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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online