Quantum Mechanics I
Quantum Mechanics I PHY 851
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This 3 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 851 at Michigan State University taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/207632/phy-851-michigan-state-university in Physics 2 at Michigan State University.
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Date Created: 09/19/15
Lecture 18 February 7 2001 Scattering in the Born approximation Scattering cross sections can be thought of giving the rate for scattering when multiplied by the particle flux v Rain kf WV 1 where V is the volume per individual projectile and v is the velocity of the projectiles Thus one can write the cross section for scattering into any state V 0 2120 ng 2 kf l 2 Ermatang Eigt 3 v7 kf One may express the sum over states an integral and write the matrix element as kfr 81km ltkvkigt writr W UV to obtain an expression for the cross section Where the volume has canceled 1 4 4 V 0 rm WE E1 d rVltrgte quotfquot zgt lt6 Tn d3 1Vltkf7kilr 2 n MW T T e i One may express this a differential cross section do m2 g 4 v 7 T m i dd quotV quote quot k 8 Thus the differential cross section is determined by the Fourier transform of the potential Where the momentum transfered to the target k k f enters the Fourier transform Born Approximation Example As a function of the scattering angle 9 find the differential cross section for particles scattered off a spherically symmetric potential Xr we dlt9 9 First calculate the Fourier transform of the potential which depends only on the magnitude Of k1 kf lq lgd3re T2Quzleiq r 10 Vwi 27r32equot1 quot2 11 Lecture 18 February 7 2001 where q k k k 1 cos92sin29 ct21 cos9 14 2k sin92 Thus the differential cross section is do 27r7ngozslr2 e4amp2 mag2 m 7Z4 16 Note that finding the total cross section would require performing a rather difficult integral over the solid angle 152 27rd cos 9 Coulomb Scattering in the Born Approximation In this case the potential is Vr 7 17 Note the cross sections will be the same whether the potential is attractive of repulsive in the Born approximation Performing the Fourier transform Vq 27r72d7quot fidxeime 18 47H2 q 2falrsinqr 19 4lr cosqr 8C 20 z lt21 2 22 The limit at r gt 00 can be realized by adding an exponential damping term to the potential 22quotquot T gt 0 and repeating the integral to see that one gets the same answer with the evaluation at 00 going to zero The expression for the differential cross section is then 10 M284 7 7 2 12 4h4k4sin4w2 3 Coincidentally this answer is identical to the Rutherford cross section in classical mechanics Note that this cross section is illbehaved at 9 gt 0 meaning that the total cross section is in nite Lecture 18 February 7 2001 Diffraction Often scattering is done within a material for the purpose of learning about the structure of the material rather than understanding the potential In that case the Fourier transform of the potential can be written as vltqgt Z di m q39nr a 24 a Z e q39a d3reiq39kaur a 25 a Mendel 26 where 3q E 2 e q39a This allows us to write the differential cross section ast do m2 2 2 7 E 8 E 27 which means that if the potential is understood one may therefore determine 8q The information from measuring the differential cross section is related to the probability that two scattering centers are separated by a distance a 801 quaa 28 NEW6a 29 621 N d36a swam63 30 mm 31 where q is the momentum transfer and N is the number of scattering centers Here 3 is known as the structure function 3x and gives theyprobability of nding two scattering centers separated by x If the material is a lattice 3q has spikes for values of q which correspond to inphase contributions from different sites Whereas if the material is a liquid the spikes disappear since there is no long range order The structure function 3x then has a hole near x 0 and perhaps a few wiggles before being flat for large x Since the momentum transfer is very high in nuclear or particle experiments one usually neglects the structure of the target in those experiments whereas it comes into play for Xray scattering or lowenergy neutron scattering Higher Order Expansions The T matrix Decays scattering cross sections and propagators next topic are all instances where one invokes Fermi s golden rule In all these instances higher order calculations can be included by replacing the V in Fermi s golden rule with the T matrix Writing the transition matrix element to second order in timedependent perturbation theory 1 t I l ltnUt oogi Vquot dt c EquotEithquott 32 h x 3
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