342-PS2sol - Solutions to Problem Set 2 Physics 342 by...

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Unformatted text preview: Solutions to Problem Set 2 Physics 342 by: Callum Quigley 1 2 d Delta-Function Potentials (i) We want to preserve the basic property that 1 = Z d 2 xδ 2 ( ~x ) = Z d r d θ rδ 2 ( ~x ) . (1.1) Thus, in polar coordinates the two dimensional delta-function becomes δ 2 ( ~x ) = 1 r δ ( r ) δ ( θ ) . (1.2) Note that 1 2 πr δ ( r ) would not work as a definition, since the delta function should also satisfy f (0 , 0) = Z d r d θ rδ 2 ( ~x ) f ( r,θ ) , (1.3) and this alternative would only work in the special cases where f = f ( r ). Then, recalling that L =- i ~ ∂ θ , we can write the Hamiltonian as H =- ~ 2 2 m ∇ 2 + gδ 2 ( ~x ) =- ~ 2 2 m ∂ rr + 1 r ∂ r + L 2 2 mr 2 + g r δ ( r ) δ ( θ ) (1.4) (ii) If ‘ = 0 then ψ = ψ ( r ), and the Schr¨ odinger equation reads 0 = ( H- E ) ψ ( r ) =- ~ 2 2 m ∂ rr + 1 r ∂ r- 2 mg ~ 2 r δ ( r ) δ ( θ ) + 2 mE ~ 2 ψ ( r ) . (1.5) Note that E =-| E | for bound states. Defining ρ = r p 2 m | E | / ~ , the Schr¨odinger equation can be written ( ρ 2 ∂ ρρ + ρ∂ ρ- ρ 2 ) ψ ( ρ ) = ρAδ ( ρ ) δ ( θ ) ψ ( ρ ) . (1.6) where A = 2 mg/ ~ 2 . For ρ 6 = 0, the RHS vanishes and we recognize what remains as the modified Bessel equation of order 0. 1 The general solution (at least away from ρ = 0), is then given by ψ ( ρ ) = BI ( ρ ) + CK ( ρ ) (1.7) 1 Recall that this is related to the ordinary Bessel equation by ρ 7→ iρ , or equivalently by flipping the sign in the last term of the LHS. 1 for some coefficients B,C ∈ C . I ( ρ ) ,K ( ρ ) are the modified Bessel functions (of order 0), which are related to their standard counterparts via I ( ρ ) = J ( iρ ). However, I is not normalizable since I ( ρ )-→ e ρ √ 2 πρ , as ρ → ∞ = ⇒ Z ∞ | I ( ρ ) | 2 ρ d ρ → ∞ (1.8) So I ( ρ ) does not correspond to a bound state. On the other hand, K ( ρ ) decays at infinity and so is normalizable. In particular, K ( ρ )-→ e- ρ √ 2 πρ , ρ → ∞- ln( ρ/ 2)- γ, ρ → (1.9) γ is the Euler-Mascheroni constant, whose precise value doesn’t matter here. Even thoughis the Euler-Mascheroni constant, whose precise value doesn’t matter here....
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This note was uploaded on 02/27/2010 for the course PHYSICS 342 taught by Professor Sghy during the Spring '10 term at King's College London.

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342-PS2sol - Solutions to Problem Set 2 Physics 342 by...

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