Basic questions on quantum scattering

[hiyok]

there are several questions I'm going to ask about scattering. however, first i got to say something on a plane wave, say, 'exp(ik*r)'. Here both k and r are vectors and the star means a scalar product. As is known, this state has a current density J proportional to 'k' but independent on 'r', which would indicate that a detector will have the same amount of counts during an interval, despite where it's placed. Is this right? If so, I'd like to go on.

Usually, in scattering problems, the incident particle is assumed in a plane wave, and so is the outgoing one. But, both the incident particle and the outgoing one should have a well-defined position, viz, its uncertainty should be smaller than the resolution of the detector. So, they should be superpositions of plane waves. They have to be wave packets. The only compromise seems that, such wave packets have a principal wave-vector, which determines the energy of the particle. Is this right?

Finally, In my opinion, scattering is a temporal event, whose treatment should involve the time-dependent perturbation theory. However, usually, these things are completely dealt with using the stationary Schrödinger equation. So, what's the relationship between these two jargons? I'm especially keen to hear a physical explanation.

Thank you very much !

 

[per.sundqvist]

there are several questions I'm going to ask about scattering. however, first i got to say something on a plane wave, say, 'exp(ik*r)'. Here both k and r are vectors and the star means a scalar product. As is known, this state has a current density J proportional to 'k' but independent on 'r', which would indicate that a detector will have the same amount of counts during an interval, despite where it's placed. Is this right? If so, I'd like to go on.

Usually, in scattering problems, the incident particle is assumed in a plane wave, and so is the outgoing one. But, both the incident particle and the outgoing one should have a well-defined position, viz, its uncertainty should be smaller than the resolution of the detector. So, they should be superpositions of plane waves. They have to be wave packets. The only compromise seems that, such wave packets have a principal wave-vector, which determines the energy of the particle. Is this right?

Finally, In my opinion, scattering is a temporal event, whose treatment should involve the time-dependent perturbation theory. However, usually, these things are completely dealt with using the stationary Schrödinger equation. So, what's the relationship between these two jargons? I'm especially keen to hear a physical explanation.

Thank you very much !

Hi, The plane wave is assumed for simplicity, but the scattered wave is a sum of outgoing spherical Hankel functions (google it). You are confusing the wave with the probability of detection which is $$\mid\Psi_{out}(\vec{r})\mid^2$$. The detection is local but not the wave function with exactly known energy, i.e., energy is fixed in the problem ("different energies does not speak with each other").

For these large plane waves the charge density at the collission center is =0, because the wave is infinite (normalization constant is infinity), so there can't be any "reaction" of the wave-package around the scattering center of interest in time. You could transform from the Energy scattering picture to the time-dependent by using Fourier-Transform.

I hope this explained something,
Per

 

[reilly]

You are quite right about wave packets, which are used in formal scattering theory. This formal theory is quite sophisticated, because it is based on some very tricky limiting procedures -- cf the Lippman-Schwinger Eq. You will find a very extensive and complete discussion of scattering in the classic book, Collision Theory by Goldberger and Watson; Weinberg discusses scattering theory in Vol. I of his field theory serioes.

The key observation: if a scattered particle interacts with a potential or the more general Interaction Hamiltonian, then the initial and final states asymptotically approach free particle states -- the interaction must vanish at infinity. So the use of plane waves can be justified physically as well as mathematically.

A standard way to obtain transition rates -- say to get scattering cross sections - uses time dependent perturbation theory to calculate the probability of a transition from initial to final states -- that's where the energy delta function comes from. The net of that time dependent work is that its OK to go to the time independent approach, which uses perturbation theory to compute the scattering wave function.

You can work out a lot of this with the problem of a single particle scattering from a finite potential well or barrier in 1 dimension.

Regards,
Reilly Atkinson

 

[hiyok]

Hi, Dear Per and Rebelli, thanks for your reply. This leads me to some new questions:

For these large plane waves the charge density at the collission center is =0, because the wave is infinite (normalization constant is infinity), so there can't be any "reaction" of the wave-package around the scattering center of interest in time. You could transform from the Energy scattering picture to the time-dependent by using Fourier-Transform.

(1) i don't think a plane wave can't react to a scatterer, since it's inappropriate to interpret its square as the prob. density when it's not normalizable. Indeed, from Born 's scattering theory, one could see a plane wave can be inflected with a finite prob. amplitude proportional to the Fourier transform of the scattering potential. how do you think about it?

(2)It seems that plane waves have a strange property in the sense that, on one hand, they do correspond to physical states, and on the other hand, they can not be normalized (they can only be normalized to Dirac function) as usual. So the Born's interpretation doesn't apply to them at all. To see how this may lead to very uncomfortable result, i'd like to consider a state |psi>=a|p_a>+b|p_b>, where |p_a> and |p_b> are eigenstates of momentum operator while a and b are coefficients. Physically, one may assert that |a|^2+|b|^2=1, since an experiment devised to measure momentum distribution can either display p_a or p_b. Now let's put |psi>=\int dp phi_p|p>, which indicates that phi_p=a\delta(p-p_a)+b\delta(p-p_b). Then from the normalizability of |psi>, i.e., 1=<psi|psi>, we infer that \int|phi_p|^2 dp=1, which, uncomfortabley, doesn't yield |a|^2+|b|^2=1. What am I missing in this argument?

