Low energy scattering quantum mechanics
We come spinning out of nothing, scattering stars like dust! Last weekI explained Rayleigh and Raman scattering from a classical point of view. In the process, I explained why the sky is blue and introduced Raman spectroscopya powerful tool for studying the structure of molecules.
This week, I fill in the gaps and explain scattering from a quantum-mechanical point of view. Before we can talk about scattering, though, we need to review some important ideas from quantum mechanics : energy levels and the Heisenberg uncertainty principle.
Energy Levels. The story of energy levels starts deep within the atom. The energy and momentum of a particle control how many times the corresponding wave wiggles within a certain distance.
If the atom is part of a molecule especially a crystalthe discrete allowed energies become so numerous that, together, they look like a continuous band. And this leads to band structure. For clarity, physicists often imagine extremely simple atoms with only two or three allowed electron orbits, each of which is allowed only at a single specific energy and a single specific momentum. Depending on the situation, they even neglect the momenta and only look at the allowed energies.
For example, the figure below shows a two-level atom with a single electron in the lowest energy state. When a photon — a light particle —hits the atom or, alternatively, passes right through itit has the potential to affect the electron. If we ignore quantum mechanics and look at this classically, the light would always accelerate the electron, since the electron is a charged particle and electromagnetic fields affect charged particles.
However, if the electron accelerated, it would gain kinetic energy. Otherwise, surprisingly, the photon passes right through the atom unmolested, as shown below. Importantly, once an electron absorbs a photon, it can sit in the higher energy level as long it likes. Astute regular readers may complain here. In the pastI said that electrons want to be in the lowest-energy state available. Both statements are true. But in the real world, electrons drop to lower-energy states through fluorescence.
This is because, in the real world—thanks to quantum field theory—there are always photons or other particles for the electron to interact with. And these other particles allow the electron to drop down to lower energy levels through stimulated emission. The uncertainty principle is a consequence of the fact that matter is both a particle and a wave. We cannot know both the position and the momentum of a particle at the same time.
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. I am very confused about the idea of the cross-section in classical mechanics, quantum mechanics, and high energy physics. It seems like in classical and quantum mechanics, scattering cross-section is often in terms of some geometric parameters for example the impact parameter in case of classical scatteringand if I am correct, the cross-section has the dimension of area.
However, in high energy physics, the cross-section is used to describe probability of certain interactions happening in a collision. So, given the cross-section obtained from let's say Born's approximation, how does it relate to the probabilistic idea of the cross-section in high energy?
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. The idea of scattering cross-section in classical mechanics, quantum mechanics, and high energy physics Ask Question. Asked 6 days ago. Active 6 days ago. Viewed 22 times.
It is an effective area intercepting your beam. The luminosity times the cross section amounts to the event rate, no? See here. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.
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The Overflow Blog. Socializing with co-workers while social distancing. Featured on Meta. Community and Moderator guidelines for escalating issues via new response…. Feedback on Q2 Community Roadmap. Related 4. Hot Network Questions.In quantum physicsthe scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.
The latter is described by the wavefunction. The dimension of the scattering amplitude is length. The scattering amplitude is a probability amplitude ; the differential cross-section as a function of scattering angle is given as its modulus squared.
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves, . Then the differential cross section is given by . The scattering length for X-rays is the Thomson scattering length or classical electron radiusr 0.
The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b. A quantum mechanical approach is given by the S matrix formalism. The scattering amplitude can be determined by the scattering length in the low-energy regime. From Wikipedia, the free encyclopedia. Probability amplitude in quantum scattering theory.
What determines whether there will be a photoelectric effect photon is absorbed, electron is released or whether there will be a Compton scattering the photon is scattered at some angle, and the electron is released with another direction? For a given system that the electron is in, the primary determinant is the energy of the photon. As DJBunk points out, this is a quantum mechanical process, so the "choice" is fundamentally random.
A given interaction will occur with a probability proportional to its cross section. Figure 1 of this lecture shows how the cross section for each possible process varies with photon energy. This plot is for the interaction of photons with electrons in copper. At low energies, the photoelectric effect is the dominant effect.
From about keV to about 10 MeV, Compton scattering is the dominant effect. Above 10 MeV, the dominant effect is pair production. At a given photon energy, the relative probability of two processes would be the ratio of their cross sections.
The dependence of each cross section on photon energy should be similar in form for any system; the exact numbers will vary from system to system. Table 2 of that lecture gives the dependence on the atomic number, for example. At low Energy Long wavelengths as in Radio waves and Microwaves, visible light Rayleigh scattering would dominant, in which the scattered photon would have the same wavelength as the incident photon.
At higher energy shorter wavelengths as in Ultraviolet spectrumthe electron would absorb the energy of the incident photon assuming the incident photon frequency is above the cutoff frequency and the electron would be ejected no photon is scattered here, it has been absorbed by the electron Photoelectric Effect would dominate. At extremely high energies X-ray and Gamma raysthe incident photon energy is too high for the electron to absorb and maintain Energy-Momentum conservation and that is why Photoelectric is observed on bounded electrons whereas Compton Scattering is observed on "Free" Electrons.Quantum scattering-1
So the electron would be ejected, Since the incident photon has transferred some of its energy to the electron which it collides with, it will be scattered but now it has less energy.This thesis addresses topics in low energy scattering in quantum mechanics, in particular, resonance phenomena. A powerful tool in the problems of low energy scattering is the point scatterer model, and will be used extensively throughout the thesis. Therefore, we begin with a thorough introduction to this model in Chapter 2.
As a rst application of the point scatterer model, we will investigate the phenomenon of the proximity resonance, which is one example of strange quantum behavior appearing at low energy.
