3DSHOs - ( 29 ( 29 1 1 2 ~ , , , ~ 1,2 , u F u F -- +-(11)...

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Consider the particle having a mass m which is moving in 3D homogeneous harmonic potential V ( r ): ( 29 2 2 1 2 V r m r ϖ = (1) The radial solution is given by: ( 29 ( 29 ( 29 ( 29 2 2 2 1 2 2 1 0 2 l l l l l m R r R r E m r R r r r + ′′ + + - - = ÷ h (2) For simplicity we let 1 m = = = h , then the equation becomes: ( 29 ( 29 ( 29 ( 29 2 1 2 2 0 l l l l l R r R r E r R r r r + ′′ + + - - = (3) We have been familiar that at r ~0 the wave function acts like: ( 29 ~ l l R r r (4) When r~infinity, the Equation (3) becomes: ( 29 ( 29 ( 29 2 2 0 ~ exp 2 l l l r R r r R r R r ′′ + = ± ÷ (5) Of course the solution exp(r 2 /2) is dropped for physical consideration and we have: ( 29 2 ~ exp 2 l r R r if - → ∞ ÷ (6) So the solution of Equation (3) could be written as: ( 29 ( 29 2 exp 2 l l r R r r u r = - ÷ (7) Substituting (7) into (3): ( 29 ( 29 ( 29 ( 29 ( 29 2 2 1 2 2 3 0 u r l r u r E l u r r ′′ + + - + - + = (8) Define ξ = r 2 , we change (8) as: ( 29 2 2 0 d u du u d d ξ γ α + - - = (9) This is the famous confluent hypergeometric function with parameters: 1 3 2 2 3 2 l E l = + - ÷ = + (10) Its solutions are:
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Unformatted text preview: ( 29 ( 29 1 1 2 ~ , , , ~ 1,2 , u F u F -- +-(11) The second solution in (11) is dropped for physical consideration and we must have: ( 29 3 3 2 ~ , , , , 2 2 l E u F F l +- = + (12) When , we have ( 29 , , ~ F e . So if we substitute such solutions into equation (7) the radial wave function will not satisfy the BC and we must require this infinite series to be truncated into a polynomial, and this requires: 1 3 , 0,1,2,. .. 2 2 l E n n = +-= -= (13) So we have: 3 3 2 , , 0,1,2,. .. 2 2 E n l N n l = + + = + = h h (14)...
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3DSHOs - ( 29 ( 29 1 1 2 ~ , , , ~ 1,2 , u F u F -- +-(11)...

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