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# ph135_hw3 - Physics 135c Homework 3 1 Construct an s-wave...

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Physics 135c Homework 3 1.) Construct an s-wave scattering length operator ˆ a for the n p system such that the matrix element ( SM | ˆ a | SM ) for two-nucleon spin states gives a s for spin-singlet states (i.e. S = 0) and a t for spin-triplet states (i.e. S = 1) - this operator should be equivalent to the diagonal matrix we discussed in class. This operator will contain the operator ˆ s n · ˆ s p with e.g. ˆ s n = 1 2 ˆ σ n as well as the values a s and a t . For unpolarized n p scattering this operator should yield a total cross section of 4 π ( a 2 ) = π (3 a 2 t + a 2 s ) Now consider very low energy neutron scattering from molecular hydrogen H 2 , such that the two protons in the molecule interact coherently with the neutron (its wavelength is larger than the spacing between the protons). Thus you can replace the ˆ s p in your operator with ˆ S H 2 . Evaluate your new operator ˆ a H 2 for unpolarized neutron scattering from H 2 in the “para-hydrogen” state (i.e. S H 2

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