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101-452: Compton scattering (or the Compton effect ) is the quantum theory of high frequency photons scattering following an interaction with a charged particle , usually an electron. Specifically, when the photon hits electrons, it releases loosely bound electrons from the outer valence shells of atoms or molecules. The effect was discovered in 1923 by Arthur Holly Compton while researching

202-480: A {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that

303-502: A laser . Non-linear inverse Compton scattering is an interesting phenomenon for all applications requiring high-energy photons since NICS is capable of producing photons with energy comparable to the charged particle rest energy and higher. As a consequence NICS photons can be used to trigger other phenomena such as pair production, Compton scattering, nuclear reactions , and can be used to probe non-linear quantum effects and non-linear QED . Quantum mechanics Quantum mechanics

404-466: A definite prediction of what the quantum state ψ ( t ) {\displaystyle \psi (t)} will be at any later time. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian . Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example,

505-510: A family of unitary operators parameterized by a variable t {\displaystyle t} . Under the evolution generated by A {\displaystyle A} , any observable B {\displaystyle B} that commutes with A {\displaystyle A} will be conserved. Moreover, if B {\displaystyle B} is conserved by evolution under A {\displaystyle A} , then A {\displaystyle A}

606-417: A high-energy photon (X-ray or gamma ray) during the interaction with a charged particle, such as an electron. It is also called non-linear Compton scattering and multiphoton Compton scattering. It is the non-linear version of inverse Compton scattering in which the conditions for multiphoton absorption by the charged particle are reached due to a very intense electromagnetic field, for example the one produced by

707-471: A loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. As described above, entanglement

808-426: A mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a ( separable ) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector

909-417: A measurement of its position and also at the same time for a measurement of its momentum . Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and

1010-401: A nucleon or alpha particle from the nucleus in a process called photodisintegration . Compton scattering is the most important interaction in the intervening energy region, at photon energies greater than those typical of the photoelectric effect but less than the pair-production threshold. By the early 20th century, research into the interaction of X-rays with matter was well under way. It

1111-427: A photon can be completely absorbed and its energy can eject an electron from its host atom, a process known as the photoelectric effect. High-energy photons of 1.022 MeV and above may bombard the nucleus and cause an electron and a positron to be formed, a process called pair production ; even-higher-energy photons (beyond a threshold energy of at least 1.670 MeV , depending on the nuclei involved), can eject

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1212-433: A potential well, or as an atom with a small ionization energy. In the former perspective, energy of the incident photon is transferred to the recoil particle, but only as kinetic energy. The electron gains no internal energy, respective masses remain the same, the mark of an elastic collision . From this perspective, Compton scattering could be considered elastic because the internal state of the electron does not change during

1313-471: A probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to

1414-465: A second expression for the magnitude of the momentum of the scattered electron, Equating the alternate expressions for this momentum gives which, after evaluating the square and canceling and rearranging terms, further yields Dividing both sides by 2 h f f ′ m e c {\displaystyle 2hff'm_{\text{e}}c} yields Finally, since fλ = f ′ λ ′ = c , It can further be seen that

1515-405: A single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of

1616-551: A single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves e i ( k x − ℏ k 2 2 m t ) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}} , which are eigenstates of

1717-473: Is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of

1818-542: Is incoherent (there is no phase relationship between the scattered photons), the MCP is representative of the bulk properties of the sample and is a probe of the ground state. This means that the MCP is ideal for comparison with theoretical techniques such as density functional theory . The area under the MCP is directly proportional to the spin moment of the system and so, when combined with total moment measurements methods (such as SQUID magnetometry), can be used to isolate both

1919-469: Is a fundamental theory that describes the behavior of nature at and below the scale of atoms . It is the foundation of all quantum physics , which includes quantum chemistry , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, but

2020-478: Is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. There are many mathematically equivalent formulations of quantum mechanics. One of

2121-424: Is a valid joint state that is not separable. States that are not separable are called entangled . If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes

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2222-405: Is conserved under the evolution generated by B {\displaystyle B} . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . The simplest example of a quantum system with a position degree of freedom is a free particle in

2323-1066: Is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The Hamiltonian H {\displaystyle H} is known as the generator of time evolution, since it defines a unitary time-evolution operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} for each value of t {\displaystyle t} . From this relation between U ( t ) {\displaystyle U(t)} and H {\displaystyle H} , it follows that any observable A {\displaystyle A} that commutes with H {\displaystyle H} will be conserved : its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A {\displaystyle A} can generate

