Phase shift of tunneling particle

[FunkyDwarf]

Hi All,

I'm sure this question has a trivial answer but for the life of me I've no idea :\

I am calculating (numerically because I am lazy and it scales/ports better to other cases) the phase shift of a particle in the non-rel Schrödinger equation tunneling through a potential step. This potential step gives rise to resonances and I would like to analyze how the energy of the resonance widths relates to an approximate calculation for tunneling time based on the WKB approximation.

So, my question is, what is the phase shift due to the decay region?

I did what i THINK is the correct method (counted total number of notes then compared to unshifted wavefunction to get delta) with the plot of phase versus energy here:

http://members.iinet.net.au/~housewrk/Phase.pdf .

Ignoring for the moment the wiggles caused by my hack-n-slash numerics, the main thing I don't understand is how i would fit a Breit-Wigner [ArcTan] line profile to these resonances given that they don't 'plateau' out on either side of the step in pi. I should clarify this result by saying I have checked that these phases do indeed shift the 'free particle' wave function to line up with the wavefunction after tunneling. Also I'm doing this in the case of standing waves so there is an exponentially growing and decaying solution in the tunneling region, where of course for on resonance energies the growing solution is suppressed.

Am I counting the total phase correctly, is this what one would expect? (doesn't look like the alpha-decay resonances I've seen, which is basically the same system without the Coulomb decay which is subtracted anyway, right?)

Hope that all made sense!
Cheers,

EDIT: Updated plot with stronger potential for more/sharper resonances, better plotrange.

 

[FunkyDwarf]

Bump? =(

Was the question poorly posed, do people need more information? I'm still a bit stuck with this :\

 

[SpectraCat]

Can you please elucidate what you mean by "counting total phase correctly"?

Also, I guess when you ask for the phase shift, you mean relative to the propagating wave in the absence of the barrier? If so, then I think the answer to your question is that the phases of the impinging and transmitted waves depend on the boundary conditions imposed by the tunneling barrier, as well as the frequency of the wave. Also, if the potential in the classically allowed regions before and after the barriers are not the same, then the frequencies of the impinging and exiting waves will not match, so "phase shift" needs to be carefully qualified in such cases.

If you are asking for a mathematical relationship between the phase shift and the physical parameters of the barrier, then I am not aware of any simple analytical relationship, but I guess one could probably figure them out for barriers with simple shapes (i.e. rectangular barriers).

Finally, I have not worked this out .. but it seems to me that the resonance might be due to certain phase matching conditions. In other words, it would make intuitive sense to me if the tunneling resonances correspond to energies where the phase shift goes to zero. I don't deal with this sort of tunneling much except to teach the simply pedagogy of it in basic QM courses. However, it is extremely important in the theory and design of certain semiconductor devices .. resonant tunneling diodes for instance. You might start searching websites/books/articles on topics related to that to see what is known about this.

 

[FunkyDwarf]

Hi SpectraCat,

Cheers for the reply!

1.) When i say counting the total phase I really only know how to explain in the terms of the above barrier case, i.e. calculating the phase accumulated by a wave from a to b. My understanding of this picture breaks down in the classical forbidden region (only allowed 0 or 1 nodes).

2.) Yes i mean phase shift of the resulting wavefunction in the second region compared to the original wavefunction without the barrier present. For simplicity i have considered the case where the potential in the classically allowed regions, and thus frequency of the wave, is the same.

3.) I'm not so much searching for the relationship, rather a check/validation of if what I have solved for (again, numerically because I'm lazy and want to input other potentials later) makes physical/mathematical sense.

As said previously, the phase shift that I computed and plotted has been back substituted into the original wavefunction to ensure it gives the correct shifted wavefunction.

It's simply that in the case of say alpha decay the few plots of phase i have seen illustrate that the phase on either side of the resonance asymptotes to some constant value whereas mine clearly doesn't. Of course this smacks of some background phase shift I have failed to take into account but I can't seem to find it :\