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QUANTUM SCATTERING BY THE ONE-DIMENSIONAL POTENTIAL BARRIER IN MAPLE

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QUANTUM SCATTERING BY THE ONE-DIMENSIONAL
POTENTIAL BARRIER IN MAPLE
 

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 barrier is solved in MAPLE.
Analytical formulas for reflection and transmission coefficients are obtained and visualized.
 

1. One-dimensional Schrodinger equation 

> restart:
 

Consider one-dimensional Schrodinger equation  [1] 

> SchrodingerEQN:=diff(psi(x),x$2)+2*m/h^2*(E-U(x))*psi(x);
 

`:=`(SchrodingerEQN, `+`(diff(psi(x), `$`(x, 2)), `/`(`*`(2, `*`(m, `*`(`+`(E, `-`(U(x))), `*`(psi(x))))), `*`(`^`(h, 2))))) (1.1)
 

where Typesetting:-mrow(Typesetting:-mi( is wave function, m is mass and E is energy of particle; Typesetting:-mrow(Typesetting:-mi( is Planck's constant . 

This equation describes one-dimensional motion of quantum particles in the field of potential barrier Typesetting:-mrow(Typesetting:-mi(: 

> U(x)=U0/(cosh(alpha*x)^2);
 

U(x) = `/`(`*`(U0), `*`(`^`(cosh(`*`(alpha, `*`(x))), 2))) (1.2)
 

We will use notations: 

> nu=expand(solve(U0=-nu*(nu+1)/(2*m)*h^2*alpha^2,nu));
 

nu = `+`(`-`(`/`(1, 2)), `/`(`*`(`^`(`+`(`*`(`^`(h, 2), `*`(`^`(alpha, 2))), `-`(`*`(8, `*`(U0, `*`(m))))), `/`(1, 2))), `*`(2, `*`(alpha, `*`(h))))) (1.3)
 

> EQ:=k=sqrt(2*m*E)/h;
E:=solve(EQ,E);
U0:=-nu*(nu+1)/(2*m)*h^2*alpha^2;
U(x):=U0/(cosh(alpha*x)^2);
 

`:=`(EQ, k = `/`(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(m, `*`(E)), `/`(1, 2)))), `*`(h))) (1.4)
 

`:=`(E, `+`(`/`(`*`(`^`(k, 2), `*`(`^`(h, 2))), `*`(2, `*`(m))))) (1.4)
 

`:=`(U0, `+`(`-`(`/`(`*`(nu, `*`(`+`(nu, 1), `*`(`^`(h, 2), `*`(`^`(alpha, 2))))), `*`(2, `*`(m)))))) (1.4)
 

`:=`(U(x), `+`(`-`(`*`(`/`(1, 2), `*`(`/`(`*`(nu, `*`(`+`(nu, 1), `*`(`^`(h, 2), `*`(`^`(alpha, 2))))), `*`(m, `*`(`^`(cosh(`*`(alpha, `*`(x))), 2))))))))) (1.4)
 

and obtain 

> SchrodingerEQN:=SchrodingerEQN;
 

`:=`(SchrodingerEQN, `+`(diff(psi(x), `$`(x, 2)), `/`(`*`(2, `*`(m, `*`(`+`(`/`(`*`(`^`(k, 2), `*`(`^`(h, 2))), `*`(2, `*`(m))), `*`(`/`(1, 2), `*`(`/`(`*`(nu, `*`(`+`(nu, 1), `*`(`^`(h, 2), `*`(`^`(a... (1.5)
 

2. Solution of Schrodinger equation 

Introduce new variable as 

> z=1/2-1/2*tanh(alpha*x);
x=solve(z=1/2-1/2*tanh(alpha*x),x);
 

z = `+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))) (2.1)
 

x = `+`(`-`(`/`(`*`(arctanh(`+`(`*`(2, `*`(z)), `-`(1)))), `*`(alpha)))) (2.1)
 

> Diff(psi(x),x)=Diff(z(x),x)*Diff(psi(z),z);
Diff(psi(x),x)=1/diff(-arctanh(2*z-1)/alpha,z)*diff(psi(z),z);
Diff(psi(x),x,x)=1/diff(-arctanh(2*z-1)/alpha,z)*diff(1/diff(-arctanh(2*z-1)/alpha,z)*diff(psi(z),z),z);
 

Diff(psi(x), x) = `*`(Diff(z(x), x), `*`(Diff(psi(z), z))) (2.2)
 

