How do I duplicate a complex plot that was done in Wolfram math sofware?

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Question:How do I duplicate a complex plot that was done in Wolfram math sofware?

Posted: Frankoldstudent 45 Product: Maple

September 24 2021

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

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

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

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

(1)

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

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

(2)

(3)

(4)

(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

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

>	psi(t, x, y, z)

(7)

Loading PDEtools

(8)

(9)

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

(10)

Classical physics example

KE = p^2/(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

(13)

(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

= "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"

=<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 ,

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

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

THE ABOVE FUNCTION WILL BE A MAXIMUM WHEN `#mover(mi("x"),mo("→"))` = `#mover(mi("p"),mo("→"))`*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("→"))` = `#mover(mi("p"),mo("→"))`*t*x*axis/m, width = sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("⁢"))`*t/(sigma*m))^2)*so*at*t and sqrt(sigma^2+(`#msup(mi("&hbar;"),mo("⁢"))`*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("⁢"))`*t/(sigma*m)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

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

(24)

(25)

I*h*ts/M

(26)

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

(27)

(28)

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

(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))))"

I*P*x/h

(30)

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

(31)

(32)

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

(33)

(34)

(35)

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

(36)

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))

(38)

(39)

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

Download Lec7QuantumWaveSE.mw

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

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

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

(1)

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

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

(2)

(3)

(4)

(5)

(6)

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

THE WAVE FUNCTION EVOLVES IN TIME ACCORDING TO SCHRODINGER EQ

>	psi(t, x, y, z)

(7)

Loading PDEtools

(8)

(9)

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

(10)

Classical physics example

KE = p^2/(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

(13)

(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

= "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"

=<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 ,

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

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

distribution,

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

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

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

(24)

(25)

I*h*ts/M

(26)

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

(27)

(28)

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

(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))))"

I*P*x/h

(30)

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

(31)

(32)

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

(33)

(34)

(35)

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

(36)

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))

(38)

(39)

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

Download Lec7QuantumWaveSE.mw

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

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

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

(1)

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

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

(2)

(3)

(4)

(5)

(6)

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

THE WAVE FUNCTION EVOLVES IN TIME ACCORDING TO SCHRODINGER EQ

>	psi(t, x, y, z)

(7)

Loading PDEtools

(8)

(9)

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

(10)

Classical physics example

KE = p^2/(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

(13)

(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

= "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"

=<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 ,

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

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

distribution,

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

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

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

(24)

(25)

I*h*ts/M

(26)

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