QUANTUM SCATTERING BY THE ONE-DIMENSIONAL POTENTIAL STEP IN MATHEMATICA

Alexei V. Tikhonenko
General Physics Department, State Technical University of Nuclear Power Engineering, Obninsk, Russia
futureprint@obninsk.com

Scattering problem of quantum particles by the one-dimensional potential step is solved in MATHEMATICA.
Analytical formulas for reflection and transmission coefficients are obtained and visualized.

1. One-dimensional Schrodinger equation

Consider one-dimensional Schrodinger equation  [1]

DifEqnQntOsc=D[ψ[x], {x, 
              2}] + (((2 (Subscript[m, 0]) )/(ℏ^2)) 
                  En  - ((2 (Subscript[m, 0]))/(ℏ^2)) V[x]) ψ[
              x] == 0

where ψ[x]  is wave function, (Subscript[m, 0]) is mass and En is energy of particle; ℏ is Planck's constant .
This equation describes one-dimensional motion of quantum particles in the field of potential barrier V[x]:

V[x] = 
      ((Subscript[U, 0])/(1 + (e^(-α x)))),

We will use notations:

k1 = ((Sqrt[2 (Subscript[m, 0]) En])/ℏ),  
    k2 = ((Sqrt[2 (Subscript[m, 0]) (En - (Subscript[U, 0]))])/ℏ),

Solve[{k1 == ((Sqrt[2 (Subscript[m, 0]) En])/ℏ), 
        k2 == ((Sqrt[2 (Subscript[m, 0]) (En - (Subscript[U, 0]))])/ℏ)}, {En, 
        (Subscript[U, 0])}]

and obtain

DifEqnQntOsc1=Simplify[ReplaceAll[DifEqnQntOsc,{V[x] -> 
                ((Subscript[U, 0])/(1 + (E^(-α x))))}]]

or

DifEqnQntOsc2=FullSimplify[ReplaceAll[DifEqnQntOsc1,{(Subscript[U, 0]) -> ((((k1^2) - (k2^2)) (ℏ^2))/(2 (Subscript[m, 0]))), En -> (((k1^2) (ℏ^2))/(2 (Subscript[m, 0])))}]]

2. Solution of Schrodinger equation

Introduce new variable as

z = -(e^(-α x)), 
    x = -Log[-z]/α,

x = -Log[-z]/α

Differentiation by x leads to

-z α (∂_z) F[z]

ψ''==Simplify[-z α D[(-z α) D[ψ[
                  z], z], z]]

In this variable Schrodinger equation is

((((k1^2) + (E^(x α)) (k2^2)) ψ[x])/(1 + (E^(x α))))+z (α^2) (ψ'[z]+z ψ''[z])==0

Thus solution of Schrodinger equation is

DSolve[((((k1^2) - ((k2^2)/z)) ψ[z])/(1 - (1/z)))+z (α^2) (ψ'[z]+z ψ''[z])==0,ψ[z],z]

3. Analysis of the solution

Note, that two lineally independent solutions of Schrodinger equation are complex conjugate each other. Now we choose solution corresponding to transmitted wave with dependence of x:

(e^(i k2 x))=\(z\^\(-\(\(i\ k2\)\/α\)\)\ \((z = \(-e\^\(\(-α\)\ x\)\), 
          x = \(-\(Log[\(-z\)]\/α\)\))\)\)

As (with positive α)

Clear[x]
Limit[-(E^(-x)), 
      x -> ∞]
Limit[
      Hypergeometric2F1[-((I k1)/α) - ((I k2)/α), ((I k1)/α) - ((I k2)/α), 
        1 - ((2 I k2)/α), -(E^(-x))], 
      x -> ∞]
Limit[
      Hypergeometric2F1[-((I k1)/α) + ((I k2)/α), ((I k1)/α) + ((I k2)/α), 
        1 + ((2 I k2)/α), -(E^(-x))], x -> ∞]