2424-868: Is conventional to measure the energies, not the wavelengths, of the scattered photons. For a given incident energy E γ = h c / λ {\displaystyle E_{\gamma }=hc/\lambda } , the outgoing final-state photon energy, E γ ′ {\displaystyle E_{\gamma ^{\prime }}} , is given by E γ ′ = E γ 1 + ( E γ / m e c 2 ) ( 1 − cos ⁡ θ ) . {\displaystyle E_{\gamma ^{\prime }}={\frac {E_{\gamma }}{1+(E_{\gamma }/m_{\text{e}}c^{2})(1-\cos \theta )}}.} A photon γ with wavelength λ collides with an electron e in an atom, which

2525-448: Is given by The operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state ψ ( 0 ) {\displaystyle \psi (0)} – it makes

2626-406: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\displaystyle \langle \psi ,P_{\lambda }\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In

2727-726: Is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . When quantum systems interact,

2828-467: Is known as the Compton wavelength of the electron; it is equal to 2.43 × 10 m . The wavelength shift λ′ − λ is at least zero (for θ = 0° ) and at most twice the Compton wavelength of the electron (for θ = 180° ). Compton found that some X-rays experienced no wavelength shift despite being scattered through large angles; in each of these cases the photon failed to eject an electron. Thus

2929-444: Is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making

3030-628: Is not sufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale. Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately

3131-815: Is part of quantum communication protocols, such as quantum key distribution and superdense coding . Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then

Compton scattering - Misplaced Pages Continue

3232-406: Is possible for the gamma rays to scatter out of the detectors used. Compton suppression is used to detect stray scatter gamma rays to counteract this effect. Magnetic Compton scattering is an extension of the previously mentioned technique which involves the magnetisation of a crystal sample hit with high energy, circularly polarised photons. By measuring the scattered photons' energy and reversing

3333-540: Is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ = 1 {\displaystyle \langle \psi ,\psi \rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent

3434-466: Is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}} . When two different quantum systems are considered together,

3535-419: Is seen in the interaction between an electron and a photon. When a high-energy photon collides with a free electron (more precisely, weakly bound since a free electron cannot participate in inelastic scattering with a photon) and transfers energy, the process is called Compton scattering. Furthermore, when an electron with relativistic energy collides with an infrared or visible photon, the electron gives energy to

3636-508: Is the Planck constant . Before the scattering event, the electron is treated as sufficiently close to being at rest that its total energy consists entirely of the mass-energy equivalence of its (rest) mass m e {\displaystyle m_{\text{e}}} , After scattering, the possibility that the electron might be accelerated to a significant fraction of the speed of light, requires that its total energy be represented using

3737-439: Is the number of spin-unpaired electrons in the system, n ↑ ( p ) {\displaystyle n_{\uparrow }(\mathbf {p} )} and n ↓ ( p ) {\displaystyle n_{\downarrow }(\mathbf {p} )} are the three-dimensional electron momentum distributions for the majority spin and minority spin electrons respectively. Since this scattering process

3838-415: Is the reduced Planck constant . The constant i ℏ {\displaystyle i\hbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation

3939-469: Is then If the state for the first system is the vector ψ A {\displaystyle \psi _{A}} and the state for the second system is ψ B {\displaystyle \psi _{B}} , then the state of the composite system is Not all states in the joint Hilbert space H A B {\displaystyle {\mathcal {H}}_{AB}} can be written in this form, however, because

4040-496: Is treated as being at rest. The collision causes the electron to recoil , and a new photon γ ′ with wavelength λ ′ emerges at angle θ from the photon's incoming path. Let e ′ denote the electron after the collision. Compton allowed for the possibility that the interaction would sometimes accelerate the electron to speeds sufficiently close to the velocity of light as to require the application of Einstein's special relativity theory to properly describe its energy and momentum. At

4141-467: The BKS theory . Compton scattering is commonly described as inelastic scattering . This is because, unlike the more common Thomson scattering that happens at the low-energy limit, the energy in the scattered photon in Compton scattering is less than the energy of the incident photon. As the electron is typically weakly bound to the atom, the scattering can be viewed from either the perspective of an electron in

Compton scattering - Misplaced Pages Continue

4242-505: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ → , ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}} , where λ → {\displaystyle {\vec {\lambda }}}

