Aim-solve the schrodinger wave equation for a proton as a Harmonic oscillator with amplitude (+1,-1) for using finite difference method.find their first ‘3’ energy eigen states in Mev and plot the first two normalized eigen function and their probability densities (with proper labeling ). K=100Mev/fm^2.

 Apparetus- Scilab software, Laptop .

Algorithm-

1. Input all the values of h,m,k,r0,rm etc.

2. Define the command of linspace .

3. Define the zeros matrix of A and v of order (n*n) .

4 write the equation of matrix A i.e,A(i,[i-1:i+1])=[1,-2,1] .

5. write the given equation of matrix v i.e., v(i,i)=0.5*k*r(i)^2 .

6. Find the value of H i.e., H=(((-h^2)/(2*m*d.^2)*A))+v

7. Find the eigen value of H using the spec(H) command .

8. Normalized the eigen function .

9. Using the subplot command and plot the graph .

INPUT-

h=197.3//Mev*fm/c

k=100//ev/fm^2

m=938.28//Mev

r0=-1//Angstrom

rm=1//Angstrom

n=200

r=linspace(r0,rm,n)

d=(rm-r0)/n

v=zeros(n,n)

A=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)=0.5*k*r(i)^2

v(1)=0.5*k*r(i)^2

end

H=(((-h^2)/(2*m*d.^2)*A))+v

[y,vl]=spec(H)

disp(vl(1,1),"ground state energy")

disp(vl(2,2),"first excited state energy")

disp(vl(3,3),"second excited state energy")

deff('z=f(r)','z=y(:,n).*y(:,n)')

nor=inttrap(r,f(r))

for i=1:3

psi(:,i)=y(:,i)/sqrt(nor)

end

subplot(2,2,1)

plot(r,psi(:,1),'b+')

xlabel('r -->','fontsize',3)

ylabel('$\psi (x)$','fontsize',3)

title('graph b/w psi(:,1) v/s r','fontsize',3)

subplot(2,2,2)

plot(r,psi(:,2),'r+')

xlabel('r -->','fontsize',3)

ylabel('$\psi (x)$','fontsize',3)

title('graph b/w psi(:,2) v/s r','fontsize',3)

subplot(2,2,3)

plot(r,(psi(:,1)).^2,'y+')

xlabel('r -->','fontsize',3)

ylabel('$\psi^{2} (x)$','fontsize',3)

title('graph b/w psi(:,1).^2 v/s r','fontsize',3)

subplot(2,2,4)

plot(r,(psi(:,2)).^2,'g+')

xlabel('r -->','fontsize',3)

ylabel('$\psi^{2} (x)$','fontsize',3)

title('graph b/w psi(:,2).^2 v/s r','fontsize',3)

OUTPUT-

ground state energy

57.194849

first excited state energy

216.96469

second excited state energy

471.87836