(((-1))^(-((I k2)/α))) (z^(-((I k2)/α))) C[
          1] Hypergeometric2F1[-((I k1)/α) - ((I k2)/α), ((I k1)/α) - ((I k2)/α), 1 - ((2 I k2)/α), 
          0] + (((-1))^(((I k2)/α))) (z^(((I k2)/α))) C[
          2] Hypergeometric2F1[-((I k1)/α) + ((I k2)/α), ((I k1)/α) + ((I k2)/α), 1 + ((2 I k2)/α), 
          0]

So we choose

\(ψ[z] = 
      z\^\(-\(\(i\ k2\)\/α\)\)\ Hypergeometric2F1[\(-\(\(i\ k1\)\/α\)\) - \(i\ k2\)\/α, \(i\ k1\)\/α - \(i\ k2\)\/α, 
          1 - \(2\ i\ k2\)\/α, z]\)

z = -(E^(-α x));
(Subscript[ψ, incom])[z] = 
      Simplify[(z^(-((I k2)/α))) Hypergeometric2F1[-((I k1)/α) - ((I k2)/α), ((I k1)/α) - ((I k2)/α), 1 - ((2 I k2)/α), z]]

or

(Subscript[ψ, incom])[z]=(e^(I k2 x)) Hypergeometric2F1[a, b, c, z]

where we used

a = -((i (k1 + k2))/α),  
    b = ((i (k1 - k2))/α),  
    c = 1 - ((2 i k2)/α)

Obtained solution must be extend to x ->-∞.  But Gauss hypergeometric function is defined on interval [0,1]. So we need to use Gauss relation [2]

Hypergeometric2F1[a, b, c, 
        z] == ((GAMMA[c]*GAMMA[b - a])/(GAMMA[b]*
                GAMMA[c - a])) (((-z))^(-b)) 
          Hypergeometric2F1[a, 1 - c + a, 1 - b + a, 
            (1/z)] + ((GAMMA[c]*GAMMA[a - b])/(GAMMA[a]*
                GAMMA[c - b])) (((-z))^(-b)) Hypergeometric2F1[b, 
            1 + b - c, 1 - a + b, (1/z)]
or
Hypergeometric2F1[a, b, c, z] == 
      C1 (((-z))^(-a)) 
          Hypergeometric2F1[a, 1 - c + a, 1 - b + a, (1/z)] + 
        C2 (((-z))^(-b)) Hypergeometric2F1[b, 1 + b - c, 1 - a + b, 
            (1/z)]
C1 = ((GAMMA[c]*GAMMA[b - a])/(GAMMA[b]*GAMMA[c - a])),  
    C2 = ((GAMMA[c]*GAMMA[a - b])/(GAMMA[a]*GAMMA[c - b]))

Clear[x]
z = -(E^(-α x));
ψ2[x] =  
      Simplify[(z^(-((I k2)/α))) (C1 (((-z))^(-a)) Hypergeometric2F1[a, 1 - c + a, 1 - b + a, (1/z)] + 
              C2 (((-z))^(-b)) Hypergeometric2F1[b, 1 + b - c, 
                  1 - a + b, (1/z)])]

As (with positive α)

Limit[-(E^x), 
      x -> -∞]
Limit[
      Hypergeometric2F1[a, 1 + a - c, 
        1 + a - b, -(E^(x ))], 
      x -> -∞]
Limit[
      Hypergeometric2F1[b, 1 + b - c, 
        1 - a + b, -(E^(x ))], 
      x -> -∞]

asymptote at x->-∞ are:

(Subscript[ψ, as])[x]=ReplaceAll[C1 (((-(E^(-x α))))^(-((I k2)/α)))  ((((E^(-x α))))^(-a)) Hypergeometric2F1[a, 
                  1 + a - c, 1 + a - b, 
                  0] + (((-(E^(-x α))))^(-((I k2)/α)))  C2 ((((E^(-x α))))^(-b)) Hypergeometric2F1[b, 
                  1 + b - c, 1 - a + b, 0],{a -> -((I (k1 + 
                          k2))/α), 
            b -> ((I (k1 - k2))/α)}]

