Simple Harmonic Oscillator
How can we visually display which transitions are allowed? We use the harmonic oscillator to model simple molecular vibration. We use reduced mass in order to “fix” one side. This allows us to assume that we have one dimensional motion of on mass along an axis.
First, the time independent Schrodinger equation will be solved using the potential energy of the harmonic oscillator. The original particle in a box code was modified such that the potential energy matrix modeled the harmonic oscillator potential energy. Then the eigenvectors and eigenvalues can be solved. We will use the dipole moment as the transition operator to determine whether or not the transition is allowed. We say that the transition dipole operator is the complex conjugate of the final wavevector state multiplied by the x position multiplied by the initial wavevector. We yield a scalar which is proportional to the likelihood of the transition from one initial state to a final state. Therefore, the likelihood of the transition is equal to
The dipole transition operator is represented with $\mu$.
On the graph below, the eigenvectors are again shifted up by their energies. With increasing energy levels, the nodes of the wavevectors become spaced closer together. The kinetic energy of the states increases with increasing energy levels, so the curvature of the graphical depiction of each energy state also increases. This is why the nodes become closer together with increasing energy level.
Each different energy level is equally separated, the spacing is where also refers to the frequency of the The zero point energy is of the first energy level, depicted by the ground state with a Gaussian shape.
The probabilty of the transition from one initial state to any other state is depicted below. The more probable transitions are indicated by colors closer to yellow, while the less probable transitions are blue. The selection rules hold in the simple harmonic oscillator, where the transition that are allowed are when the change in energy level is equal to 1.
## Code for Simple Harmonic Oscillator Transitions
First a number of different constants must be defined. Here, the mass of the particle (m),the charge (q), and ${\hbar^2}$ are all defined as 1 in order to simplify the problem. K refers to the spring constant in the problem. The number points used is set equal to 250. With n defined as one, the first 10 energy levels are plotted.
function selection(n)
m = 1;
hbarsq = 1;
pts = 250;
k = 1e3;
q = 1;
n=10;
The number of elements in the x vector is now defined. The linspace command is used to create a vector of x values which span from zero to the box length and are composed of pts number of elements. These x elements are evenly spaced between 0 and L. The change in x between all of these points is constant, as the points are evenly spaced and defined here as the difference between the second x element and the first x element.
x = linspace(-1, 1, pts);
dx = x(2)-x(1);
The potential vector Vvec can now be constructed with the potential energy of the simple harmonic oscillator. By putting the entries of Vvec on the diagonal of a new matrix, V, the potential energy matrix has been created.
Vvec=.5*k*(x.^2);
V=diag(Vvec);
We’ve then created a matrix which will find the kinetic energy of a different matrix. The second derivative of a function can be thought of as how the change in slope of a graph changes, and a similar thought is used here to consider how the change in entries of a vector change. This second derivative matrix shown below is then multiplied by ${1/dx^2}$ and that resulting matrix is then multiplied by $\frac{\hbar^2}{2* m}$ to determine the total kinetic energy of the vector. A new matrix H is then defined as the sum of the potential energy matrix and the kinetic energy matrix, as the Hamiltonian takes both the potential energy and kinetic energy in account in order to solve the total energy of a state.
D2 = -2*eye(pts) + diag(ones(pts-1,1), 1) + diag(ones(pts-1,1),-1);
T = 1/(dx^2)*(-hbarsq/(2*m))*D2;
H=T+V;
The [vecs, vals] command creates two new matrices which are the eigenvectors and eigenvalues of the matrix, H. The vecs matrix has the eigenvectors of H as columns in the matrix, and the vals matrix has all of the eigenvalues for H on the diagonal of the matrix. The srtvecs commmand, described here puts the eigenvectors and eigenvalues together in ascending order of energy level.
[vecs, vals] = eig(H);
[srtvecs, srtvals] = eigsort(vecs, vals);
Each energy level graph will then be shifted up by their own energies so that these different energy levels can be better visualized. A new matrix, called repvals, will be used in order to shift up the graphs. This shift must also be scaled by a factor in order for it to be better visualized. The shiftvecs matrix is a new matrix where the entries of the shifted eigenvectors have each been shifted up by their corresponding energy eigenvalue.
k = diag(srtvals);
l = ones(pts, 1);
repvals = l*k';
sc = 100;
shiftvecs = repvals + sc*srtvecs;
The new variable transitions defines the probability of the transition by multiplying the vecs by the transition dipole operator by vecs again as described above. The result is a scalar which defines the probabiltiy of the transition. The probabilities of these transitions can then be plotted.
transitions=vecs'*(q*diag(x))*vecs;
figure(1);pcolor(abs(transitions(1:n,1:n).^2))
axis square
figure(2);plot(x,shiftvecs(:,1:n),x,Vvec);
axis([-inf inf 0 200]);
end