Diff(psi(x), x) = `+`(`-`(`*`(`/`(1, 2), `*`(`*`(`+`(1, `-`(`*`(`^`(`+`(`*`(2, `*`(z)), `-`(1)), 2)))), `*`(alpha, `*`(diff(psi(z), z)))))))) (2.2)
 

Diff(psi(x), `$`(x, 2)) = `+`(`-`(`*`(`/`(1, 2), `*`(`*`(`+`(1, `-`(`*`(`^`(`+`(`*`(2, `*`(z)), `-`(1)), 2)))), `*`(alpha, `*`(`+`(`-`(`*`(`/`(1, 2), `*`(`*`(`+`(`-`(`*`(8, `*`(z))), 4), `*`(alpha, `*... (2.2)
 

In this variable Schrodinger equation is 

> SchrodingerEQN:=-1/2*(1-(2*z-1)^2)*alpha*(-1/2*(-8*z+4)*alpha*diff(psi(z),z)-1/2*(1-(2*z-1)^2)*alpha*diff(psi(z),`$`(z,2)))+simplify(subs(x=-arctanh(2*z-1)/alpha,2*m/h^2*(1/2*k^2*h^2/m+1/2*nu*(nu+1)/m*h^2*alpha^2/cosh(alpha*x)^2)))*psi(z);
 

`:=`(SchrodingerEQN, `+`(`-`(`*`(`/`(1, 2), `*`(`*`(`+`(1, `-`(`*`(`^`(`+`(`*`(2, `*`(z)), `-`(1)), 2)))), `*`(alpha, `*`(`+`(`-`(`*`(`/`(1, 2), `*`(`*`(`+`(`-`(`*`(8, `*`(z))), 4), `*`(alpha, `*`(dif...
`:=`(SchrodingerEQN, `+`(`-`(`*`(`/`(1, 2), `*`(`*`(`+`(1, `-`(`*`(`^`(`+`(`*`(2, `*`(z)), `-`(1)), 2)))), `*`(alpha, `*`(`+`(`-`(`*`(`/`(1, 2), `*`(`*`(`+`(`-`(`*`(8, `*`(z))), 4), `*`(alpha, `*`(dif...
(2.3)
 

Thus solution of Schrodinger equation is 

> dsolve(SchrodingerEQN,psi(z));
 

psi(z) = `+`(`*`(_C1, `*`(hypergeom([`+`(`-`(nu)), `+`(nu, 1)], [`/`(`*`(`+`(alpha, `*`(k, `*`(I)))), `*`(alpha))], z), `*`(`^`(`+`(z, `-`(1)), `+`(`-`(`/`(`*`(`+`(`*`(`/`(1, 2), `*`(I))), `*`(k)), `*... (2.4)
 

> z:=1/2-1/2*tanh(alpha*x);
psi(x)=_C1*hypergeom([-nu,nu+1],expand([1/alpha*(alpha+k*I)]),z)*z^(1/2*I*k/alpha)*(z-1)^(-1/2*I*k/alpha)+_C2*(z*(z-1))^(-1/2*I*k/alpha)*hypergeom([expand((-nu*alpha-I*k)/alpha),expand(1/alpha*(nu*alpha+alpha-I*k))],[expand(1/alpha*(alpha-I*k))],z);
 

`:=`(z, `+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x)))))))) (2.5)
 

psi(x) = `+`(`*`(_C1, `*`(hypergeom([`+`(`-`(nu)), `+`(nu, 1)], [`+`(1, `/`(`*`(k, `*`(I)), `*`(alpha)))], `+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x)))))))), `*`(`^`(`+`(`/`(1, 2), `...
psi(x) = `+`(`*`(_C1, `*`(hypergeom([`+`(`-`(nu)), `+`(nu, 1)], [`+`(1, `/`(`*`(k, `*`(I)), `*`(alpha)))], `+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x)))))))), `*`(`^`(`+`(`/`(1, 2), `...
(2.5)
 

3. Analysis of the solution 

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

Typesetting:-mrow(Typesetting:-msup(Typesetting:-mo(. 