(Subscript[ψ, as])[x] = 
      FullSimplify[
        PowerExpand[
          C2 (((-(E^(-x α))))^(-((I k2)/α))) ((((E^(-x α))))^(-((I (k1 - k2))/α))) + 
            C1 (((-(E^(-x α))))^(-((I k2)/α))) ((((E^(-x α))))^(((I (k1 + k2))/α)))]]

4. Probability flux densities

Physical interpretation of obtained solutions (and its asymptotes) are

1) incoming wave:

(Subscript[ψ, incom])[x] = 
      FullSimplify[
        PowerExpand[
          C2 (((-(E^(-x α))))^(-((I k2)/α))) ((((E^(-x α))))^(-((I (k1 - k2))/α))) Hypergeometric2F1[b, 1 + b - c, 
              1 - a + b, -(E^(x α))]]]
(Subscript[ψ, incomAS])[x] = 
      PowerExpand[
        C2 (((-(E^(-x α))))^(-((I k2)/α))) ((((E^(-x α))))^(-((I (k1 - k2))/α))) ]

(Subscript[ψ, incomAS])[x] = 
      C2 (E^(I k1 x))

2) reflected wave:

(Subscript[ψ, reflec])[x] = 
      FullSimplify[
        PowerExpand[(((-(E^(-x α))))^(-((I k2)/α))) C1 ((((E^(-x α))))^(((I (k1 + k2))/α))) Hypergeometric2F1[a, 
              1 + a - c, 
              1 + a - b, -(E^(x α))]]]
(Subscript[ψ, reflecAS])[x] = 
      PowerExpand[(((-(E^(-x α))))^(-((I k2)/α))) C1 ((((E^(-x α))))^(((I (k1 + k2))/α))) Hypergeometric2F1[a, 
            1 + a - c, 1 + a - b, 0]]

(Subscript[ψ, reflecAS])[x] = 
      C1 (E^(-I k1 x))

3) transmitted wave:

(Subscript[ψ, trans])[
        x] = (E^(I k2 x)) Hypergeometric2F1[a, 
          b, c, -(E^(-x α))]
(Subscript[ψ, transAS])[
        x] = (E^(I k2 x)) Hypergeometric2F1[a, 
          b, c, 0]

So probability flux densities for asymptotes are [1]:

(Subscript[Flow, incom])[x] = 
      FullSimplify[
        ComplexExpand[((I ℏ)/(2 (Subscript[m, 0]))) ((Subscript[ψ, incomAS])[
                    x] Conjugate[(∂_x) (Subscript[ψ, incomAS])[
                        x]] - (∂_x) (Subscript[ψ, incomAS])[
                      x] Conjugate[(Subscript[ψ, incomAS])[
                      x]])]]
(Subscript[Flow, reflec])[x] = 
      FullSimplify[
        ComplexExpand[((I ℏ)/(2 (Subscript[m, 0]))) ((Subscript[ψ, reflecAS])[
                    x] Conjugate[(∂_x) (Subscript[ψ, reflecAS])[
                        x]] - (∂_x) (Subscript[ψ, reflecAS])[
                      x] Conjugate[(Subscript[ψ, reflecAS])[
                      x]])]]
(Subscript[Flow, trans])[x] = 
      FullSimplify[
        ComplexExpand[((I ℏ)/(2 (Subscript[m, 0]))) ((Subscript[ψ, transAS])[
                    x] Conjugate[(∂_x) (Subscript[ψ, transAS])[
                        x]] - (∂_x) (Subscript[ψ, transAS])[
                      x] Conjugate[(Subscript[ψ, transAS])[x]])]]

