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A particle of mass mis located in a spherically symmetrical potential well U(r )=0 for r lt r_(0) and U(r )=U_(0) for r gt r_(0). (a) By means of the substitution Psi(r )= xi(r )//r find the equation defining the proper values of energy E of the particle for E lt U_(0), when its motion is described by a wave function Psi(r ) depending only on r. Reduce that equation to the form sinkr_(0)+-kr_(0)sqrt( ħ^(2)//2mr_(0)^(2)U_(0)), where k=sqrt(2mE//) ħ (b) Calculate the value of the quantity r_(0)^(2)U_(0) at which the first level appears. |
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Answer» Solution :If we put `Psi=(CHI(r ))/(r )` the equation for `chi(r )` has the from `chi''k^(2)chi=0,0 LE r lt r_(0)` and `chi''-alpha^(2)chi=0r_(0) lt r lt oo` where `k^(2)=(2mE)/( ħ^(2)),alpha^(2)=(2M(U_(0)-E))/( ħ^(2))` The boundary condition is `{:(chi(0),,,=,0),(and chi,chi',are conti n uous at,r=,r_(0)):}}` These are exactly same as in the one dimensional problem in problem (6.85) We therefore omit further details. |
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