In this tutorial we revisit the quantum mechanics problem discussed in class. Recall that the ODE system that we need to solve is \[ \frac{du}{dx}=w \] \[ \frac{dw}{dx} = (-1 - E)u \] with initial conditions \[ u(0) = 0 \] \[ \frac{du}{dx}(0) = 1 \] We then need to adjust \( E \) so that \[ \lim_{x\to\infty} u(x) = 0 \]
Download the script file deut.m and function file
fcn_deut.m
DOWNLOAD SOURCE
The energy eigenvalue for the case \( x_0 = 2\) (the case coded in
deut.m) satisfies
\[
-0.11 \le E \le -0.10
\]
EXERCISE
First, run deut.m with \( E = -0.11 \) and \( E = -0.10 \) so that you
can see the two different types of behaviour at large \( x \).
Then, using bisection, repeatedly modify deut.m and run it until you have
found a value for \( E \) accurate to about 6 decimal digits.
Recall that we have \[ u = r \psi \] and that the normalization on \( \psi \) is \[ \int \psi \psi^* dV = \int r^2 \psi \psi^* dr = 1 \] so, in terms of \( u \) (which is real) \[ \int u^2 dr = 1 \] Let us assume that we have adopted units so that \( x = r \). Then we also have \[ \int u^2 dx = 1 \] Now, let \( \bar u \) be a properly normalized wave function. If we compute the integral \[ N = \int_{0}^{x_{\rm max}} u^2 dx \] where \( u \) is unnormalized, then we can normalize using \[ \bar u = \frac{u}{\sqrt{N}} \] One approach to normalizing \( u \) is based on the observation that the above integral normalization condition can be recast as the differential equation \[ \frac{dI}{dx} = u^2 \] with the initial condition \[ I(0) = 0 \] Then \[ N = I(x_{\rm max}) \] EXERCISE
Modify deut.m and fcn_deut.m to include the normalization ODE so that
the wavefunction can be normalized after an integration.
Can you see any flaw in this implementation of normalization?
deut.mEXERCISE
As the section heading states, incorporate automatic bisection in deut.m
so that given an initial bracket
\[
E_{\rm LO} \le E \le E_{\rm HI}
\]
the script automatically determines the eigenvalue \( E \) to some
specified tolerance.
Use your script to find eigenvalues and eigenfunctions for several values of \( x_0 \).