5. Reflection and transmission coefficients

Reflection and transmitted coefficient are calculated by the formulas [1]:

Reflection1 = 
      Abs[(Subscript[Flow, reflec])[x]/(Subscript[Flow, incom])[x]]

     Transmitted1 = Abs[(Subscript[Flow, trans])[x]/(Subscript[Flow, incom])[x]]

Now we us relations

a = -((I (k1 + 
                  k2))/α) 
b = ((I (k1 - k2))/α)
 c = 1 - ((2 I k2)/α)
C1 = ((GAMMA[c]*GAMMA[b - a])/(GAMMA[b]*GAMMA[c - a]))
C2 = ((GAMMA[c]*GAMMA[a - b])/(GAMMA[a]*GAMMA[c - b]))
 Reflection1 = 
      Abs[(Subscript[Flow, reflec])[x]/(Subscript[Flow, incom])[x]]
 Transmitted1 = Abs[(Subscript[Flow, trans])[x]/(Subscript[Flow, incom])[x]]

and obtain

Reflection = 
      FullSimplify[((GAMMA[1 - ((I (k1 - k2))/α) - ((2 I k2)/α)] GAMMA[((I (k1 - k2))/α) + ((I (k1 + k2))/α)] GAMMA[-((I (k1 + k2))/α)])/(GAMMA[-((I (k1 - k2))/α) - ((I (k1 + k2))/α)] GAMMA[1 - ((2 I k2)/α) + ((I (k1 + k2))/α)] GAMMA[((I (k1 - k2))/α)])) ((GAMMA[1 - ((-I (k1 - k2))/α) - ((-2 I k2)/α)] GAMMA[((-I (k1 - k2))/α) + ((-I (k1 + k2))/α)] GAMMA[-((-I (k1 + k2))/α)])/(GAMMA[-((-I (k1 - k2))/α) - ((-I (k1 + k2))/α)] GAMMA[1 - ((-2 I k2)/α) + ((-I (k1 + k2))/α)] GAMMA[((-I (k1 - k2))/α)]))]

    
 Transmitted = 
      FullSimplify[((k2 GAMMA[1 - ((I (k1 - k2))/α) - ((2 I k2)/α)] GAMMA[-((I (k1 + k2))/α)])/(k1 GAMMA[1 - ((2 I k2)/α)] GAMMA[-((I (k1 - k2))/α) - ((I (k1 + k2))/α)])) (( GAMMA[1 - ((-I (k1 - k2))/α) - ((-2 I k2)/α)] GAMMA[-((-I (k1 + k2))/α)])/(GAMMA[1 - ((-2 I k2)/α)] GAMMA[-((-I (k1 - k2))/α) - ((-I (k1 + k2))/α)]))]

Transorm these expressions with formulas [2]

GAMMA[ξ] GAMMA[1 - ξ] =  (π/Sin)[π ξ],  
    GAMMA[-ν] GAMMA[
          1 + ν] =  (π/Sin)[π (ν + 1)]
\(GAMMA[i\ ϕ]\ GAMMA[\(-i\)\ ϕ] = π\/\(ϕ\ Sinh[π\ ϕ]\), \ 
    GAMMA[1 + i\ ω]\ GAMMA[
          1 - i\ ω] = \(π\ ω\)\/Sinh[π\ ω]\)

Thus we have analytical formulas for reflection and transmission coefficients:

Reflection = 
      FullSimplify[(((π/Sin)[π ((I (k1 + k2))/α)] (π/Sin)[π (((I (k1 + k2))/α) + 1)])/((π/Sin)[π ((I (k1 - k2))/α)] (π/Sin)[π (((I (k1 - k2))/α) + 1)]))]
  Transmitted = 
      FullSimplify[
        (k2/k1) (((π/Sin)[π ((I (k1 + k2))/α)] (π/Sin)[π (((I (k1 + k2))/α) + 1)])/((π/(((2 I k1)/α) Sinh[π ((2 k1)/α)])) ((π ((2 I k2)/α))/Sinh)[π ((2 k2)/α)]))]

