Aim-Solve the s-wave Schrodinger equation for the ground state and the first excited state of the hydrogen atom: Here, m is the reduced mass of the electron. Obtain the energy eigenvalues and plot the corresponding wavefunctions. Remember that the ground state energy of the hydrogen atom is -13.6 eV. Take e = 3.795 (eVÅ)1/2, ħc = 1973 (eVÅ) and m = 0.511x106 eV/c2.

 Apparatus- scilab software,Laptop .

Algorithm- 1. Write the all given values of h,m,n,r0,e .

2.Use the linspace command .

3. Give the matrix of ‘A’and ‘V’ .

4. Using the for loop .

5. write the equation of v(i,i) .

6. write the equation of ‘H’ .

7. Using the ‘spec(H)’ command for eigen value .

8.Using the ‘subplot’ command and plot the potential graph and other three graphs.

9. Using the ‘legend’ command .

10.Give the display command and show the ‘eigen value’ .

INPUT-1

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-3)),'g')

xlabel("X-AXIS")

ylabel("Y-AXIS")

subplot(2,2,5)

legend('y,ground state n=1,y2 first n=2,l=0')

OUTPUT- 1.

EIGEN VALUE -

13.448465

3.3668089

1.2713873