Physics); interface(imaginaryunit = i)

Setup(hermitianoperators = {H, O__H, O__S}, realobjects = {`&hbar;`, m, t, x_}, combinepowersofsamebase = true, mathematicalnotation = true)

[combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`&hbar;`, m, t, x_}]

(1)

with(Physics[Vectors])

WAVE FUNCTION OF PARITCLE. t =time , x= some position vector

x_ = `<,>`(x, y, z)

x_ = Vector[column](%id = 36893489722226370188)

(2)

psi(t, x_)

psi(t, x_)

(3)

psi(t, x, y, z)

psi(t, x, y, z)

(4)

p_

p_

(5)

ψ is xome complex number with a real and imaginary number, the absolute value of abs(psi(t, x_))^2
`and`((is*the*probabilty*density*of*finding*the)*particle*at*postion*vector*x, at*time*t), ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, int(abs(psi(t, x_))^2, V) = 1

abs(psi(t, x_))^2

 

is*the^2*probabilty*density*of*finding*particle*at*postion*vector*x and at*time*t, ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, abs(psi(t, x_))^2*V = 1

(6)

THE ABOVE EQUATION IS TRUE IF THE WAVE FUNCTION IS "PROPERLY NORMALIZED"

THE WAVE FUNCTION EVOLVES IN TIME ACCORDING TO SCHRODINGER EQ

"`i&hbar;`(&PartialD;Psi)/(&PartialD;t)= -(`&hbar;`^(2))/(2 m)(((&PartialD;)^2)/((&PartialD;)^( )x^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )y^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )z^2)(psi))",
diff(psi, x, x)+diff(psi, y, y)+((diff(psi, z, z))*IS*THE*KENETIC*ENERGY*OF*THE)*PARTICLE

restart

psi(t, x, y, z)

psi(t, x, y, z)

(7)

Loading PDEtools

I*`&hbar;`*(diff(Ket(psi, t), t))

I*`&hbar;`*(diff(Ket(psi, t), t))

(8)

SElns := I*`&hbar;`*(diff(Ket(psi, t), t)); 'SElns'

SElns

(9)

SErhs1 := `&hbar;`^2*(diff(psi, t, x, x)+diff(psi, t, y, y)+diff(psi, t, z, z))/(2*m)

0

(10)

``

 

``

Classical physics example

KE = p^2/(2*m)

KE = (1/2)*p^2/m

(11)

KE = (p[x]^2+p[y]^2+p[z]^2)/(2*m)

KE = (1/2)*(p[x]^2+p[y]^2+p[z]^2)/m

(12)

QUANTUM MECHANICS

``

p[x] = `&hbar;`*(Diff(p[x], x))/(I)

p[x] = -I*`&hbar;`*(Diff(p[x], x))

(13)

 

p[x]^2 = Diff(rhs(p[x] = -I*`&hbar;`*(Diff(p[x], x))))

p[x]^2 = Diff(-I*`&hbar;`*(Diff(p[x], x)))

(14)

substituting each value of p[x], p[y],p[z]  IN EQ 11 is the KE

ONE VERY IMPORTANT SOLUTUION TO SE IS LISTED BELOW

psi(t, `#mover(mi("x"),mo("&rarr;"))`)= "Pi^(-3/(4)) sigma^(-3/(2))C(t)^(-3/(2))Exp((-1/(2))(((x)^(2))/(sigma^(2 )C(t)))+ ((i p *x-(i( p)^(2)(t/(`&hbar;`^())))/(2 m))/(C(t)))"

