Matrix method to find coefficients of 1-d S.E.

[davidbenari]

I haven't taken a course on quantum mechanics yet, but I was asked to solve (numerically)

$[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)]\phi(x)=E\phi(x)$

$V(x)=2000(x-0.5)^2$

by supposing the solution is $\sum_{0}^{\infty} a_n \phi_n(x)$ and $\phi_n(x)$ is the typical solution to the a square potential $\phi_n(x) = \sqrt{\frac{2}{L}} sin(\frac{n \pi x}{L})$.

Now, to solve this I've done the approximation that my sum is actually a finite sum. Doing some manipulations one can show that you can find coefficients via the matrix equation

$\mathbf{M}\mathbf{a}=E\mathbf{a}$.

where $M_{mn} = E_m \delta_{mn} + \int_{0}^{L} \phi_m \phi_n V(x) dx$.

And $E_m= \frac{\hbar^2 m^2 \pi^2}{2 M L^2}$

Now, I've implemented a MATLAB program to solve eigenvalues and eigenvectors for $\mathbf{M}$ and used those coefficients to construct the solution to this problem.

Now, my question is how do I mathematically know $\phi(x)$ is normalized as well? Regarding the eigenvalues $E$, does QM say that my system can have any of the eigenvalues as energies when I'm not observing? Is that the interpretation?

 

[andrewkirk]

The vectors $|\psi\rangle$ used in QM (in QM they are usually called 'kets') are normalised by assumption. There can be problems with kets that are based on the Dirac delta function, but the energy eigenfunctions $|\phi_n\rangle$ that are in the above problem are not of that kind.

Your second question can't really be answered without knowing something about the physical system. All we have above is a differential equation, with no initial conditions and no physical description.

 

[davidbenari]

I guess you mean $\phi_n$ are normalised by assumption, but I see no mathematical reason that justifies why a linear superposition (that satisfies the DE) of them should also be normalised. Perhaps I didn't understand what you said correctly.

Your second question can't really be answered without knowing something about the physical system.

What information would be important to look for here? How does the typical problem statement go? I thought superposition of energy states was relevant to QM.

 

[andrewkirk]

I guess you mean $\phi_n$ are normalised by assumption, but I see no mathematical reason that justifies why a linear superposition (that satisfies the DE) of them should also be normalised.

If $\phi$ is a solution of the DE then so is $k\phi$, for any complex constant $k$. So we can just choose the solution ket that has unit norm.

In QM, physical states are mapped to one-dimensional vector subspaces, not to individual vectors (kets). So $|\psi\rangle$ and $k|\psi\rangle$ correspond to the same physical state.

 

[vanhees71]

To use the box eigenstates as basis for the harmonic oscillator is a very bad idea! The box is of finite extent, and there the given sine functions are proper square-integrable functions. Of course they are not proper square-integrable functions on the state, where space is over the entire real axis. Here, a good choice for bound state problems can be the harmonic-oscillator energy eigenstates, which you are supposed to calculate. Where does this problem come from, i.e., who "asked you" to do this problem in this quite questionable way?

 

[Chandra Prayaga]

To use the box eigenstates as basis for the harmonic oscillator is a very bad idea! The box is of finite extent, and there the given sine functions are proper square-integrable functions. Of course they are not proper square-integrable functions on the state, where space is over the entire real axis. Here, a good choice for bound state problems can be the harmonic-oscillator energy eigenstates, which you are supposed to calculate. Where does this problem come from, i.e., who "asked you" to do this problem in this quite questionable way?

Absolutely, vanhees71! The original Schrödinger equation is that of a (displaced) harmonic oscillator. Trying to solve it with box eigenstates will not work.

 

[davidbenari]

Where does this problem come from, i.e., who "asked you" to do this problem in this quite questionable way?

Its a class on computational physics; and it was my professor. To be fair though, I think the method does work since it gave me the correct functions (which I think are Hermite's, right?) I could annex a plot if anyone is interested.

Plus: the method produces 300 coefficients in 0.4 secs, which seems pretty decent. It pretty much ties up with a finite element method. Perhaps this is only true in this case.

 

[vanhees71]

Interesting. So I guess, you've put the system in a finite box of length $L$? The exact oscillator eigenfunctions are Hermite polynomials times a Gaussian:

https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator#One-dimensional_harmonic_oscillator

 

[davidbenari]

The exact oscillator eigenfunctions are Hermite polynomials times a Gaussian:

Yeah I'm pretty sure I got those. My particle is confined to [0,L].

I haven't been able to show if the superposition will be normalized or not. I think andrekirk was implying that I could change the coefficient of my solution (which is true), but I guess it isn't guaranteed the superposition (w/o having to adjust the proper coefficient) will satisfy the normalization condition, which makes sense...

$(\sum a_n \phi_n) ( \sum a^*_n \phi*_n) = \sum \sum a^*_m a_n \phi_n \phi*_m$

Integrating

$\sum \sum a_n a^*_m \delta_{mn} = \sum |a_n|^2 = ?$

Doesn't have to be 1 for sure I guess.

 

[vanhees71]

I see, then this is the most simple basis you can choose, and of course you can just multiply your wave function with corresponding normalization factor $\sqrt{1/\sum_n |a_n|^2}$ to normalize it.

 

[davidbenari]

I was thinking this shouldn't be too surprising as its very closely related to a Fourier sine series and this gives pretty good convergence to a lot of functions.