PH 216, Problem set 1, Due 4/6 1. In class, we estimated the ground-state energy of the 3D SHO (see Shankar 12.6.42) using a ‘guess’ of ψ0 ( a ; °r )= R ( a ; r ) Y 00( θ,φ ) , by varying a and minimizing ° ψ ( a ) | H | ψ ( a ) ± . Suppose we try to get the Frst energy E 1 using the trial function ψ 1 = R ( a ; r ) Y 01 , Where the Y enforces orthogonality with our ψ 0. A trial function with zero nodes that has the right asymptotics ( ∝ r +1 as r → 0 and → 0as r →∞ ) is ψ 2 , 1 , 0= N ( r/a 0) exp( − r/ 2 a 0)cos θ. what then is E 1 , and how does this compare with the Frst level of the actual 3D SHO? 2. (Shankar 16.2.8) Consider the ± = 0 radial equation for the three-dimensional Coulomb problem. Since V ( r ) is singular at the turning point r = 0, we cannot use n + 3 4 as in eq. (16.2.42) of Shankar. (a) Will the additive constant be more or less than 3 4 ? (Only heuristic reasoning is required, but you are welcome to derive the result formally if you like.) (b) By analyzing the exact equation near r = 0, it can be shown that the constant equals 1. Using this constant, show that the WKB energy levels agree with the exact results.
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