As 

> 1/2-1/2*tanh(alpha*x)=simplify((convert(1/2-1/2*tanh(alpha*x),exp)));
-1/2-1/2*tanh(alpha*x)=simplify((convert(-1/2-1/2*tanh(alpha*x),exp)));
hypergeom([-nu, nu+1],[1+k/alpha*I],0);
hypergeom([-nu-I*k/alpha, nu+1-I*k/alpha],[1-I*k/alpha],0);
 

`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))) = `/`(1, `*`(`+`(exp(`+`(`*`(2, `*`(alpha, `*`(x))))), 1))) (3.1)
 

`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))) = `+`(`-`(`/`(`*`(exp(`+`(`*`(2, `*`(alpha, `*`(x)))))), `*`(`+`(exp(`+`(`*`(2, `*`(alpha, `*`(x))))), 1))))) (3.1)
 

1 (3.1)
 

1 (3.1)
 

we can calculate 

> psi(x)=_C1*hypergeom([-nu,nu+1],[1+k/alpha*I],0)*(1/2-1/2*tanh(alpha*x))^(1/2*I*k/alpha)*(-1/2-1/2*tanh(alpha*x))^(-1/2*I*k/alpha)+_C2*((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*hypergeom([-nu-I*k/alpha, nu+1-I*k/alpha],[1-I*k/alpha],0);
 

psi(x) = `+`(`*`(_C1, `*`(`^`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `/`(`*`(`*`(`/`(1, 2), `*`(I)), `*`(k)), `*`(alpha))), `*`(`^`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), ... (3.2)
 

> psi[as](x)=_C1*exp(-2*alpha*x)^(1/2*I*k/alpha)*(-1)^(-1/2*I*k/alpha)+_C2*(exp(-2*alpha*x)*(-1))^(-1/2*I*k/alpha);
 

psi[as](x) = `+`(`*`(_C1, `*`(`^`(exp(`+`(`-`(`*`(2, `*`(alpha, `*`(x)))))), `/`(`*`(`*`(`/`(1, 2), `*`(I)), `*`(k)), `*`(alpha))), `*`(`^`(-1, `+`(`-`(`/`(`*`(`+`(`*`(`/`(1, 2), `*`(I))), `*`(k)), `*... (3.3)
 

> psi[as](x) = _C1*exp(-x*I*k)*(-1)^(-1/2*I*k/alpha)+_C2*(-exp(x*I*k));
 

psi[as](x) = `+`(`*`(_C1, `*`(exp(`+`(`-`(`*`(`+`(I), `*`(x, `*`(k)))))), `*`(`^`(-1, `+`(`-`(`/`(`*`(`+`(`*`(`/`(1, 2), `*`(I))), `*`(k)), `*`(alpha)))))))), `-`(`*`(_C2, `*`(exp(`*`(x, `*`(k, `*`(I)... (3.4)
 

So we have: 

> psi[trans](x):=((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*hypergeom([-nu-I*k/alpha,nu+1-I*k/alpha],[1-I*k/alpha],1/2-1/2*tanh(alpha*x));
 

`:=`(psi[trans](x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`... (3.5)
 

or 

> psi[trans](x):=((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*hypergeom([a,b],[c],1/2-1/2*tanh(alpha*x));
 

`:=`(psi[trans](x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`... (3.6)
 

where used realtions: 

> a=-nu-I*k/alpha;
b=nu+1-I*k/alpha;
c=1-I*k/alpha;
 

a = `+`(`-`(nu), `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha)))) (3.7)
 

b = `+`(nu, 1, `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha)))) (3.7)
 

c = `+`(1, `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha)))) (3.7)
 

Obtained solution must be extend to x ---> -infinity. But Gauss hypergeometric function 

Typesetting:-mrow(Typesetting:-mi(  

is defined on interval Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn(. So we need to use Gauss relation [2] 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

or 

> C1=(GAMMA(c)*GAMMA(c-a-b))/(GAMMA(c-b)*GAMMA(c-a));
C2=(GAMMA(c)*GAMMA(a+b-c))/(GAMMA(a)*GAMMA(b));
hypergeom([a, b],[c],y)=C1*hypergeom([a, b],[a+b-c+1],1-y)+C2*[(1-y)^(c-a-b)]*hypergeom([b, 1-c+b],[1-a+b],1-y);
 

C1 = `/`(`*`(GAMMA(c), `*`(GAMMA(`+`(c, `-`(a), `-`(b))))), `*`(GAMMA(`+`(c, `-`(b))), `*`(GAMMA(`+`(c, `-`(a)))))) (3.8)
 