Proximity resonances will be addressed theoretically in Chapter 3, and experimentally in Chapter 4. Threshold resonances, another type of low energy scattering resonance, are considered in Chapter 5, along with their connection to the E mov and Thomas eects, and scattering in the presence of an external con ning potential. Although the point scatterer model will serve us well in the work presented here, it does have its limitations. These limitations will be removed in Chapter 6, where we describe how to extend the model to include higher partial waves.
In Chapter 7, we extend the model one step further, and illustrate how to treat vector wave scattering with the model.
Finally, in Chapter 8 we will depart from the topic of low energy scattering and investigate the inuence of diraction on an open quantum mechanical system, again both experimentally and theoretically. Documents: Advanced Search Include Citations. Citations: 1 - 0 self. Abstract This thesis addresses topics in low energy scattering in quantum mechanics, in particular, resonance phenomena.
Powered by:.A blackbody is an ideal thermal object that absorbs all radiation falling on it at low temperatures and is also a perfect radiator. The curves of radiation intensity versus wavelength could not be explained by classical physics. Max Planck — developed an equation for blackbody radiation that agreed with the data.
This derivation required two assumptions:. The radical nature of Planck's vision is the assumption of quantized energy states. The terms discrete and quantum referred to considering the energy as coming in packets instead of as a continuous flow; thus, the molecule will change energy states only if the amount of energy absorbed or radiated is a discrete amount of energy.
The phot oelectric effect is the emission of electrons from certain metals when light shines on the metallic surface. The emitted electrons are called photoelectrons. A number of aspects of the effect were puzzling:. Einstein explained the photoelectric effect, using Planck's quantum hypothesis and the conservation of energy. His equation is K. The work function is the energy required to release the electron from a particular metal.
The energy of the incoming photons, or quanta, is hf ; therefore, the photoelectric equation simply states that the energy of the ejected electrons is the difference between the energy absorbed from the quanta of light and the energy required to escape from the material. The unexplained observations described above can be illuminated by the following arguments. Compton scattering. Additional evidence for the quantized nature of electromagnetic waves came from the Compton effect, named for Arthur Compton — The scattered photon has less energy than the original photon, which can be seen as a change in wavelength.
Compton explained this by assuming that the photon behaves like a particle when interacting with the electron. The conservation of momentum and energy used for elastic collisions of billiard balls could mathematically explain the experimental observations.
The scattering effect is dependent upon the angle but not the wavelength. The small shift in wavelength would be too difficult to detect with less energetic photons, such as light photons. Particle-wave duality. The photoelectric effect and the Compton effect again point to the duality of the nature of electromagnetic radiation. The models of light as a wave and also as a particle complement each other.
When the photons of electromagnetic radiation are of relatively high energy, the wavelengths are short. Then the photon acts more like a particle than a wave. When the photons of electromagnetic radiation have relatively low energy, the wavelengths are long. Radio waves are an example of less energetic photons that act more like waves than particles.
De Broglie waves. Louis de Broglie — postulated that because photons have both wave and particle characteristics, perhaps particles also have wave characteristics. From the energy of the photon.This material is covered in Gasiorowicz Chapter Scattering of one object from another is perhaps our best way of observing and learning about the microscopic world. Indeed it is the scattering of light from objects and the subsequent detection of the scattered light with our eyes that gives us the best information about the macroscopic world.
We can learn the shapes of objects as well as some color properties simply by observing scattered light. There is a limit to what we can learn with visible light. In Quantum Mechanics we know that we cannot discern details of microscopic systems like atoms that are smaller than the wavelength of the particle we are scattering.
Since the minimum wavelength of visible light is about 0. The physics of atoms, nuclei, subatomic particles, and the fundamental particles and interactions in nature must be studied by scattering particles of higher energy than the photons of visible light.
Scattering is also something that we are familiar with from our every day experience. For example, billiard balls scatter from each other in a predictable way. We can fairly easily calculate how billiard balls would scatter if the collisions were elastic but with some energy loss and the possibility of transfer of energy to spin, the calculation becomes more difficult.
Let us take the macroscopic example of BBs scattering from billiard balls as an example to study. We will motivate some of the terminology used in scattering macroscopically. Assume we fire a BB at a billiard ball. If we miss the BB does not scatter. If we hit, the BB bounces off the ball and goes off in a direction different from the original direction.
Assume our aim is bad and that the BB has a uniform probability distribution over the area around the billiard ball. The area of the projection of the billiard ball into two dimensions is just if is the radius of the billiard ball. Assume the BB is much smaller so that its radius can be neglected for now. We can then say something about the probability for a scattering to occur if we know the area of the projection of the billiard ball and number of BBs per unit area that we shot.
In normal scattering experiments, we have a beam of particles and we know the number of particles per second. We measure the number of scatters per second so we just divide the above equation by the time period to get rates. Clearly there is more information available from scattering than whether a particle scatters or not.
For example, Rutherford discovered that atomic nucleus by seeing that high energy alpha particles sometimes backscatter from a foil containing atoms. The atomic model of the time did not allow this since the positive charge was spread over a large volume.
We measure the probability to scatter into different directions. This will also happen in the case of the BB and the billiard ball.
We can measure the scattering into some small solid angle. The part of the cross section that scatters into that solid angle can be called the differential cross section. The integral over solid angle will give us back the total cross section.
The idea of cross sections and incident fluxes translates well to the quantum mechanics we are using. If the incoming beam is a plane wavethat is a beam of particles of definite momentum or wave number, we can describe it simply in terms of the number or particles per unit area per second, the incident flux.
The scattered particle is also a plane wave going in the direction defined by.