C(t) = 1 +  "(`i&hbar;` t))/((m sigma^(2))), sigma= initial position uncertainty of the particle,m,mass of particle"

```#mover(mi("p"),mo("&rarr;"))`=<p[x],p[y],p[z]>...just 3 Real no...this is NOT THE SAME AS EQ 14

There are 3 parmeters in the above eq σ (NOT PAULI), m ,`#mover(mi("p"),mo("&rarr;"))`

abs(psi(t, `#mover(mi("x"),mo("&rarr;"))`))^2 = 1/Pi^(3/2)Exp(-(`#mover(mi("x"),mo("&rarr;"))`-i*`#mover(mi("p"),mo("&rarr;"))`(t/m))^2/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2))/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)^(3/2)

int(abs(psi(t, `#mover(mi("x",fontweight = "bold"),mo("&rarr;",fontweight = "bold"))`))^2, V) = 1

THE ABOVE FUNCTION WILL BE A MAXIMUM WHEN `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t/m

distribution,

n. 1. (Statistics) the set of possible values of a random variable, or points in a sample space , considered in terms of their theoretical or observed frequency . 2. also called generalized function. a generalization of the concept of a function, defined as continuous linear functionals  over spaces of infinitely differentiable functions, introduced so that all continuous functions possess partial distributional derivatives (also called Schwartzian derivatives) that are again distributions. This leads to so-called weak solutions of differential equations and is of importance in the theory of partial differential equations ...

THE EQ YIEDS A GAUSSIAN "BELL SHAPE IN 2 D ONLY, CANNOT BE DRAWN 3D, 
abs(psi)^2*y*axis, `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t*x*axis/m, width = sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t and sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t = ((0*this)*will*equal*sigma*initial*uncertainity*of*the*position*of)*the*particle

t→∞ width = `#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m)

``

restart

with(Student[VectorCalculus])

with(VectorCalculus)

with(plots)

 

ln(1) := ts

ts

(15)

ts := 2

2

(16)

sigma := 4

4

(17)

h := 1

1

(18)

M := 1

1

(19)

P := 1

1

(20)

x

x

(21)

psi(x, t)

psi(x, t)

(22)

I = sqrt(-1)

I = I

(23)

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

h, I, ts, M

1, I, 2, 1

(24)

sigma^2

16

(25)

I*h*ts/M

2*I

(26)

 

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

2+2*I

2+2*I

(27)

(-1/2)*(x^2)

-(1/2)*x^2

(28)

"((-1/2)((x^2)/((sigma^2+(I*h*(t/M))))"

NULL

-(1/2)*x^2/(2+2*I)

(-1/8+(1/8)*I)*x^2

(29)

``

``

"Exp(((-1/2)((x^2)/((sigma^2+(I*h*(t/M)))))+(((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

NULL

I*P*x/h

I*x

(30)

 

I*P^2*t/((2*M)*h)

((1/2)*I)*t

(31)

I*x-((1/2)*I)*t

I*x-((1/2)*I)*t

(32)

"((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

``

NULL

I*h*t

I*t

(33)

sigma^2+M

17

(34)

1+I*t*(1/17)

1+((1/17)*I)*t

(35)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(36)

NULL

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(37)

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(38)

simplify(psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)))

psi(x, t) = exp(-I*(-2*x+t)*(I*t+34)/((2*I)*t+34))

 

Explore(psi(x, t) = exp(VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(I, `-`(2*x)+t), VectorCalculus:-`+`(VectorCalculus:-`*`(I, t), 34)), 1/VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`*`(2, I), t), 34)))), parameters = [[t = 0 .. 40, controller = slider], [x = -20 .. 80, controller = slider]], loop = never, size = NoUserValue, numeric = false, echoexpression = true)

(39)

with(plots)

implicitplot(abs(psi(x, ts*t)), Re(psi(x, ts*t)), Im(psi(x, ts*t)), x = -20 .. 80, scaling = constrained)

Error, (in plots/implicitplot) invalid input: lhs received Im(psi(x, 2*t)), which is not valid for its 1st argument, expr

 

NULL


 

Download Lec7QuantumWaveSE.mw
 

NULL

BELOW IS ADRESSING PARTICLE MOTION AND HOW ELECTRONS MOVE AROUND AND HOW MAGNETIC FIELDS AFFECTION THE MOTION OF ELECTRONS

 

PARTICLE MOTION "IGNORING SPIN" WILL COME BACK TO THAT

restart

with(Physics); interface(imaginaryunit = i)

Setup(hermitianoperators = {H, O__H, O__S}, realobjects = {`&hbar;`, m, t, x_}, combinepowersofsamebase = true, mathematicalnotation = true)

[combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`&hbar;`, m, t, x_}]

(1)

with(Physics[Vectors])

WAVE FUNCTION OF PARITCLE. t =time , x= some position vector

x_ = `<,>`(x, y, z)

x_ = Vector[column](%id = 36893489722226370188)

(2)

psi(t, x_)

psi(t, x_)

(3)

psi(t, x, y, z)

psi(t, x, y, z)

(4)

p_

p_

(5)

ψ is xome complex number with a real and imaginary number, the absolute value of abs(psi(t, x_))^2
`and`((is*the*probabilty*density*of*finding*the)*particle*at*postion*vector*x, at*time*t), ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, int(abs(psi(t, x_))^2, V) = 1

abs(psi(t, x_))^2

 

is*the^2*probabilty*density*of*finding*particle*at*postion*vector*x and at*time*t, ASSUME*SOME*POSITION*VECTOR*x, with*an*infintesmial*cube*around*it, infintesimal*volume*dV, PROBABILY*DENSITY*IS*abs(psi(t, x_))^2*dV, abs(psi(t, x_))^2*V = 1

(6)

THE ABOVE EQUATION IS TRUE IF THE WAVE FUNCTION IS "PROPERLY NORMALIZED"

THE WAVE FUNCTION EVOLVES IN TIME ACCORDING TO SCHRODINGER EQ

"`i&hbar;`(&PartialD;Psi)/(&PartialD;t)= -(`&hbar;`^(2))/(2 m)(((&PartialD;)^2)/((&PartialD;)^( )x^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )y^2)(psi)+((&PartialD;)^2)/((&PartialD;)^( )z^2)(psi))",
diff(psi, x, x)+diff(psi, y, y)+((diff(psi, z, z))*IS*THE*KENETIC*ENERGY*OF*THE)*PARTICLE

restart

psi(t, x, y, z)

psi(t, x, y, z)

(7)

Loading PDEtools

I*`&hbar;`*(diff(Ket(psi, t), t))

I*`&hbar;`*(diff(Ket(psi, t), t))

(8)

SElns := I*`&hbar;`*(diff(Ket(psi, t), t)); 'SElns'

SElns

(9)

SErhs1 := `&hbar;`^2*(diff(psi, t, x, x)+diff(psi, t, y, y)+diff(psi, t, z, z))/(2*m)

0

(10)

``

 

``

Classical physics example

KE = p^2/(2*m)

KE = (1/2)*p^2/m

(11)

KE = (p[x]^2+p[y]^2+p[z]^2)/(2*m)

KE = (1/2)*(p[x]^2+p[y]^2+p[z]^2)/m

(12)

QUANTUM MECHANICS

``

p[x] = `&hbar;`*(Diff(p[x], x))/(I)

p[x] = -I*`&hbar;`*(Diff(p[x], x))

(13)

 

p[x]^2 = Diff(rhs(p[x] = -I*`&hbar;`*(Diff(p[x], x))))

p[x]^2 = Diff(-I*`&hbar;`*(Diff(p[x], x)))

(14)

substituting each value of p[x], p[y],p[z]  IN EQ 11 is the KE

ONE VERY IMPORTANT SOLUTUION TO SE IS LISTED BELOW

psi(t, `#mover(mi("x"),mo("&rarr;"))`)= "Pi^(-3/(4)) sigma^(-3/(2))C(t)^(-3/(2))Exp((-1/(2))(((x)^(2))/(sigma^(2 )C(t)))+ ((i p *x-(i( p)^(2)(t/(`&hbar;`^())))/(2 m))/(C(t)))"

C(t) = 1 +  "(`i&hbar;` t))/((m sigma^(2))), sigma= initial position uncertainty of the particle,m,mass of particle"