C2 = `/`(`*`(GAMMA(c), `*`(GAMMA(`+`(a, b, `-`(c))))), `*`(GAMMA(a), `*`(GAMMA(b)))) (3.8)
 

hypergeom([a, b], [c], y) = `+`(`*`(C1, `*`(hypergeom([a, b], [`+`(a, b, `-`(c), 1)], `+`(1, `-`(y))))), `*`(C2, `*`([`^`(`+`(1, `-`(y)), `+`(c, `-`(a), `-`(b)))], `*`(hypergeom([b, `+`(1, `-`(c), b)]... (3.8)
 

Now solution will have form: 

> psi(x):=subs(y=1/2-1/2*tanh(alpha*x),((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*(C1*hypergeom([a, b],[a+b-c+1],1-y)+C2*[(1-y)^(c-a-b)]*hypergeom([b, 1-c+b],[1-a+b],1-y)));
 

`:=`(psi(x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`*`(`/`(...
`:=`(psi(x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`*`(`/`(...
(3.9)
 

or 

> psi(x):=((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*(C1*hypergeom([a, b],[a+b-c+1],1/2+1/2*tanh(alpha*x))+C2*subs(a=-nu-I*k/alpha,b=nu+1-I*k/alpha,c=1-I*k/alpha,(1/2+1/2*tanh(alpha*x))^(c-a-b))*hypergeom([b, 1-c+b],[1-a+b],1/2+1/2*tanh(alpha*x)));
 

`:=`(psi(x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`*`(`/`(...
`:=`(psi(x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`*`(`/`(...
(3.10)
 

and 

> psi(x):=((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*C1*hypergeom([a, b],[a+b-c+1],1/2+1/2*tanh(alpha*x))+((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*C2*(1/2+1/2*tanh(alpha*x))^(k/alpha*I)*hypergeom([b, 1-c+b],[1-a+b],1/2+1/2*tanh(alpha*x));
 

`:=`(psi(x), `+`(`*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`*`(...
`:=`(psi(x), `+`(`*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`*`(...
(3.11)
 

4. Probability flux densities 

As 

> 1/2-1/2*tanh(alpha*x)=simplify((convert(1/2-1/2*tanh(alpha*x),exp)));
-1/2-1/2*tanh(alpha*x)=simplify((convert(-1/2-1/2*tanh(alpha*x),exp)));
hypergeom([a, b],[a+b-c+1],0);
hypergeom([b, 1-c+b],[1-a+b],0);
 

`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))) = `/`(1, `*`(`+`(exp(`+`(`*`(2, `*`(alpha, `*`(x))))), 1))) (4.1)
 

`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))) = `+`(`-`(`/`(`*`(exp(`+`(`*`(2, `*`(alpha, `*`(x)))))), `*`(`+`(exp(`+`(`*`(2, `*`(alpha, `*`(x))))), 1))))) (4.1)
 

1 (4.1)
 

1 (4.1)
 

we obtained asymptotics solution at x = - infinity: 

> psi[as](x) := (1*(-exp(2*alpha*x)))^(-1/2*I*k/alpha)*C1*hypergeom([a, b],[a+b-c+1],0)+(1*(-exp(2*alpha*x)))^(-1/2*I*k/alpha)*C2*exp(2*alpha*x)^(k/alpha*I)*hypergeom([b, 1-c+b],[1-a+b],0);
 

`:=`(psi[as](x), `+`(`*`(`^`(`+`(`-`(exp(`+`(`*`(2, `*`(alpha, `*`(x))))))), `+`(`-`(`/`(`*`(`+`(`*`(`/`(1, 2), `*`(I))), `*`(k)), `*`(alpha))))), `*`(C1)), `*`(`^`(`+`(`-`(exp(`+`(`*`(2, `*`(alpha, `... (4.2)
 

or 

> psi[as](x):=simplify((-exp((-x)*I*k))*C1+(-exp((-x)*I*k))*C2*(-exp(2*x*I*k)));
 

`:=`(psi[as](x), `+`(`-`(`*`(exp(`+`(`-`(`*`(`+`(I), `*`(x, `*`(k)))))), `*`(C1))), `*`(C2, `*`(exp(`*`(x, `*`(k, `*`(I)))))))) (4.3)
 

Physical  interpretation of obtained solutions (and its asymptotics) are 

1) incoming wave: 

> psi[incom](x):=((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*C2*(1/2+1/2*tanh(alpha*x))^(k/alpha*I)*hypergeom([b, 1-c+b],[1-a+b],1/2+1/2*tanh(alpha*x));
psi[incom_as](x):=C2*exp(k*x*I)*hypergeom([b, 1-c+b],[1-a+b],0);
 