Thus we have analytical formulas for reflection and transmission coefficients:

and ascertain

FullSimplify[FunctionExpand[Reflection + Transmitted]]

Now we can make up inverse transformation for constants and obtain reflection coefficient:

k1 = ((Sqrt[2 (Subscript[m, 0]) En])/ℏ)
k2 = ((Sqrt[2 (Subscript[m, 0]) (En - (Subscript[U, 0]))])/ℏ)

and get

Reflection = 
      Csch[(((k1 + k2) π)/α)]^2 Sinh[(((k1 - k2) π)/α)]^2
Transmitted = 
      Csch[(((k1 + k2) π)/α)]^2 Sinh[((2 k1 π)/α)] Sinh[((2 k2 π)/α)]

6. Visualization of reflection and transmitted coefficients

<< Graphics`Graphics`
<< Graphics`Graphics3D`

Specify systems parameter and visualized functions:

(Subscript[U, 0]) = 10;
(Subscript[m, 0]) = 1;
\(\(ℏ = 1;\)
\)
Reflection = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) - (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2
Transmitted = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((2 (Sqrt[2]) π (Sqrt[En (Subscript[m, 0])]))/(α ℏ))] Sinh[((2 (Sqrt[2]) π (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/(α ℏ))]

α = 1
V1 = 
      ((Subscript[U, 0])/(1 + (E^(-α x))))
Reflection1 = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) - (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2
Transmitted1 = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((2 (Sqrt[2]) π (Sqrt[En (Subscript[m, 0])]))/(α ℏ))] Sinh[((2 (Sqrt[2]) π (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/(α ℏ))]

α = 2
V2 = 
      ((Subscript[U, 0])/(1 + (E^(-α x))))
Reflection2 = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) - (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2
Transmitted2 = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((2 (Sqrt[2]) π (Sqrt[En (Subscript[m, 0])]))/(α ℏ))] Sinh[((2 (Sqrt[2]) π (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/(α ℏ))]

α = 4
V3 = 
      ((Subscript[U, 0])/(1 + (E^(-α x))))
Reflection3 = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) - (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2
Transmitted3 = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((2 (Sqrt[2]) π (Sqrt[En (Subscript[m, 0])]))/(α ℏ))] Sinh[((2 (Sqrt[2]) π (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/(α ℏ))]

α = 8
V4 = 
      ((Subscript[U, 0])/(1 + (E^(-α x))))
Reflection4 = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) - (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2
Transmitted4 = 
      Csch[((π ((((Sqrt[2]) (Sqrt[En (Subscript[m, 0])]))/ℏ) + (((Sqrt[2]) (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/ℏ)))/α)]^2 Sinh[((2 (Sqrt[2]) π (Sqrt[En (Subscript[m, 0])]))/(α ℏ))] Sinh[((2 (Sqrt[2]) π (Sqrt[(Subscript[m, 0]) (En - (Subscript[U, 0]))]))/(α ℏ))]

Visualization of potential barrier at some values of α

Plot[{V1, V2, V3, V4}, {x, -4, 
        4}, 
PlotRange -> {0, 
          11}, 
Background -> 
        RGBColor[0.95, 0.95, 
          1], 
PlotStyle -> {{Thickness[0.006], 
            RGBColor[1, 0, 0]}, {Thickness[0.006], 
            RGBColor[0, 0.45, 0]}, {Thickness[0.006], 
            RGBColor[0, 0, 1]}, {Thickness[0.006], 
            RGBColor[0.5, 0.5, 1]}}, 
Frame -> True,  
      AspectRatio -> 0.4,  GridLines -> Automatic,  
      ImageSize -> {600, 
          250}, 
PlotLabel -> "POTENTIAL \
STEP"]