```#mover(mi("p"),mo("&rarr;"))`=<p[x],p[y],p[z]>...just 3 Real no...this is NOT THE SAME AS EQ 14

There are 3 parmeters in the above eq σ (NOT PAULI), m ,`#mover(mi("p"),mo("&rarr;"))`

abs(psi(t, `#mover(mi("x"),mo("&rarr;"))`))^2 = 1/Pi^(3/2)Exp(-(`#mover(mi("x"),mo("&rarr;"))`-i*`#mover(mi("p"),mo("&rarr;"))`(t/m))^2/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2))/(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)^(3/2)

int(abs(psi(t, `#mover(mi("x",fontweight = "bold"),mo("&rarr;",fontweight = "bold"))`))^2, V) = 1

THE ABOVE FUNCTION WILL BE A MAXIMUM WHEN `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t/m

distribution,

n. 1. (Statistics) the set of possible values of a random variable, or points in a sample space , considered in terms of their theoretical or observed frequency . 2. also called generalized function. a generalization of the concept of a function, defined as continuous linear functionals  over spaces of infinitely differentiable functions, introduced so that all continuous functions possess partial distributional derivatives (also called Schwartzian derivatives) that are again distributions. This leads to so-called weak solutions of differential equations and is of importance in the theory of partial differential equations ...

