# Discrepancies between numerical and analytical solutions

• Ziezi

#### Ziezi

OP warned about not using the homework template
The analytical solutions are:

\psi(x) =
\begin{cases}
Ce^{\alpha x}, \text{if } x < -\frac{L}{2}\\
Asin(kx) + Bcos(kx), \text{if } -\frac{L}{2} \leq x \leq \frac{L}{2}\\
Fe^{-\alpha x} , \text{if } x > \frac{L}{2}
\end{cases}

Next, ##\psi(x)## must be continuous and differentiable, which means that the value of the functions and their derivatives must match at the boundaries, ##-\frac{L}{2}## and ##\frac{L}{2}##. Depending on whether the solution is symmetric, ##\psi(-x) = \psi(x)##, or anti-symmetric, ##\psi(-x) = -\psi(x)##, ##A = 0##, ##C = F## or ##B = 0##. ##C = -F##, respectively. For the symmetric case:

\begin{array}{|l@{}}
Fe^{-\alpha L/2} = Bcos(k L/2)\\
-\alpha F e^{-\alpha L/2} = -\alpha Bcos(k L/2)
\end{array}

taking the ratio of the above two give us ##\alpha = k tan(k L/2)##. Similarly, from the anti-symmetric case we get ##\alpha = -k cot(k L/2)##.Both ##\alpha## and ##k## depend on the energy, ##E##, making the above equations transcendental. To solve them we make the following substitutions:

\begin{array}{|l@{}}
\eta = \frac{L}{2}\frac{\sqrt{2mV_0}}{\hbar}\\
\xi = \frac{L}{2}\frac{\sqrt{2mE}}{\hbar} = k \frac{L}{2} \quad (1)
\end{array}

With a small rearrangement our equations for the symmetric and anti-symmetric become:

\begin{array}{|l@{}}
\alpha = k tan(k L/2) \iff \sqrt{\frac{\eta^2}{\xi^2} - 1} = tan(\xi)\\
\alpha = -k cot(k L/2) \iff \sqrt{\frac{\eta^2}{\xi^2} - 1} = -cot(\xi)
\end{array}

To calculate the above I use the following Octave code:
Matlab:
    x = -6 : 0.1 : 46;                                  % coordinates  interval.
w = 1;                                              % well width.
U_0 = 40;                                           % well depth.
hbar = 1;                                           % Planck's constant.
m = 1;                                              % particle mass.

eta = ( w*sqrt(2*m*U_0) ) / (2*hbar);   xi = @(E) ( w*sqrt(2*m * E) ) / (2*hbar);
lhs = @(E) sqrt( ( (eta*eta) ./ (xi(E).^2) )  - 1);
rhs_s = @(E) tan( xi(E) );                          % symmetric case.
rhs_a = @(E) -cot( xi(E) );                         % anti-symmetric case.

% Solve the transcendental equation using root-finding algorithm.
func_1 = @(E) lhs(E) - rhs_s(E);  func_2 = @(E) lhs(E) - rhs_a(E);

x_0_s = 2;                              x_0_a =  10;
root_xi_x_s = fzero(func_1, x_0_s);     root_xi_x_a = fzero(func_2, x_0_a);
root_xi_y_s = real(rhs_s(root_xi_x_s)); root_xi_y_a = real(rhs_a(root_xi_x_a));

E_0 = (2*root_xi_x_s^2) / (m * w^2); E_1 =  (2*root_xi_x_a^2) / (m * w^2)

subplot(1, 2, 1);
plot(x, real(rhs_s(x)), 'b', x, real(lhs(x)), 'r', root_xi_x_s, root_xi_y_s, 'go')
axis([-6 46 -2 4])'; xlabel('\xi');
label3 = ['(', sprintf("%.2f", root_xi_x_s), ', ', sprintf("%.2f", root_xi_y_s), ')'];
legend('tan(\xi)', '\surd(\eta^2 / \xi^2 - 1)', label3);

subplot(1, 2, 2);
plot(x, real(rhs_a(x)), 'b', x, real(lhs(x)), 'r', root_xi_x_a, root_xi_y_a, 'go')
axis([-6 46 -2 4]); xlabel('\xi');
label4 = ['(', sprintf("%.2f", root_xi_x_a), ', ', sprintf("%.2f", root_xi_y_a), ')'];
legend('cot(\xi)', '\surd(\eta^2 / \xi^2 - 1)', label4);

From $$(12)$$ and the two values for $$\xi$$, the results for the energies are:
> 331.98<br>
> 21.50<br>

Now, for the numerical solution I use the following:
Matlab:
    % Parameters for solving problem in the interval -L < x < L.
L = 5;                                               % Interval Length.
N = 1000;                                            % No of points.
x = linspace(-L, L, N)';                             % Coordinate vector
dx = x(2) - x(1);                                    % Coordinate step

% Finite square well of width 2w and depth given by U_0.
U_0 = 40;                           w = L / 10;
U = -U_0*(heaviside(x + w) - heaviside(x - w));

% Discretized (three-point finite-difference) representation of the Laplacian.
e = ones(N, 1);
Lap = spdiags([e  -2*e e], [-1 0 1], N, N)

% Total Hamiltonian
hbar = 1;
m = 1;
H = -1/2*(hbar^2 / m)*Lap + spdiags(U, 0, N, N);  % T = H + U.

% Find (first 3) (smalest algebraic) eigenvalues.
nmodes = 3;             options.disp = 0;
[V, E] = eigs(H, nmodes, 'sa' , options);
[E, ind] = sort(diag(E));
E

The results for the energies are:
>36.72<br>
>27.14<br>
>12.27

I'm expecting some error but in this case, the energies differ in an order of magnitude.

Clearly, I am doing something wrong but I don't know whether the error is in the analytical or numerical solution. What am I doing wrong here? Any hint will be appreciated.

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I am not a MATLAB expert, but I know enough to suggest putting the two sets of solutions back into your equations (4) and see which one (if any) satisfies them.

Ziezi
I am not a MATLAB expert, but I know enough to suggest putting the two sets of solutions back into your equations (4) and see which one (if any) satisfies them.
That is not a bad idea, however, equation (12) is useful assuming that the values of ##\xi## are valid. The second value of ##\xi## produces an energy, ##E = 331.98##, making it a bit ambiguous given that the potential is only ##U_0 = 40##.

Most probably there are mistakes in both, analytical and numerical, methods.

Wait a minute. Look at your equations (3). What are your definitions of ##k## and ##\alpha##? If this is a particle in a finite square well problem, then if you redefine dimensionless constants ##\eta## and ##\xi##, it must be true that ##\eta^2+\xi^2 = const.##

Ziezi
Wait a minute. Look at your equations (3). What are your definitions of ##k## and ##\alpha##? If this is a particle in a finite square well problem, then if you redefine dimensionless constants ##\eta## and ##\xi##, it must be true that ##\eta^2+\xi^2 = const.##

You are right, I've omitted a crucial part:

Inside the box, the wavefunction is:

\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x) \iff
\frac{d^2 \psi(x)}{dx^2} = k^2 \psi(x)

where ##k = \frac{\sqrt{2mE}}{\hbar}##. Outside the box:

-\frac{\hbar^2}{m} \frac{\partial^2 \psi(x)}{\partial x^2} = (E - V_0) \psi(x) \iff \frac{\partial^2 \psi(x)}{\partial x^2} = \alpha^2 \psi(x)

where ##\alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar}##.

P.S.: I can't edit the OP more. So, I'm placing what should be at the beginning as a response