`:=`(psi[incom](x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`... (4.4)
 

`:=`(psi[incom_as](x), `*`(C2, `*`(exp(`*`(x, `*`(k, `*`(I))))))) (4.4)
 

2) reflected wave: 

> psi[reflec](x):=((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*C1*hypergeom([a, b],[a+b-c+1],1/2+1/2*tanh(alpha*x));
psi[reflec_as](x):=-exp(-I*x*k)*C1;
 

`:=`(psi[reflec](x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(... (4.5)
 

`:=`(psi[reflec_as](x), `+`(`-`(`*`(exp(`+`(`-`(`*`(`+`(I), `*`(x, `*`(k)))))), `*`(C1))))) (4.5)
 

3) transmitted wave: 

> psi[trans](x):=((1/2-1/2*tanh(alpha*x))*(-1/2-1/2*tanh(alpha*x)))^(-1/2*I*k/alpha)*hypergeom([a, b],[c],1/2-1/2*tanh(alpha*x));
psi[trans_as](x):=exp(x*k*I);
 

`:=`(psi[trans](x), `*`(`^`(`*`(`+`(`/`(1, 2), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))), `*`(`+`(`-`(`/`(1, 2)), `-`(`*`(`/`(1, 2), `*`(tanh(`*`(alpha, `*`(x))))))))), `+`(`-`(`/`(`*`(`+`(`... (4.6)
 

`:=`(psi[trans_as](x), exp(`*`(x, `*`(k, `*`(I))))) (4.6)
 

So probability flux densities for asymptotics are [1] 

> with(tensor):
J[incom](x):=simplify(evalc((I*h/(2*m))*(psi[incom_as](x)*conj(diff(psi[incom_as](x),x))-conj(psi[incom_as](x))*diff(psi[incom_as](x),x))));
J[reflec](x):=simplify(evalc((I*h/(2*m))*(psi[reflec_as](x)*conj(diff(psi[reflec_as](x),x))-conj(psi[reflec_as](x))*diff(psi[reflec_as](x),x))));
J[trans](x):=simplify(evalc((I*h/(2*m))*(psi[trans_as](x)*conj(diff(psi[trans_as](x),x))-conj(psi[trans_as](x))*diff(psi[trans_as](x),x))));
 

`:=`(J[incom](x), `/`(`*`(h, `*`(`^`(C2, 2), `*`(k))), `*`(m))) (4.7)
 

`:=`(J[reflec](x), `+`(`-`(`/`(`*`(h, `*`(`^`(C1, 2), `*`(k))), `*`(m))))) (4.7)
 

`:=`(J[trans](x), `/`(`*`(k, `*`(h)), `*`(m))) (4.7)
 

5. Reflection and transmitted coefficients 

Reflection coefficient is calculated by the formula [1]: 

> Reflection:=abs(J[reflec](x)/J[incom](x));
Transmitted:=abs(J[trans](x)/J[incom](x));
 

`:=`(Reflection, abs(`/`(`*`(`^`(C1, 2)), `*`(`^`(C2, 2))))) (5.1)
 

`:=`(Transmitted, `/`(1, `*`(`*`(`^`(abs(C2), 2))))) (5.1)
 

and using relations 

> a:=-nu-I*k/alpha; b:=nu+1-I*k/alpha; c:=1-I*k/alpha;
C1:=(GAMMA(c)*GAMMA(c-a-b))/(GAMMA(c-b)*GAMMA(c-a));
C2:=(GAMMA(c)*GAMMA(a+b-c))/(GAMMA(a)*GAMMA(b));
 

`:=`(a, `+`(`-`(nu), `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha))))) (5.2)
 

`:=`(b, `+`(nu, 1, `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha))))) (5.2)
 

`:=`(c, `+`(1, `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha))))) (5.2)
 

`:=`(C1, `/`(`*`(GAMMA(`+`(1, `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha))))), `*`(GAMMA(`/`(`*`(k, `*`(I)), `*`(alpha))))), `*`(GAMMA(`+`(`-`(nu))), `*`(GAMMA(`+`(nu, 1)))))) (5.2)
 