THE EQ YIEDS A GAUSSIAN "BELL SHAPE IN 2 D ONLY, CANNOT BE DRAWN 3D, 
abs(psi)^2*y*axis, `#mover(mi("x"),mo("&rarr;"))` = `#mover(mi("p"),mo("&rarr;"))`*t*x*axis/m, width = sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t and sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m))^2)*so*at*t = ((0*this)*will*equal*sigma*initial*uncertainity*of*the*position*of)*the*particle

t→∞ width = `#msup(mi("&hbar;"),mo("&InvisibleTimes;"))`*t/(sigma*m)

``

restart

with(Student[VectorCalculus])

with(VectorCalculus)

with(plots)

 

ln(1) := ts

ts

(15)

ts := 2

2

(16)

sigma := 4

4

(17)

h := 1

1

(18)

M := 1

1

(19)

P := 1

1

(20)

x

x

(21)

psi(x, t)

psi(x, t)

(22)

I = sqrt(-1)

I = I

(23)

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

h, I, ts, M

1, I, 2, 1

(24)

sigma^2

16

(25)

I*h*ts/M

2*I

(26)

 

about(I)

I:
  All numeric values are properties as well as objects.
  Their location in the property lattice is obvious,
  in this case complex(extended_numeric).
 

 

2+2*I

2+2*I

(27)

(-1/2)*(x^2)

-(1/2)*x^2

(28)

"((-1/2)((x^2)/((sigma^2+(I*h*(t/M))))"

NULL

-(1/2)*x^2/(2+2*I)

(-1/8+(1/8)*I)*x^2

(29)

``

``

"Exp(((-1/2)((x^2)/((sigma^2+(I*h*(t/M)))))+(((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

NULL

I*P*x/h

I*x

(30)

 

I*P^2*t/((2*M)*h)

((1/2)*I)*t

(31)

I*x-((1/2)*I)*t

I*x-((1/2)*I)*t

(32)

"((I *P* (x/h))- (I* ((P^2)/(2*M))*(t/h))/((1+(((I*h*t))/((M+sigma^2))))"

``

NULL

I*h*t

I*t

(33)

sigma^2+M

17

(34)

1+I*t*(1/17)

1+((1/17)*I)*t

(35)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)

(36)

NULL

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(37)

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t))

(38)

simplify(psi(x, t) = exp(I*x-((1/2)*I)*t+(I*x-((1/2)*I)*t)/(1+((1/17)*I)*t)))

psi(x, t) = exp(-I*(-2*x+t)*(I*t+34)/((2*I)*t+34))

 

Explore(psi(x, t) = exp(VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(I, `-`(2*x)+t), VectorCalculus:-`+`(VectorCalculus:-`*`(I, t), 34)), 1/VectorCalculus:-`+`(VectorCalculus:-`*`(VectorCalculus:-`*`(2, I), t), 34)))), parameters = [[t = 0 .. 40, controller = slider], [x = -20 .. 80, controller = slider]], loop = never, size = NoUserValue, numeric = false, echoexpression = true)

(39)

with(plots)

implicitplot(abs(psi(x, ts*t)), Re(psi(x, ts*t)), Im(psi(x, ts*t)), x = -20 .. 80, scaling = constrained)

Error, (in plots/implicitplot) invalid input: lhs received Im(psi(x, 2*t)), which is not valid for its 1st argument, expr

 

NULL


 

Download Lec7QuantumWaveSE.mw

 

 

I am trying to plot a Runge-Kutta method for 1+ tsin(tx). When ever I try to do the plot data command an empty graph shows up. I noticed that some of my values from the algorithm did not calculate properly. How do I fix this problem? This is what I typed in for the calculations. 

f := (t, x) -> 1 + t*sin(x);
t[0] := 0;x[0] := 0;
h := 0.1;
 

for n to 20 do
    t[n] := n*h;
    m1 := f(t[n - 1], x[n - 1]);
    m2 := f(t[n - 1] + h/2, x[n - 1] + m1*h/2);
    m3 := f(t[n - 1] + h/2, x[n - 1] + m2*h/2);
    m4 := f(t[n - 1] + h, h*m3 + x[n - 1]);
    x[n] := x[n - 1] + h/6*(m1 + 2*m2 + 2*m3 + m4);
end;
 

Hi

I am trying to follow this paper

http://euclid.trentu.ca/aejm/V4N1/Chatterjee.V4N1.pdf

Lets start with the easier problem, equal poles. Assume that the length of the cable is 120m and the two poles have equal height of 50m. Our goal is to determine the minimum distance between the two poles that will prevent the cable from touching the ground.

