wavefunction and adjusting the value of E until the condition is satisfied. For the lowest energy state (ground state) E1, the wavefunction is zero only at the extreme values for x and therefore, the number of crossing is zero. The 1st excited state E2 will have only 1 crossing and the nth excited state En+1 will have n crossings as shown in Figure 3. In the Finite Difference Method, we start with where N is the maximum number of x coordinates, x(1) = xmin and x(N) = xmax and then as x is incremented, the other values of are calculated from the equation for c = 2 to N-1 When a physically accepted solution is found for the nth state the wavefunction is normalized by numerically integrating the wavefunction using Simpson’s 1/3 rule The m-script se_fdm.m was used to find the total energies and its corresponding wavefunctions for a potential well of depth -400 eV and width 0.1 nm. The range for the x-coordinates was from -0.1 nm to + 0.1 nm. The value of E was manually adjusted to find the physically acceptable solutions as shown in Figure 3. For this potential well, there are four bound states. The total energy for the ground state, n = 1 is E 1 = -373.84 eV. Thus, the binding energy of the electron or its ionization energy (energy need to free the electron from its bound state) is EB = +373.84 eV. Fig. 3. The four states for an electron confined by a potential well of -400 eV and width 0.10 nm with xmin = -0.10 nm, xmax = +0.10 nm. [se_fdm.m] The smaller binding energy, the less accurate are the results because the of x values is not large enough to accurately define the exponential convergence to zero of the wavefunctions as x approaches xmax. For the case n = 4 as shown in Figure 3, E4 = -21 eV when xmax = 0.1 nm and one can see that the exponential tail is not very well defined. When xmax is increased to 0.2 nm, the exponential tail is better defined and the total
… for c = 1 : N-2 U_matrix (c,c) = U(c); end Next, we have to represent the operator as a matrix of size (N-2)(N-2). From the definitions of the first and second derivatives of the function y(x), we can approximate them by the equations Hence, the second derivative matrix for N = 6 can be written as a 44 The SD matrix size is (N-2)(N-2) and not NN because the second derivative of the function can’t be evaluated at the end points, n = 1 and n = N. The kinetic energy matrix [K] is then defined as We can now define the Hamiltonian matrix as A sample code [se_solve.m] for the generating the Hamiltonian matrix is % Make Second Derivative Matrix ------------------------------------------ off = ones(num-3,1); SD_matrix = (-2*eye(num-2) + diag(off,1) + diag(off,-1))/dx2; % Make KE Matrix K_matrix = Cse * SD_matrix; % Make Hamiltonian Matrix H_matrix = K_matrix + U_matrix; Therefore, the Schrodinger Equation in matrix form is This is an eigenvalue equation in matrix form the action of the Hamilton matrix results in each value to be the vector being multiplied by a multiplied by the set of
numbers En. The set of numbers En are called the eigenvalues and set of vectors are the eigenvectors. This is a single Matlab function that finds both the eigenvalues and eigenvectors. The syntax of the command is [e_funct, e_values] = eig(H) where e_funct is an (N-2)(N-2) matrix with the nth column corresponding to the nth eigenfunction and e_values is a column vector for the N eigenvalues in increasing order. Only the negative values of e_values are significant. To obtain the complete eigenvector we need to include the end points where . A sample Malab code [se_solve.m] to obtain the discrete set of eigenvalues and normalized eigenfunction is % All Eigenvalues 1, 2 , ... n where E_N < 0 flag = 0; n = 1; while flag == 0 E(n) = e_values(n,n); if E(n) > 0, flag = 1; end; % if n = n + 1; end % while E(n-1) = []; n = n-2; % Corresponding Eigenfunctions 1, 2, ... ,n: Normalizing the wavefunction for cn = 1 : n psi(:,cn) = [0; e_funct(:,cn); 0]; area = simpson1d((psi(:,cn) .* psi(:,cn))',xMin,xMax); psi(:,cn) = psi(:,cn)/sqrt(area); prob(:,cn) = psi(:,cn) .* psi(:,cn); end % for A potential well as shown in Figure (5) has a minimum at x = 0 and tends to zero away from the origin. A classical particle would be trapped by this potential well and oscillate to and fro about x = 0 because the force on the particle is always directed towards origin for position since F = -dU/dx.
One meaning of the wavefunction is that its magnitude measures the probability amplitude of the particle found at some point. If the wavefunction corresponds to an eigenfunction , then the probability distribution probn(x) is independent of time ()and is given by and the probability Pn of finding the particle between x1 and x2 is We can also calculate probability distributions for observable quantities in a similar manner to above. The probability distribution is characterized by two measures – its expectation value (estimate of the mean value of the distribution) and its uncertainty (spread of values around the mean). Suppose that an observable quantity A has a discrete set of possible values a1, a2, … . Then, we can prepare N identical systems, all in the same state and measure the value of A in each system. In general, we will get a spread of results. The best estimate of the quantity A is its mean value <A> and the standard deviation can be used as a measure of the spread or uncertainty in the measurements. The expectation value of an observable A is the quantum-mechanical prediction for the mean value. In a stationary state described by the wavefunction , the expectation value of an observable A is given by (9) where is the quantum-mechanical operator corresponding to A. The operator is ‘sandwiched’ between the two wavefunction and , and for this reason we call the right hand side of Equation (9) a sandwich integral. The first part of the integrand is called the bra () and the second part the ket (). Some of the dynamical properties of a system that can be calculated are shown in Table 1. Total probability Position Momentum* Kinetic Energy Potential Energy Total energy where H is the Hamiltonian