Frankoldstudent

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These are questions asked by Frankoldstudent

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 = {`ℏ`, m, t, x_}, combinepowersofsamebase = true, mathematicalnotation = true)

[combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`ℏ`, 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

 

 

Hi,

I have had trouble with a  .mw worksheet..which I will attach.(I am using Maple 2021)..This is second time it got corrupted..I used the

"CompleteMiniCourseComputerAlgebraPhysics" and copy parts to worksheets so I can work the problems and I do 

Alot of markups and highlighting as well as add coments a copy as Text from help on certain commands..

I read some other posts that discuss corrupt file thats use XML I think in the worksheet....I checked this in the area I was working and did not find anything (I completed about half of the course saving as I went and had no problems)....so any guidance would be appreciated..

ThanksMyMinitCourseComputerAlgebraForPhysicsPart2_-_Copy_-_Copy.zipMyMinitCourseComputerAlgebraForPhysicsPart2_-_Copy_-_Copy.zip

Frank McFee

Hi,

As I am continuing to use Maple, I find I use multiple customized worksheets and documents as well as a maple student guide or AEM guide.......How can I save the workspace with these other documents so that I don't have to rebuild them everytime I use that customized Maple system?

Thanks for your help!

Frank

Hi,

I very new to Maple....I had problems getting the value of a slider and someone helped me with that

but how do a setup "commands,etc" to always get the latest value of the slider??  whenever I move the slider..
Thanks so much

Frank....I am also attaching screenshot

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/CreatingSlider_ac_-_Copy.mw .
 

Download CreatingSlider_ac_-_Copy.mw

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/CreatingSlider_ac_-_Copy.mw .

Download CreatingSlider_ac_-_Copy.mw

I have tried everthing I know

Any suggestions would be appreciated

Hi,

I have have been trying to creat a slider and use the value as a "variable" to vary the sin function..

I tried many things and can't seem to get iit to work...any suggestions woul be appreciated..I am still brand new to Maple

I have searched Application center..and spend hours trying to get this right...I am attacing my worksheet

Thanks for your help!

Frank

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/CreatingSlider.mw .
 

Download CreatingSlider.mw

Just in case the worksheet won't work I am attaching a jpeg

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