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HAtom4

# HAtom4 - Physical Chemistry Course Number C362 12.4 Using...

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Physical Chemistry Course Number: C362 12.4 Using the boundary conditions to enforce physical bounds on the quan- tum numbers the structure of the periodic table 1. Now, as i → ∞ , C i +1 c i /i . For large i , F ( ρ ) diverges. Note: for large ρ C i +1 c i /i which yields F ( ρ ) i (1 /i !) ρ i exp ρ which diverges for large values of ρ . This is clearly non-physical since the wavefunction should be finite. Hence, we need to somehow truncate the series for some finite value of the summation index, say truncate at i max , so as to control F ( ρ ) from blowing up. [I hope that you are reminded of our derivation in the harmonic oscillator case at this time.] How do we do we truncate this series? We could do that by requiring that the numerator of Eq. ( 12.47 ) go to zero for i = i max . That is, ( i max + l +1) λ =0 . (12.46) Note what happens if you do this. Using Eq. ( 12.46 ) in Eq. ( 12.47 ), for i = i max : c i max +1 = ( i max + l +1) λ ( i max +1)( i max +2 l +2) c i max =0 (12.47) This implies, c i max +2 = c i max +3 = · · · =0 . Thus the power series is truncated

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