- #1

FunkyDwarf

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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 Schrodinger 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.

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 Schrodinger 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.

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