To solve the s wave Schrodinger equation for the ground state & the first excited state of the hydrogen atom d2y/dx2=A(r)U(r); A(r)=8mPie2/h2[V(r)-E];V(r)=-e2/r, where is m is reduced mass of electron. Find the energy eigen value & plot the corresponding wave functions. Take ground state energy of hydrogen atom is -13.6eV, hc=12400eVA0 & m=0.511x106 eV/c2.

To solve the s wave Schrodinger equation for the ground state & the first excited state of the hydrogen atom d2y/dx2=A(r)U(r); A(r)=8mPie2/h2[V(r)-E];V(r)=-e2/r, where is m is reduced mass of electron. Find the energy eigen value & plot the corresponding wave functions. Take ground state energy of hydrogen atom is -13.6eV,  hc=12400eVA0 & m=0.511x106 eV/c2. 

Coding:

h=1973
m=0.511e6
n=200
r0=1e-15
rm=10
e=3.795
r=linspace(r0,rm,n)
d=(rm-r0)/n
A=zeros(n,n)
V=zeros(n,n)
A(1,[1:2])=[-2,1];
A(n,[n-1:n])=[1,-2];
for i=2:n-1
A(i,[i-1:i+1])=[1,-2,1];
end
for i=1:n
V(i,i)=((e.^2)/r(i));
V1(i)=(-(e.^2)/r(i));
end
H=((h.^2)/(2*m*d*d)*A)+V
[y,eig]=spec(H)
subplot(2,2,1)
for i=10:n
plot(r(i),V1(i),'y+')
xlabel("X-AXIS")
ylabel("Y-AXIS")
end
disp(eig(n-1,n-1))
disp(eig(n-2,n-2))
disp(eig(n-3,n-3))
subplot(2,2,2)
plot(r,abs(y(:,n-1)),'r')
xlabel("X-AXIS")
ylabel("Y-AXIS")
subplot(2,2,3)
plot(r,abs(y(:,n-2)),'b')
xlabel("X-AXIS")
ylabel("Y-AXIS")
subplot(2,2,4)
plot(r,abs(y(:,n-2)),'g')
xlabel("X-AXIS")
ylabel("Y-AXIS")
subplot(2,2,5)
legend('y,ground state n=1,y2 first n=2,l=0')

OUTPUT:

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