NRQM10sol_4

NRQM10sol_4 - Physics 216 Solutions to Problem Set 4 Spring...

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Unformatted text preview: Physics 216 Solutions to Problem Set 4 Spring 2010 1. (a) Consider the Born approximation as the first term of the Born series. Show that: (i) the Born approximation for the forward scattering amplitude [i.e. at θ = 0] is purely real, and therefore (ii) the Born approximation fails to satisfy the optical theorem. Do not assume that the potential is spherically symmetric. However, you may assume that the potential is hermitian. Consider the Born series for the transition operator T , T = V + V 1 E − H + iǫ V + V 1 E − H + iǫ V 1 E − H + iǫ V + ··· , where H ≡ vector p 2 / (2 m ) is the free particle Hamiltonian. The scattering amplitude is given by f ( vector k , vector k ′ ) = − 4 π 2 m planckover2pi1 2 ( vector k ′ | T | vector k ) , and satisfies the optical theorem, σ T = 4 π k Im f ( vector k , vector k ) , (1) where f k ( θ = 0) ≡ f ( vector k , vector k ) is the forward scattering amplitude (where vector k ′ = vector k ). The first Born approximation of the scattering amplitude [denoted by f (1) ( vector k , vector k ′ )] corresponds to taking T ≃ V , in which case, f (1) ( vector k , vector k ′ ) = − 4 π 2 m planckover2pi1 2 ( vector k ′ | V | vector k ) . It follows that f (1) ( vector k , vector k ) = − 4 π 2 m planckover2pi1 2 ( vector k | V | vector k ) is manifestly real , assuming that V is an hermitian operator. Thus, Im f (1) ( vector k , vector k ) = 0, in violation of the optical theorem. This behavior can be understood as follows. The total cross section is given by: σ T = integraldisplay dσ d Ω d Ω = integraldisplay | f k ( θ ) | 2 d Ω . It is convenient to replace V with λV in the Born series as a bookkeeping device, as we can keep track of the various orders of the Born series. Clearly, the Born series for 1 σ T in powers of λ starts off with a term of O ( λ 2 ) since the first Born approximation, f (1) k ( θ ), is of O ( λ ). Thus, the optical theorem given by eq. (1) implies that Im f k ( θ ) is of O ( λ 2 ). Since f k ( θ ) = λf (1) k ( θ ) + λ 2 f (2) k ( θ ) + ··· , it follows that Im f (1) k ( θ ) = 0, which confirms the result obtained above. REMARK: If the potential is spherically symmetric, then we can write V ( vector r ) = V ( r ), in which case as derived in class, f (1) k ( θ ) = − 2 m planckover2pi1 2 integraldisplay ∞ sin( qr ) q rV ( r ) dr , where q = 2 k sin( θ/ 2) . (2) In this case, the first Born approximation to the scattering amplitude, f (1) k ( θ ), is manifestly real for all values of θ (and not just θ = 0 as proven above) if V ( r ) is a real potential. (b) Consider the Yukawa potential: V = − ge − μr r . (3) In class, we computed the (first) Born approximation to the scattering amplitude....
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NRQM10sol_4 - Physics 216 Solutions to Problem Set 4 Spring...

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