Aim-Consider an electron trapped on 1D infinite square well of width 1Angstrom compute the first two energy eigen values in ev and plot the normalized eigen function and their probability densities .

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 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 .

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=1973//eV Angstrom

k=100//eV/Angstrom.^2

m=0.511e6//eV/c.^2

e=3.795//(eV Angstrom).^1/2

r0=0

rm=1 // A

n=200

a=1//Angstrom

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

V1(i)=0

end

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

[y,V1]=spec(H)

disp(V1(1,1),"ground state energy")

disp(V1(2,2),"first state energy")

subplot(2,2,1)

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

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

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

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

subplot(2,2,2)

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

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

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

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

subplot(2,2,3)

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

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

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

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

subplot(2,2,4)

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

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

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

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

OUTPUT-

ground state energy

37.218773

first state energy

148.866