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Find the possible values of energy of a paraticle of mass m located in s spheri |
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Answer» Solution :The schrodinger equation is `grad^(2)Psi+(2m)/ (ħ^(2))(E-U(r ))Psi=0` when `Psi` depends on `t` only `grad^(2)Psi=(1)/(r^(2))(d)/(dr)(r^(2)(dPsi)/(dr))` If we put`Psi=(chi(r ))/(r ),(dPsi)/(dr)=(chi')/(r )-(chi)/(r^(2))` and `grad^(2)Psi=(chi')/(r )`. Thus we GET `(d^(2)chi)/(dr^(2))+(2m)/( ħ^(2))(E-U(r ))chi=0` The solution is `chi=A sin KR,r LT r_(0)` `k^(2)=(2mE)/( ħ^(2))` and `chi=0r gt r_(0)` (for `r lt r_(0)` we have rejected a TERM `B cos kr` as it does not vanish at `r=0`). Continuity of the wavefunction at `r=r_(0)` requires `kr_(0)=npi` Hence `E_(n)=(n^(2)pi^(2) ħ^(2))/(2mr_(0)^(2))`. |
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