Dependences of reflection and transmitted coefficients by energy En (2d plots)

Plot[{Reflection1, Reflection2, Reflection3, 
        Reflection4}, {En, (Subscript[U, 0]), (Subscript[U, 0]) + 2}, 
      PlotRange -> {0, 1}, 
Background -> 
        RGBColor[0.95, 0.95, 
          1], 
PlotStyle -> {{Thickness[0.006], 
            RGBColor[1, 0, 0]}, {Thickness[0.006], 
            RGBColor[0, 0.45, 0]}, {Thickness[0.006], 
            RGBColor[0, 0, 1]}, {Thickness[0.006], 
            RGBColor[0.5, 0.5, 1]}}, 
AspectRatio -> 
        0.4,  GridLines -> Automatic,  
      ImageSize -> {600, 
          250}, 
PlotLabel -> "REFLECTION \
COEFFICIENTS"]


     Plot[{Transmitted1, Transmitted2, Transmitted3, Transmitted4}, {En, 
        (Subscript[U, 0]), (Subscript[U, 0]) + 2}, 
      PlotRange -> {0, 1}, 
Background -> 
        RGBColor[0.95, 0.95, 
          1], 
PlotStyle -> {{Thickness[0.006], 
            RGBColor[1, 0, 0]}, {Thickness[0.006], 
            RGBColor[0, 0.45, 0]}, {Thickness[0.006], 
            RGBColor[0, 0, 1]}, {Thickness[0.006], 
            RGBColor[0.5, 0.5, 1]}}, 
AspectRatio -> 
        0.4,  GridLines -> Automatic,  
      ImageSize -> {600, 
          250}, 
PlotLabel -> "TRANSMITTED \
COEFFICIENTS"]

Dependences of reflection and transmitted coefficients by energy En  and α (3d surface plots):

Plot3D[
      Reflection, {En, (Subscript[U, 0]), (Subscript[U, 0]) + 2}, {α, 0.1, 
        50}, 
PlotRange -> {0, 1},  
      PlotPoints -> 100,  Mesh -> False,  
      ViewPoint -> {1.123,  -2.488,  
          2.000}, 
AspectRatio -> 0.6,  
      ImageSize -> {500, 300}]


     Plot3D[
      Transmitted, {En, (Subscript[U, 0]), (Subscript[U, 0]) + 2}, {α, 0.1, 
        50}, 
PlotRange -> {0, 1},  
      PlotPoints -> 100,  Mesh -> False,  
      ViewPoint -> {1.123,  -2.488,  
          2.000}, 
AspectRatio -> 0.6,  
      ImageSize -> {500, 300}]

Dependences of reflection and transmitted coefficients by energy En  and α (Contour plots):

ContourPlot[Reflection,{En, (Subscript[U, 0]), (Subscript[U, 0]) + 2},{α, 0.1, 50},
PlotPoints -> 35, AspectRatio -> 0.6, Contours -> 45, ImageSize -> {600, 500}]

 ContourPlot[Transmitted,{En, (Subscript[U, 0]), (Subscript[U, 0]) + 2},{α, 0.1, 50},
PlotPoints -> 35, AspectRatio -> 0.6, Contours -> 45, ImageSize -> {600, 500}]

Dependences of reflection and transmitted coefficients by energy En and α (Density plots):

DensityPlot[
      Reflection, {En, (Subscript[U, 0]), (Subscript[U, 0]) + 3}, {α, 0.01, 
        30}, 
PlotPoints -> 50, Mesh -> False, 
      AspectRatio -> 0.6,  
      ImageSize -> {500, 300}]


     DensityPlot[
      Transmitted, {En, (Subscript[U, 0]), (Subscript[U, 0]) + 3}, {α, 0.01, 
        30}, 
PlotPoints -> 50, Mesh -> False, 
      AspectRatio -> 0.6,  ImageSize -> {500, 300}]