`:=`(C2, `/`(`*`(GAMMA(`+`(1, `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha))))), `*`(GAMMA(`+`(`-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha))))))), `*`(GAMMA(`+`(`-`(nu), `-`(`/`(`*`(`+`(I), `*`(k)), `*`(alpha)))... (5.2)
 

we obtain 

> nu=(-1/2*h*alpha+1/2*(h^2*alpha^2-8*U0*m)^(1/2))/alpha/h;
Reflection:=simplify((C1/C2)*conj(C1/C2));
Transmitted:=simplify((1/C2)*conj(1/C2));
 

nu = `/`(`*`(`+`(`-`(`/`(`*`(h, `*`(alpha)), `*`(2))), `/`(`*`(`^`(`+`(`*`(`^`(h, 2), `*`(`^`(alpha, 2))), `*`(4, `*`(nu, `*`(`+`(nu, 1), `*`(`^`(h, 2), `*`(`^`(alpha, 2))))))), `/`(1, 2))), `*`(2))))... (5.3)
 

`:=`(Reflection, `/`(`*`(`^`(sin(`*`(Pi, `*`(`+`(nu, 1)))), 2)), `*`(sin(`/`(`*`(Pi, `*`(`+`(`*`(nu, `*`(alpha)), alpha, `-`(`*`(`+`(I), `*`(k)))))), `*`(alpha))), `*`(sin(`/`(`*`(Pi, `*`(`+`(`*`(nu, ... (5.3)
 

`:=`(Transmitted, `/`(`*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(k)), `*`(alpha))), 2)), `-`(1))), `*`(sin(`/`(`*`(Pi, `*`(`+`(`*`(nu, `*`(alpha)), `-`(`*`(`+`(I), `*`(k)))))), `*`(alpha))), `*`(sin(`/`(`*`... (5.3)
 

or 

> Reflection := simplify(expand(sin(Pi*(nu+1))^2/sin(Pi/alpha*(nu*alpha+alpha-I*k))/sin(Pi*(nu*alpha+alpha+k*I)/alpha)));
Transmitted := simplify(expand((cosh(Pi/alpha*k)^2-1)/sin(Pi*(nu*alpha-I*k)/alpha)/sin(Pi*(nu*alpha+k*I)/alpha)));
 

`:=`(Reflection, `+`(`-`(`/`(`*`(`+`(`-`(1), `*`(`^`(cos(`*`(Pi, `*`(nu))), 2)))), `*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(k)), `*`(alpha))), 2)), `-`(`*`(`^`(cos(`*`(Pi, `*`(nu))), 2))))))))) (5.4)
 

`:=`(Transmitted, `/`(`*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(k)), `*`(alpha))), 2)), `-`(1))), `*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(k)), `*`(alpha))), 2)), `-`(`*`(`^`(cos(`*`(Pi, `*`(nu))), 2))))))) (5.4)
 

and ascertain 

> T+R=simplify(convert(Reflection+Transmitted,exp));
 

`+`(T, R) = 1 (5.5)
 

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

> restart; Reflection := -(-1+cos(Pi*nu)^2)/(cosh(Pi/alpha*k)^2-cos(Pi*nu)^2):
Transmitted := (cosh(Pi/alpha*k)^2-1)/(cosh(Pi/alpha*k)^2-cos(Pi*nu)^2):
REFLACTION(E,alpha):=simplify(subs([k=sqrt(2*m*E)/h,nu = -1/2+1/2/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2)],Reflection));
TRASMITTED(E,alpha):=simplify(subs([k=sqrt(2*m*E)/h,nu = -1/2+1/2/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2)],Transmitted));
T+R=simplify(convert(REFLACTION(E,alpha)+TRASMITTED(E,alpha),exp));
 

`:=`(REFLACTION(E, alpha), `/`(`*`(`^`(cos(`+`(`/`(`*`(Pi, `*`(`^`(`+`(`*`(`^`(h, 2), `*`(`^`(alpha, 2))), `-`(`*`(8, `*`(U0, `*`(m))))), `/`(1, 2)))), `*`(2, `*`(alpha, `*`(h)))))), 2)), `*`(`+`(`*`(... (5.6)
 

`:=`(TRASMITTED(E, alpha), `/`(`*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(m, `*`(E)), `/`(1, 2))))), `*`(alpha, `*`(h)))), 2)), `-`(1))), `*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(... (5.6)
 

`+`(T, R) = 1 (5.6)
 

6. Visualization of reflection and transmitted coefficients 

Specify systems parameter and visualized functions: 