I was trying to get Maple to agree with their derived formula, namely

y(x)= 11*cosh(1/11*x) - 61,

but I think I have not set the IC's correctly. or provided for the length of the cable.

restart:
int(sqrt(1+diff(y(t),t)^2),t=0..x)=120/2

//can't solve the above directly, or maybe someone clever here can

//a is a constant
DIV := diff(y(x), x, x) = a*sqrt(1 + diff(y(x), x)^2);
RV := y(0) = 0, D(y)(0) = -50;
dsolve({DIV});
Opl := dsolve({DIV, RV}, y(x));
allvalues(%)
 

 

Dear all,

Reversion of series---computing a series for the functional inverse of a function---has been in Maple since forever, but many people are not aware of how easy it is.  Here's an example, where we are looking for "self-reverting" series---which I called "ambiverts".  Anyway have fun.

 

https://maple.cloud/app/5974582695821312/Series+Reversion%3A+Looking+for+ambiverts

PS There looks to be some "code rot" in the branch point series for Lambert W in Maple, which we encounter in that worksheet.  Or, I may simply have not coded it very well in the first place (yeah, that was mine, once upon a time).  Checking now.  But there is a workaround (albeit an ugly one) shown in that worksheet.

 

Hi I have experienced another Maple 2021 error with those of my students who Maple 2021 Mac edition. 

Lets say their have saved a .mw on their main drive and tries to open the file from inside Maple. Maple gives an error like "file cannot be opened - please try to another". This also happens when trying to open the file from outside Maple. 

This never happens on the Windows version. So any idea what could be causing this ?

 

 

 

 

 

"D1(s,t) :=P- (alpha1-beta*S) +  alpha2 + beta2 *q(t)^();"

proc (s, t) options operator, arrow; P+beta*S-alpha1+alpha2+beta2*q(t) end proc

(1)

"(->)"

dem

(2)

``

ode1 := diff(q(t), t)+theta*q(t)/(1+N-t) = -D1(s, t)

diff(q(t), t)+theta*q(t)/(1+N-t) = -P-beta*S+alpha1-alpha2-beta2*q(t)

(3)

fn1 := q(t)

q(t)

(4)

ic1 := q(T) = 0

q(T) = 0

(5)

sol1 := simplify(dsolve({ic1, ode1}, fn1))

q(t) = (-S*beta-P+alpha1-alpha2)*(Int(exp(beta2*_z1)*(1+N-_z1)^(-theta), _z1 = T .. t))*exp(-beta2*t)*(1+N-t)^theta

(6)

NULL

Download data.mw

Hello all. I'm trying to solve the following first-order differential equation. 

Please help in understanding why the equation (6) doesn't contain proper solution for the function q(t) on solving the ode1 with the given initial condition

How (can I?) display the value in a legend in Engineering format -- 10^3, 10^-6, etc?

Lres := 1/((2*Pi*freq)^2*Cres);
ftest := 10e6;
p1 := plot(eval(subs(freq = ftest, Lres)), Cres = 0.10000000 .. 0.10000000, labels = [Cres, 'Lres'], legend = ftest, color = red, title = 'Inductance*Value*as*a*Function*of*Resonant*Capacitance', axis = [gridlines = [default]]);

I would like the legend to display 10^6 rather than 1^7.

I've tried changing the default number format for the whole worksheet to Engineering, but that doesn't seem to apply to legends.

 

thank you.

Hi,

I'd like to know, if it is possible to define any sort of range for parameters in NonlinearFit. E. g. I know that one of parameters should be somewhere between 0.2 - 0.4. I know there is a possibility of initalvalues, but using it doesn't lead into this range.

Thanks.

Dear all,

Recently I discovered the noncommuting variables in the Physics package due to Edgardo Cheb-Terrab; doubtless there are many posts here on Maple Primes describing them.  Here is one more, which shows how to use this package to prove the Schur complement formula.

https://maple.cloud/app/6080387763929088/Schur+Complement+Proof+in+Maple

I guess I have a newbie's question: how well-integrated are Maple Primes and the Maple Cloud?  Anyway that seemed the easiest way to share this.

-r

Dear all;

Some of you will have heard of the new open access (and free of page charges) journal Maple Transactions https://mapletransactions.org which is intended to publish expositions on topics of interest to the Maple community. What you might not have noticed is that it is possible to publish your papers as Maple documents or as Maple workbooks.  The actual publication is on Maple Cloud, so that even people who don't have Maple can read the papers.

Two examples: one by Jürgen Gerhard, https://mapletransactions.org/index.php/maple/article/view/14038 on Fibonacci numbers

and one by me, https://mapletransactions.org/index.php/maple/article/view/14039 on Bohemian Matrices (my profile picture here is a Bohemian matrix eigenvalue image).

I invite you to read those papers (and the others in the journal) and to think about contributing.  You can also contribute a video, if you'd rather.

I look forward to seeing your submissions.

Rob Corless, Editor-in-Chief, Maple Transactions

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