Where W′ is a variable canonically conjugate to t′ and H′ commutes with t′, and then using H′ as an ordinary Hamiltonian function of a dynamical, system that has W′ for its energy and t′ for its time variable. These matrices were obtained by writing the Hamiltonian equation of the system in the form The main point of this modification is that, whereas in the non-relativity theory one deals with matrices whose elements vary with the time according to the law e iwt, in the relativity theory the elements of the matrices must vary according to the law e iwt′ where t′ = t − ( l 1 x 1 + l 2 x 2 + l 3 x 3)/ c if they are to determine correctly the radiation emitted in the direction specified by the direction cosines ( l 1, l 2, l 3), x 1 x 2 and x 3 being the coordinates of the electron at the time t. This wavelength is called Compton Wavelength, which has good agreement with the experimental results.The problem of the scattering of radiation by a free electron has been treated by the author on the basis of Heisenberg's matrix mechanics, which was first modified to be in agreement with the principle of relativity. Substituting the values of h, m 0, and c we have This implies that at θ = 0, the scattering is absent and the out coming radiation has the same wavelength or frequency as that of the incident radiation. Thus the shift in wavelength or Compton shift purely depends upon the angle of scattering. = Y-component Momentum of the scattered PhotonĮquation 18 represents the shift in wavelength, i.e., Compton shift which is independent of the incident radiation as well as the nature of the scattering substance. Total Y _component of momentum before collision = 0 - > 7 Y – component of the electron at rest = 0 Y – component momentum of the incident photon = 0 II. Total Momentum before collision = Total Momentum after collision We know according to the law of conservation of momentum Total X-component of Momentum after collision X-component Momentum of recoil electron = mv cos ϕ Ii X-component Momentum of recoil electron an be calculated as follows After the collision the photon has energy hf / and the electron has acquired a kinetic energy K. ![]() X-component of the scattered photon can be calculated from the figure. Some of the energy and momentum is transferred to the electron (this is known as the Compton effect), but both energy and momentum are conserved in this elastic collision. Total X-component of Momentum before collision = ɦv/c - (3) X-component momentum of the electron at rest = 0 X-component momentum of the incident photon = ɦv/c X-component of Momentum before collision : Total energy before collision = Total energy after collision We know according to law of conservation of energy Let us find the energy and momentum components before and after collision. The phenomena of compton scattering (As well as the inverse process, or a relativistic electron upscattering a photon) are cornerstones of both particle physics. The scattered photon moves with an energy h ( lesser than hγ), at an angle θ with respect to the original direction. With these assumptions, let us consider a photon energy hγ colliding within electron at rest.ĭuring the collision process, a part of kinetic energy is given to the electron, which in turn increases the kinetic energy of the electron and hence it recoils at an angle of φ as shown. The electron is free and is at rest before collision with the incident photon. The collision occurs between the photon and an electron in the scattering materialĢ. Then by applying the laws if conservation of energy and momentum, the expression for Compton wavelength is derived.ġ. In Compton scattering the collision between a photon and an electron is considered. Thus as a result of Compton scattering, we get (i) unmodified radiations (ii) modified radiations and (iii) a recoil electron. ![]() This effect is called Compton Effect and the shift in wavelength is called Compton shift. Since the electron gains energy, it recoils with velocity’s’. Therefore the scattered photon will have lesser energy or lower frequency or higher wavelength compared to the wavelength of incident photon. When a photon of energy ‘hγ’ collides with an electron of a scatterer at rest the photon gives its energy to the electron. The other component having lower frequency or higher wavelengthĬompared to incident radiation, so called modified radiation One component having the same frequency or wavelength as that of the incident photon, so called unmodified radiation When a beam of monochromatic radiation such as x- rays, gamma rays etc of high frequency is allowed to fall on the scatterer, the beam is scattered into two components
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