Particle on a circle - Benzene application

Looks good to me, although in this model isn't the ground state m=0 case?

So dont we disregard m = 0?

The boundary condition for the circle is that $\psi (0)=\psi (2 \pi)$

A constant wavefunction satisfies this and so I think should be fine.

No, if you go through all the maths, you should get

Ah OK, but then with the particle in a box (whether its 1D, 2D, 3D) why is n=0 disallowed? Surely we can apply the same logic as above but just with a different normalisation constant (root(1/2)) ?

Is it because of the uncertainty principle?

No, it's because the solution for a particle in a 1D box is

$\psi_n = \displaystyle \sqrt{\frac{2}{L}}\sin\left( \frac{n\pi x}{L}\right)$

so that $\psi_0 = 0$. This means that the probability of the particle being found somewhere in the box is zero when n=0, i.e. the particle does not exist.

Ah OK, but then with the particle in a box (whether its 1D, 2D, 3D) why is n=0 disallowed? Surely we can apply the same logic as above but just with a different normalisation constant (root(1/2)) ?

Is it because of the uncertainty principle?

You see why this is, right? If you sub in $n=0$, you have $\sin (0) = 0$, so $\psi_0=0$, i.e. no probability

No, it's because the solution for a particle in a 1D box is

$\psi_n = \displaystyle \sqrt{\frac{2}{L}}\sin\left( \frac{n\pi x}{L}\right)$

so that $\psi_0 = 0$. This means that the probability of the particle being found somewhere in the box is zero when n=0, i.e. the particle does not exist.

No, if you go through all the maths, you should get

$\psi(\theta) = \displaystyle \sqrt{\frac{1}{2\pi}} e^{im\theta}$

so that

$\psi(0) = \displaystyle \sqrt{\frac{1}{2\pi}}$

And so the wavefn is non-zero. This won't change your answer, though, because the excitation you're looking at is now from the m=3 HSOMO to the m=4 LUMO.



Of course, the boundary condition is $\psi(\theta) = \psi(\theta+2\pi)$, or we could define some wavefn such that it meets up at $\theta \equiv 0\ \mathrm{mod}\ 2\pi$ but not at any other values.