4343-713: The canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators

4444-423: The photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called

4545-446: The scalar product yields the square of its magnitude, In anticipation of p γ c {\displaystyle p_{\gamma }c} being replaced with h f {\displaystyle hf} , multiply both sides by c 2 {\displaystyle c^{2}} , After replacing the photon momentum terms with h f / c {\displaystyle hf/c} , we get

4646-562: The wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to

4747-431: The Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on

4848-411: The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\displaystyle {\mathcal {H}}_{A}} and H B {\displaystyle {\mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system

4949-432: The Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by

5050-489: The Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses

5151-468: The Schrödinger equation for the particle in a box are or, from Euler's formula , Inelastic scattering#Photons When an electron is the incident particle, the probability of inelastic scattering, depending on the energy of the incident electron, is usually smaller than that of elastic scattering. Thus in the case of gas electron diffraction (GED), reflection high-energy electron diffraction (RHEED), and transmission electron diffraction, because

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5252-573: The Sunyaev–Zel'dovich effect provide a nearly redshift-independent means of detecting galaxy clusters. Some synchrotron radiation facilities scatter laser light off the stored electron beam. This Compton backscattering produces high energy photons in the MeV to GeV range subsequently used for nuclear physics experiments. Non-linear inverse Compton scattering (NICS) is the scattering of multiple low-energy photons, given by an intense electromagnetic field, in

5353-403: The analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. One consequence of

5454-417: The angle φ of the outgoing electron with the direction of the incoming photon is specified by Compton scattering is of prime importance to radiobiology , as it is the most probable interaction of gamma rays and high energy X-rays with atoms in living beings and is applied in radiation therapy . Compton scattering is an important effect in gamma spectroscopy which gives rise to the Compton edge , as it

5555-606: The basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy

5656-404: The collection of probability amplitudes that pertain to another. One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for

5757-451: The collision. The electron's momentum change involves a relativistic change in the energy of the electron, so it is not simply related to the change in energy occurring in classical physics. The change of the magnitude of the momentum of the photon is not just related to the change of its energy; it also involves a change in direction. Solving the conservation of momentum expression for the scattered electron's momentum gives Making use of

5858-467: The conclusion of Compton's 1923 paper, he reported results of experiments confirming the predictions of his scattering formula, thus supporting the assumption that photons carry momentum as well as quantized energy. At the start of his derivation, he had postulated an expression for the momentum of a photon from equating Einstein's already established mass-energy relationship of E = m c 2 {\displaystyle E=mc^{2}} to

5959-626: The continuous case, these formulas give instead the probability density . After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in

6060-431: The dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}}

6161-414: The electric field to accelerate a charged particle to a relativistic speed will cause radiation-pressure recoil and an associated Doppler shift of the scattered light, but the effect would become arbitrarily small at sufficiently low light intensities regardless of wavelength . Thus, if we are to explain low-intensity Compton scattering, light must behave as if it consists of particles. Or the assumption that

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6262-470: The electron can be treated as free is invalid resulting in the effectively infinite electron mass equal to the nuclear mass (see e.g. the comment below on elastic scattering of X-rays being from that effect). Compton's experiment convinced physicists that light can be treated as a stream of particle-like objects (quanta called photons), whose energy is proportional to the light wave's frequency. As shown in Fig. 2,

6363-429: The energy of the X ray photon (≈ 17 keV) was significantly larger than the binding energy of the atomic electron, so the electrons could be treated as being free after scattering. The amount by which the light's wavelength changes is called the Compton shift . Although nucleus Compton scattering exists, Compton scattering usually refers to the interaction involving only the electrons of an atom. The Compton effect

6464-442: The energy of the incident electron is high, the contribution of inelastic electron scattering can be ignored. Deep inelastic scattering of electrons from protons provided the first direct evidence for the existence of quarks . When a photon is the incident particle, there is an inelastic scattering process called Raman scattering . In this scattering process, the incident photon interacts with matter (gas, liquid, and solid) and

6565-710: The frequency of the light. In his paper, Compton derived the mathematical relationship between the shift in wavelength and the scattering angle of the X-rays by assuming that each scattered X-ray photon interacted with only one electron. His paper concludes by reporting on experiments which verified his derived relation: λ ′ − λ = h m e c ( 1 − cos ⁡ θ ) , {\displaystyle \lambda '-\lambda ={\frac {h}{m_{\text{e}}c}}(1-\cos {\theta }),} where The quantity ⁠ h / m e c ⁠

