problem1: Solve the s-wave Schrödinger equation for the ground state and the first excited state of the hydrogen atom where, d^2u/dr^2=A(r)u(r),A(r)=(2m/ℏ^2)[V(r)−E],V(r)=−e^2/r.

 Solve the s-wave Schrödinger equation for the ground state and the first excited state of the hydrogen atom where, d^2u/dr^2=A(r)u(r),A(r)=(2m/ℏ^2)[V(r)−E],V(r)=−e^2/r. where, m is the reduced mass of the electron. Obtain the energy eigenvalues and plot the corresponding wave functions. Remember that the ground state energy of the hydrogen atom is ≈-13.6 eV. Take e=3.795 (eVÅ), ħc= 1973(eVÅ) and m=0.511×10^6  eV/c^2 .

 input:

clf;clc;
charge = 3.795 // in (eV.A)^1/2
c_hcut = 1973 // in (eV.A)
m = 0.511e6 //in (eV/C^2)

r_min =0.01
r_max = 10
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 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
function y=potential(r)
    y=-(charge^2)./r

endfunction
V = diag(potential(r))
H = K+V
[U,E]= spec(H)
E = diag(E)
disp("The ground state energy is : "+string(E(1))+ " eV .") 
disp("The first excited state energy is : "+string(E(2))+ " eV .")
U_ground =U(:,1)./r'
U_1st_excited = U(:,2)./r'
U_ground = abs(U_ground)/max(abs(U_ground))
U_1st_excited =abs(U_1st_excited)/max(abs(U_1st_excited))

plot(r' , U_ground, '+r')
plot(r',U_1st_excited,'-b')
xlabel("r(angstrom)","font_size" , 4)
ylabel("wave function","font_size" , 4)
j=legend(["ground state", "1st excited state" ])
j.font_size = 4
title("H atom with Coulomb Potential ", "font_size", 4)

out put:

The ground state energy is : -13.612724 eV ."

  "The first excited state energy is : -3.4026108 eV ."




 solve the s-wave radial schrödinger equation for an atom where, d^2y/dr^2=a(r)u(r),a(r)=2m/ħ^2[v(r)−e],v(r)=−e^2/r where m is the reduced mass of the system , for the screened coulomb potential v(r)=(−e^2/r)*e^−r/a 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=0.511×10^6 ev/c^2 . the ground state energy is expected to be above "-12" ev in all three cases.