> restart:
U0:=10; m:=1; h:=1;
alpha:=1;
U1:=U0/(cosh(alpha*x)^2):
REFLACTION1:=(cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2)):
TRASMITTED1:=(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1)/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2):
alpha:=2;
U2:=U0/(cosh(alpha*x)^2):
REFLACTION2:=cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2):
TRASMITTED2:=(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1)/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2):
alpha:=4;
U3:=U0/(cosh(alpha*x)^2):
REFLACTION3:=cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2):
TRASMITTED3:=(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1)/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2):
alpha:=8;
U4:=U0/(cosh(alpha*x)^2):
REFLACTION4:=cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2):
TRASMITTED4:=(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1)/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2):
 

`:=`(U0, 10) (6.1)
 

`:=`(m, 1) (6.1)
 

`:=`(h, 1) (6.1)
 

`:=`(alpha, 1) (6.1)
 

`:=`(alpha, 2) (6.1)
 

`:=`(alpha, 4) (6.1)
 

`:=`(alpha, 8) (6.1)
 

Visualization of potential barrier at some values of Typesetting:-mrow(Typesetting:-mi(: 

> plot([U1,U2,U3,U4],x=-4..4,color=[red,blue,gold,blue],thickness=[2,2,2,2]);
 

Plot_2d
 

Dependences of reflection and transmitted coefficients by energy E: 

> plot([REFLACTION1,REFLACTION2,REFLACTION3,REFLACTION4],E=0..U0+12,color=[red,blue,gold,blue],thickness=[2,2,2,2]);
plot([TRASMITTED1,TRASMITTED2,TRASMITTED3,TRASMITTED4],E=0..U0+12,color=[red,blue,gold,blue],thickness=[2,2,2,2]);
 

Plot_2d
 

Plot_2d
 

Dependences of reflection and transmitted coefficients by energy E and Typesetting:-mrow(Typesetting:-mi( (3d surface plots): 

> restart:
U0:=10; m:=1; h:=1;
REFLACTION:=cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2);
TRASMITTED:=(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1)/(cosh(Pi/alpha*2^(1/2)*(m*E)^(1/2)/h)^2-1+cos(1/2*Pi/alpha/h*(h^2*alpha^2-8*U0*m)^(1/2))^2);
 

`:=`(U0, 10) (6.2)
 

`:=`(m, 1) (6.2)
 

`:=`(h, 1) (6.2)
 

`:=`(REFLACTION, `/`(`*`(`^`(cos(`+`(`/`(`*`(Pi, `*`(`^`(`+`(`*`(`^`(alpha, 2)), `-`(80)), `/`(1, 2)))), `*`(2, `*`(alpha))))), 2)), `*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(`^`(2, `/`(1, 2)), `*`(`^`(E, ... (6.2)
 

`:=`(TRASMITTED, `/`(`*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(`^`(2, `/`(1, 2)), `*`(`^`(E, `/`(1, 2))))), `*`(alpha))), 2)), `-`(1))), `*`(`+`(`*`(`^`(cosh(`/`(`*`(Pi, `*`(`^`(2, `/`(1, 2)), `*`(`^`(E, `... (6.2)
 

> plot3d(REFLACTION,E=U0..U0+12,alpha=1..25, orientation=[-55,75],axes=BOXED,lightmodel=light4,style=PATCHCONTOUR, orientation=[-15,55]);
plot3d(TRASMITTED,E=U0..U0+12,alpha=1..25, orientation=[-55,75],axes=BOXED,lightmodel=light4,style=PATCHCONTOUR, orientation=[-15,55]);
 

Plot
 

Plot
 

Dependences of reflection and transmitted coefficients by energy E and Typesetting:-mrow(Typesetting:-mi( (Contour plots): 

> with(plots):
contourplot(REFLACTION,E=U0..U0+12,alpha=1..25, contours=32,filled=true,coloring=[white,blue],grid=[35,35]);
contourplot(TRASMITTED,E=U0..U0+12,alpha=1..25, contours=32,filled=true,coloring=[white,blue],grid=[35,35]);
 

Plot_2d
 

Plot_2d
 

References 

1. Landau L.D., Lifshits E.V. Theoretical physics. V. 3. Quantum mechanics. - Moscow. Fizmatlit. 2003. 

2. Handbook of Mathematical Functions. Edited by Milton Abramowitz and Irene A. Stegun. Issued June 1964. Tenth Printing, December. 1972. 

 

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