6666-421: The frequency of the photon is shifted towards red or blue. A red shift can be observed when part of the energy of the photon is transferred to the interacting matter, where it adds to its internal energy in a process called Stokes Raman scattering. The blue shift can be observed when internal energy of the matter is transferred to the photon; this process is called anti-Stokes Raman scattering. Inelastic scattering

6767-415: The general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments . In the decades after the formulation of quantum mechanics,

6868-611: The interaction between an electron and a photon results in the electron being given part of the energy (making it recoil), and a photon of the remaining energy being emitted in a different direction from the original, so that the overall momentum of the system is also conserved. If the scattered photon still has enough energy, the process may be repeated. In this scenario, the electron is treated as free or loosely bound. Experimental verification of momentum conservation in individual Compton scattering processes by Bothe and Geiger as well as by Compton and Simon has been important in disproving

6969-462: The interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior

7070-430: The light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves;

7171-406: The lost energy from the photon is transferred to the recoiling particle (such an electron would be called a "Compton Recoil electron"). This implies that if the recoiling particle initially carried more energy than the photon, the reverse would occur. This is known as inverse Compton scattering , in which the scattered photon increases in energy. In Compton's original experiment (see Fig. 1),

7272-998: The magnetisation of the sample, two different Compton profiles are generated (one for spin up momenta and one for spin down momenta). Taking the difference between these two profiles gives the magnetic Compton profile (MCP), given by J mag ( p z ) {\displaystyle J_{\text{mag}}(\mathbf {p} _{z})} – a one-dimensional projection of the electron spin density. J mag ( p z ) = 1 μ ∬ − ∞ ∞ ( n ↑ ( p ) − n ↓ ( p ) ) d p x d p y {\displaystyle J_{\text{mag}}(\mathbf {p} _{z})={\frac {1}{\mu }}\iint _{-\infty }^{\infty }(n_{\uparrow }(\mathbf {p} )-n_{\downarrow }(\mathbf {p} ))d\mathbf {p} _{x}d\mathbf {p} _{y}} where μ {\displaystyle \mu }

7373-437: The magnitude of the shift is related not to the Compton wavelength of the electron, but to the Compton wavelength of the entire atom, which can be upwards of 10000 times smaller. This is known as "coherent" scattering off the entire atom since the atom remains intact, gaining no internal excitation. In Compton's original experiments the wavelength shift given above was the directly measurable observable. In modern experiments it

7474-432: The momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k} . The coefficients of the superposition are ψ ^ ( k , 0 ) {\displaystyle {\hat {\psi }}(k,0)} , which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\displaystyle \psi (x,0)} . It

7575-401: The neutron interacts with the nucleus and the kinetic energy of the system is changed. This often activates the nucleus, putting it into an excited, unstable, short-lived energy state which causes it to quickly emit some kind of radiation to bring it back down to a stable or ground state. Alpha, beta, gamma, and protons may be emitted. Particles scattered in this type of nuclear reaction may cause

7676-421: The nucleus to recoil in the other direction. Inelastic scattering is common in molecular collisions. Any collision which leads to a chemical reaction will be inelastic, but the term inelastic scattering is reserved for those collisions which do not result in reactions. There is a transfer of energy between the translational mode (kinetic energy) and rotational and vibrational modes. If the transferred energy

7777-426: The nucleus, or with just an electron. Pair production and the Compton effect occur at the level of the electron. When a high frequency photon scatters due to an interaction with a charged particle, there is a decrease in the energy of the photon and thus, an increase in its wavelength . This tradeoff between wavelength and energy in response to the collision is the Compton effect. Because of conservation of energy ,

7878-413: The oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude

7979-412: The one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written With the differential operator defined by with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle. The general solutions of

8080-455: The original quantum system ceases to exist as an independent entity (see Measurement in quantum mechanics ). The time evolution of a quantum state is described by the Schrödinger equation: Here H {\displaystyle H} denotes the Hamiltonian , the observable corresponding to the total energy of the system, and ℏ {\displaystyle \hbar }

8181-506: The photon's frame-invariant velocity c . For a photon, its momentum p = h f / c {\displaystyle p=hf/c} , and thus hf can be substituted for pc for all photon momentum terms which arise in course of the derivation below. The derivation which appears in Compton's paper is more terse, but follows the same logic in the same sequence as the following derivation. The conservation of energy E {\displaystyle E} merely equates

