Problem - 2 Solve the s-wave radial Schrödinger equation for an atom

Solve the s-wave radial Schrödinger equation for an atom

where, d2ydr2=A(r)u(r),A(r)=2mℏ2[V(r)−E],V(r)=−e2r(7)(8)(9)$$\begin{array}{}\end{array}$$

V(r)=−e2re−r/a

Where m is the reduced mass of the system (which can be chosen to be the mass of an electron), for the screened Coulomb potential

V(r)=−e2re−r/a(10)$$\begin{array}{}\text{<br>}\end{array}$$

Find the energy (in eV) of the ground state of the atom to an accuracy of three significant digits. Also, plot the corresponding wave function. Take e=3.795 (eVÅ), and a=3 Å, 5 Å, and 7 Å in the units of ħc = 1973(eVÅ) and m=0.511×106${}^{}$eV/c^22
. The ground state energy is expected to be above -12 eV in all three cases${\mathrm{<span\; style="background-color:\; transparent;"> <a\; href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizr24ii\_ZdP5tLkReB9wZudZ\_RWI0OL4JubRIVsaoCu2VefD3YWnzOiUOdlf3TAGlWx\_p2tnKW\_kuxmQ7vMuTrJsy7zw2W7sKWivqC0U\_Ri5fPYciZspw4rHEwsp8dLQRBAI3Qld2MKOD9tJV8k-MJWoRRKXzX7A4q11jHg8fSOI9Kp0ISDpNz9TZ66g/s259/download.png"\; style="font-size:\; medium;\; margin-left:\; 1em;\; margin-right:\; 1em;\; text-align:\; center;\; white-space:\; normal;"><img\; border="0"\; data-original-height="194"\; data-original-width="259"\; height="194"\; src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizr24ii\_ZdP5tLkReB9wZudZ\_RWI0OL4JubRIVsaoCu2VefD3YWnzOiUOdlf3TAGlWx\_p2tnKW\_kuxmQ7vMuTrJsy7zw2W7sKWivqC0U\_Ri5fPYciZspw4rHEwsp8dLQRBAI3Qld2MKOD9tJV8k-MJWoRRKXzX7A4q11jHg8fSOI9Kp0ISDpNz9TZ66g/s1600/download.png"\; width="259"></a><div\; class="separator"\; style="clear:\; both;\; text-align:\; center;"><br></div><br></span>}}^{}$

Input:
clf;clc;
e = 3.795 // in (eV.angstrom)^1/2
c_hcut = 1973 // in (eV.angstrom)
m = 0.511e6 //in (eV/C^2)
r_min = 0.01 
r_max = 5 
N = 1000 
r = linspace(r_min,r_max,N) 
d = r(2)-r(1) 
s = (c_hcut^2)/(2*m*d^2) 
//now build the  matix K 
//   0.   0.   0.   0.   0.   0.
//   1.   0.   0.   0.   0.   0.
//   0.   1.   0.   0.   0.   0.=diag(ones(N-1, 1),-1)[(where N=6);]
//   0.   0.   1.   0.   0.   0.
//   0.   0.   0.   1.   0.   0.
//   0.   0.   0.   0.   1.   0.
// 
//  0.   1.   0.   0.   0.   0.
//   0.   0.   1.   0.   0.   0.
//   0.   0.   0.   1.   0.   0.
//   0.   0.   0.   0.   1.   0.=diag(ones(N-1,1),1))[(where N=6);]
//   0.   0.   0.   0.   0.   1.
//   0.   0.   0.   0.   0.   0.
//   
//   -2.   0.   0.   0.   0.   0.
//   0.  -2.   0.   0.   0.   0.
//   0.   0.  -2.   0.   0.   0.=-2*eye(N,N)[(where N=6);]
//   0.   0.   0.  -2.   0.   0.
//   0.   0.   0.   0.  -2.   0.
//   0.   0.   0.   0.   0.  -2
K = -s*((diag(ones(N-1,1),-1))-2*(eye(N,N))+(diag(ones(N-1,1),1)))
//now build the  matix V
function y=potential(r, a) 
    y = -(((e^2)./r).*exp(-r./a)) 
endfunction 
a0 = [3,5,7] //given the value of a in question
for a = a0 
    V = diag(potential(r,a)) 
    H = K+V 
    [U,E] = spec(H) 
    E = diag(E) 
    disp("The ground state energy is : "+string(E(1))+ " eV ,when a = "+ string(a)+ " Å .") 
    U_ground =U(:,1)./r'
U_ground = abs(U_ground)/max(abs(U_ground)) 
    scf() 
    plot(r' , U_ground, 'r')
 xlabel("r(angstrom)","font_size" , 4)
    ylabel("wave function","font_size" , 4)
    legend(["ground state" ])
 title("An atom with screened Coulomb Potential, when a = "+string(a)+" angstrom" ,"font_size", 4) 

end
output:
"The ground state energy is : -8.9428111 eV ,when a = 3 Å ."
"The ground state energy is : -10.492118 eV ,when a = 5 Å ."
"The ground state energy is : -11.208588 eV ,when a = 7 Å ."