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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 Hermann-Bernoulli-Laplace-Hamilton-Gibbs-Runge- Lenz-Pauli Vector: The object that was independently discovered 5 different times, and not by Runge or Lenz....
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