HYDROGEN ENERGY

 // FOR HYDROGEN ENERGY .

Input –

function y1=f(x)

y1=((2/((a0).^(3/2)))*(exp(-x/a0)));

endfunction

function y2=g(x)

y2=(1/((2*sqrt(2))*((a0).^(3/2)))*(exp(-x/(2*a0)))*(2-(x/a0)))

endfunction

function y3=h(x)

y3=(2/((81*sqrt(3))*((a0.^(3/2)))))*(27-

18*(x/a0)+2*((x^2)/(a0.^2)))*(exp(-x/(3*a0)))

endfunction

a0=5

x=0:25*a0;

subplot(2,2,1)

plot(x,f)

xlabel("X-AXIS")

ylabel("Y-AXIS")

subplot(2,2,2)

plot(x,g)

xlabel("X-AXIS")

ylabel("Y-AXIS")

subplot(2,2,3)

plot(x,h)

xlabel("X-AXIS")

ylabel("Y-AXIS")

Aim:- To solve the s-wave radial schrodinger equation for the vibrations of Hydrogen molecule.. where is the reduced mass of thr two atom system for the morse potential Find the lowest vibrational energy (in MeV) of the molecule to an accuracy of three digits. Also plot the corresponding wave function. Take , m=940x106 eV/C2 D=0.755501 eV α=1.44 r0=0.131349 Å

 Apparatus:- a laptop with installed scilab

Algorithm:-

1. Put all the given constants(eg.- h,m,n,r0,e)

2. Use linspace command for linearly spaced vectors

3. give the zero matrix of 'A' and 'v'.

4. apply the loop.

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

6. write the equation of H.

7. find the eigen value using 'spec(H)' command.

8. plot the potential graph.

9.show the eigen value by 'disp()' command.

Program:-

clc;

clear()

clf()

h=197.3;

m=940;

d=0.755501

k=100;

b=0;

a=1.44

q0=3.795

r0=0.131349;

rm=10;

n=500;

r=linspace(r0,rm,n)

d=(r-r0)/r

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)=d*((exp(-2*a*(r(i)-r0)/r(i))))

-exp(-a*(r(i)-r0)/r(i))

end

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

[R,y]=spec(H)

w=find(H>>0.0)

disp(H(w(1)))

disp(y(3,3))

disp("energy first exited state=")

disp(H(w(2)))

R(1,:)=R(:,1)'

R(2,:)=R(:,2)'

plot(r,(R(1,:)),'red')

plot(r,(R(2,:)),'green')

xgrid()

output:-

energy first exited state=

0.0080949

AIM-Solve the s-wave radial Schrodinger equation for a particle of mass m: For the anharmonic oscillator potential for the ground state energy (in MeV) of particle to an accuracy of three significant digits. Also, plot the corresponding wave function. Choose m = 940 MeV/c2, k = 100 MeV fm-2, b = 0, 10, 30 MeV fm-3In these units, cħ = 197.3 MeV fm. The ground state energy I expected to lie between 90 and 110 MeV for all three cases.

 APPARATUS- sci lab 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.Dispaly the ground state energy for three value 0,10,3

INPUT-

h=197.3

m=940

k=100

b=0

n=100

r0=-5e-15

rm=5

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)=(0.5*k*r(i)*r(i)+0.3*b*r(i)*r(i)*r(i))

v1(i)=(0.5*k*r(i)*r(i)+0.3*b*r(i)*r(i)*r(i))

end

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

[y,eig]=spec(H)

subplot(2,2,1)

for i=1:n

plot(r(i),v1(i),'r+')

xlabel("X-AXIS")

ylabel("Y-AXIS")

end

disp(eig(1,1))

disp(eig(2,2))

disp(eig(3,3))

subplot(2,2,2)

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

xlabel("X-AXIS")

ylabel("Y-AXIS")

subplot(2,2,3)

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

xlabel("X-AXIS")

ylabel("Y-AXIS")

subplot(2,2,4)

plot(r,y(:,3)','g')

xlabel("X-AXIS")

ylabel("Y-AXIS")

OUTPUT-

for b=0;

1. 92.969987 (ground state energy)

2. 220.52532

3. 348.52647

for b=10

1. 96.080573 (ground state energy)

2. 231.55534

3. 370.15555

for b=30

1. 101.68673 (ground state energy)

2. 250.72637

3. 406.79431

Aim-Solve the s-wave radial Schrodinger equation for an atom: where m is the reduced mass of the system (which can be chosen to be the mass of an electron), for the screened coulomb potential

 Find the energy (in eV) of the ground state of the atom to an accuracy of

three

significant digits. Also, plot the corresponding wavefunction. Take e =

3.795

(eVÅ)1/2, m = 0.511x106 eV/c2, and a = 3 Å, 5 Å, 7 Å. In these units ħc

= 1973

(eVÅ). The ground state energy is expected to be above -12 eV in all

three cases.

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 - 2

a=3,5,7

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))*(exp(-r(i)/a));

V1(i)=(-(e.^2)/r(i))*(exp(-r(i)/a));

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 – 2

EIGEN VALUES-*for a=3, *for a=7,

1. 9.22776 1. 11.503524

2. 0.4609445 2. 1.7146218

3. - 0.9688559 3. - 0.1459806

*for a=5,

1. 10.784776

2. 1.2427585

3. - 0.4712703

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