8282-404: The photon. This process is called inverse Compton scattering . Neutrons undergo many types of scattering, including both elastic and inelastic scattering. Whether elastic or inelastic scatter occurs is dependent on the speed of the neutron, whether fast or thermal , or somewhere in between. It is also dependent on the nucleus it strikes and its neutron cross section . In inelastic scattering,

8383-428: The position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For

8484-402: The quantized photon energies of h f {\displaystyle hf} , which Einstein had separately postulated. If m c 2 = h f {\displaystyle mc^{2}=hf} , the equivalent photon mass must be h f / c 2 {\displaystyle hf/c^{2}} . The photon's momentum is then simply this effective mass times

8585-400: The question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation ). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that

8686-412: The relativistic energy–momentum relation Substituting these quantities into the expression for the conservation of energy gives This expression can be used to find the magnitude of the momentum of the scattered electron, Note that this magnitude of the momentum gained by the electron (formerly zero) exceeds the energy/ c lost by the photon, Equation (1) relates the various energies associated with

8787-413: The result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables quantum computing and

8888-566: The results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. Accordingly, this article will present

8989-463: The same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {\displaystyle L^{2}(\mathbb {C} )} , while

9090-486: The scattering of X-rays by light elements, and earned him the Nobel Prize for Physics in 1927. The Compton effect significantly deviated from dominating classical theories, using both special relativity and quantum mechanics to explain the interaction between high frequency photons and charged particles. Photons can interact with matter at the atomic level (e.g. photoelectric effect and Rayleigh scattering ), at

9191-430: The scattering process. In the latter perspective, the atom's state is change, constituting an inelastic collision . Whether Compton scattering is considered elastic or inelastic depends on which perspective is being used, as well as the context. Compton scattering is one of four competing processes when photons interact with matter. At energies of a few eV to a few keV, corresponding to visible light through soft X-rays,

9292-445: The spin and orbital contributions to the total moment of a system. The shape of the MCP also yields insight into the origin of the magnetism in the system. Inverse Compton scattering is important in astrophysics . In X-ray astronomy , the accretion disk surrounding a black hole is presumed to produce a thermal spectrum. The lower energy photons produced from this spectrum are scattered to higher energies by relativistic electrons in

9393-402: The sum of energies before and after scattering. Compton postulated that photons carry momentum; thus from the conservation of momentum , the momenta of the particles should be similarly related by in which ( p e {\displaystyle {p_{e}}} ) is omitted on the assumption it is effectively zero. The photon energies are related to the frequencies by where h

9494-625: The superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\displaystyle \psi _{A}} and ϕ A {\displaystyle \phi _{A}} are both possible states for system A {\displaystyle A} , and likewise ψ B {\displaystyle \psi _{B}} and ϕ B {\displaystyle \phi _{B}} are both possible states for system B {\displaystyle B} , then

9595-532: The surrounding corona . This is surmised to cause the power law component in the X-ray spectra (0.2–10 keV) of accreting black holes. The effect is also observed when photons from the cosmic microwave background (CMB) move through the hot gas surrounding a galaxy cluster . The CMB photons are scattered to higher energies by the electrons in this gas, resulting in the Sunyaev–Zel'dovich effect . Observations of

9696-441: The theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space

9797-437: The universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 when predicting the magnetic properties of an electron. A fundamental feature of

9898-526: The value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained

9999-504: The wavelength of the scattered rays was longer (corresponding to lower energy) than the initial wavelength. In 1923, Compton published a paper in the Physical Review that explained the X-ray shift by attributing particle-like momentum to light quanta ( Albert Einstein had proposed light quanta in 1905 in explaining the photo-electric effect, but Compton did not build on Einstein's work). The energy of light quanta depends only on

10100-662: Was observed by Arthur Holly Compton in 1923 at Washington University in St. Louis and further verified by his graduate student Y. H. Woo in the years following. Compton was awarded the 1927 Nobel Prize in Physics for the discovery. The effect is significant because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering , the classical theory of an electromagnetic wave scattered by charged particles, cannot explain shifts in wavelength at low intensity: classically, light of sufficient intensity for

10201-448: Was observed that when X-rays of a known wavelength interact with atoms, the X-rays are scattered through an angle θ {\displaystyle \theta } and emerge at a different wavelength related to θ {\displaystyle \theta } . Although classical electromagnetism predicted that the wavelength of scattered rays should be equal to the initial wavelength, multiple experiments had found that

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