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
Unformatted text preview: Solutions to Problem Set 4 Physics 342 by: Callum Quigley 1 Spherical Potential Well Bound States (i) For bound states we must have E < U . Defining u ( r ) = rψ ( r ), then for the ‘ = 0 states we have u 00 ( r ) = 2 m ~ 2 ( E U ( r )) u ( r ) = 2 m ~ 2 E u ( r ) ≡  k 2 u ( r ) , r < a 2 m ~ 2 ( U E ) u ( r ) ≡ κ 2 u ( r ) , r > a . (1.1) Demanding regularity u (0) = 0, and normalizability u ( r ) → , as r → ∞ fixes the wave function to be of the form ψ ( r ) = A sin( kr ) /r, r < a Be κr /r, r > a (1.2) Next, we impose continuity of ψ ( r ) and ψ ( r ) at r = a . This leads us to the quantization condition κ = r 2 mU ~ 2 k 2 = k cot( ka ) . (1.3) The energy levels of the particle are then given by E = ~ 2 k 2 / 2 m , where the allowed k satisfy the above transcendental equation. (ii) In the limit U → ∞ , the LHS of (1 . 3) blows up. This occurs for the RHS whenever k = nπ a , ∀ n ∈ Z . ⇒ E n = ~ 2 n 2 π 2 2 ma 2 (1.4) exactly as in the 1 d case (with total width a , instead of just the radius). (iii) In the infinite well, allowing ‘ > 0 yields the Schr¨ odinger equation u 00 ( r ) + ‘ ( ‘ + 1) r 2 u ( r ) = 2 m ~ 2 E u ( r ) ≡  k 2 u ( r ) , r < a (1.5) The solutions to this equation are well known and called the spherical Bessel functions of order ‘ (see p. 348). (We omit the Neumann solutions since they are not regular at the origin.) Thus the wavefunctions are ψ E‘m ( ~ r ) = j ‘ ( kr ) Y ‘m ( θ,φ ) with energies E = ~ 2 k 2 / 2 m , where k is fixed by the condition that ψ ( r = a ) = 0. This gives j ‘ ( ka ) = 0 . (1.6) 1 2 The HermannBernoulliLaplaceHamiltonGibbsRunge LenzPauli Vector: The object that was independently discovered 5 different times, and not by Runge or Lenz....
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.
 Spring '10
 Sghy
 Physics, mechanics

Click to edit the document details