(3) if the above argument is persuasive, one way out may be to discretize the continuous state manifold, by for instance smearing the delta-function. After all, in practice, people don't need point-like resolution, viz, don't need to pin a particle absolutely. Do you have any other suggestions?

i'm crazy about these puzzles. Please help me out! Thank you!

Yours,
hiyok

 

[hiyok]

hi, i'd like to add too my last post the following:

It seems that, although it's meaningless to speak of prob. density as the wave function is not normalizable, it's quite sensible to talk about prob. current density, and all physically reasonable results should be defined using it.

Regards,
hiyok

 

[malawi_glenn]

Interaction is <Final state| Scattering potential | Initial state >

Works fine for plane waves, re write it as a partial wave decomposition.

 

[reilly]

Hi, Dear Per and Rebelli, thanks for your reply. This leads me to some new questions:



(1) i don't think a plane wave can't react to a scatterer, since it's inappropriate to interpret its square as the prob. density when it's not normalizable. Indeed, from Born 's scattering theory, one could see a plane wave can be inflected with a finite prob. amplitude proportional to the Fourier transform of the scattering potential. how do you think about it?

(2)It seems that plane waves have a strange property in the sense that, on one hand, they do correspond to physical states, and on the other hand, they can not be normalized (they can only be normalized to Dirac function) as usual. So the Born's interpretation doesn't apply to them at all. To see how this may lead to very uncomfortable result, i'd like to consider a state |psi>=a|p_a>+b|p_b>, where |p_a> and |p_b> are eigenstates of momentum operator while a and b are coefficients. Physically, one may assert that |a|^2+|b|^2=1, since an experiment devised to measure momentum distribution can either display p_a or p_b. Now let's put |psi>=\int dp phi_p|p>, which indicates that phi_p=a\delta(p-p_a)+b\delta(p-p_b). Then from the normalizability of |psi>, i.e., 1=<psi|psi>, we infer that \int|phi_p|^2 dp=1, which, uncomfortabley, doesn't yield |a|^2+|b|^2=1. What am I missing in this argument?

(3) if the above argument is persuasive, one way out may be to discretize the continuous state manifold, by for instance smearing the delta-function. After all, in practice, people don't need point-like resolution, viz, don't need to pin a particle absolutely. Do you have any other suggestions?

i'm crazy about these puzzles. Please help me out! Thank you!

Yours,
hiyok

These are not puzzles; most of scattering theory was developed by 1940, and is so standard that it is discussed in many, many books and articles.

1. the fact that <pf|V|pi> is the Fourier transform of the potential, with respect to
exp(i(pi-pf)r) generally gives a non-zero interaction. And this is the transition matrix used in the Born formulation; or more generally used in Heisenberg's S-matrix. The difference (pf-pi) is called the momentum transfer. This approach is 60-70 years old, and has never been found wanting. The issue of normalizing continuous spectra wave functions in practical circumstances was basically solved by 1940 or so, and is covered in many, many QM books.
What's to worry?


2. The fact that plane waves are tricky, and the fact that in real experiments the particles are somewhat localized provides justification for the use of wave packets.To show that plane waves are perfectly legitimate for scattering is difficult and requires considerable mathematical sophistication. Among other things, one can show that the Born approx is
perfectly OK to use with plane waves -- this has been known since the 1920s. Also all this limiting stuff. (I'm not going to go thorough your algebra, as your result is incorrect. You have to deal with probability theory for continuous events. Again, try a real problem --scattering from a 1D potential well or barrier. Note that this problem is in virtually any QM book ever published. Do this and you will get answers that should convince you that your puzzles are not puzzles.)

3. The way folks "smear"{ the delta function is, for all practical purposes) to use wave packets. In a sense this is the most important aspect of scattering theory -- showing how to use plane waves. The demonstration is difficult, and involves some very subtle limiting procedures (the so-called "i epsilon" analysis and limits.). The primary result from this analysis is called the Lippman-Schwinger Eq.


|W> = |W0> + (1/(E-H0 + ia))V|W>

Where a is an infinitesimal real constant, H0 and V are the free Hamiltonian (usually the KE), V is the interaction Hamiltonian |W> is the exact wave function;|W0> is the initial, free, wave function. The ia term indicates,, how to deal with the singularity -- just like the electrical engineers do with contour integration for filters and for the imposition of causality.

People sometimes use so-called box normalization to beat the continuum issues.

This stuff is not easy: in Goldberger and Watson's Collision Theory, the theory of scattering
takes about 160 pages of detailed analysis.

Again, do some problems and read about scattering theory -- it is at least summarized in countless books -- cf. Landau and Lifschits, for example.

Your questions are good ones. You can find all the answers you need in the literature as the answers are very well known.

Regards,
Reilly Atkinson

 

[hiyok]

Hi, Reilly, Thank you very much! I now see where I was wrong.

Regards,
hiyok

 

[Anglea]

can anyone please show me the difference between the A causal Green's function causal Green's function ?? and the difference between nodal lines and the scars?