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Theory of Scattering Lecture 4
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Scattering of identical spinless bosons particles:
S.E. For two particle system of equal masses in C.M. Frame (1)
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Classical result for cross-section:
(2) Asymptotic solution for scattering: (3) Wave function corresponding to identical particles in quantum mechanics must be symmetrised
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For spinless bosons wave function must be
symmetric under the interchange of spatial co-ordinates of two particles implies replace In polar coordinates replaced by Note that Eq. (3) is not sysmmetric under the above conditions .
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Symmetric combination has required symmetry
Above equation is also sol of S.E. (1)
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Asymptotic form of sol is
Scattering amplitude: Differential cross-section:
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Last eq. Can be written as
Note the presence of interference term in above. Total cross-section will be For central potential:
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For , quantum differential cross-section
will be For the classical case it will be Quantum cross-section is two times bigger than the classical case.
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Mott formula for coulomb scattering : dashed lines
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Scattering of two identical spin ½ fermions
Two identical spin ½ fermions interacting through central force Singlet state S = Triplet state S = 1 Full wave function (including spatial and spin part) for fermionic system must be Anti symmetric.
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For S = 0 state, Spin part will be antisymmetric
and therefore, spatial part must be symmetric under the interchange of position vectors Thus, symmetrised scattering amplitude will be And differential cross-section will be
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For S = 1 state, Spin part will be symmetric
and therefore, spatial part must be antisymmetric under the interchange of position vectors Thus, symmetrised scattering amplitude will be And differential cross-section will be
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If the particles of incident beams and target are
randomly oriented i.e. unpolarised particles probability of particles in triplet state will be three times in singlet state
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Scattering for spin ½ fermions from coulomb potential
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