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schmidt_orthog - Harmonic Oscillator wavefunctions are the...

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Unformatted text preview: Harmonic Oscillator wavefunctions are the set of othonormal functions that result when the Schmidt Orthoganalization - . _, 2 . Procedure IS used on the set of functions x"e " ’2 on the Interval from -oo to +00 [See p. 32 in J.M. Anderson, Mathematics for Quantum Chemistry (Benjamin, New York, 1966)] Schmidt Othogonalization Procedure Start with a set of linearly independent functions fi Lat ([30 be f0 normalized. For k21 first determine the next normalization factor k—1 Nk :< fklfk > ‘Zk (lefk >I2 j=0 then use this normalization factor to determine the next othonormal function with Inc—1 (PK 2 ”£1,2[fk _ Z< (lefk > (pg) j=0 You must specify and interval q1,q2 , the integrals become q < a|b >2 " abdt Q1 §imgle Example of Schmidt Orthggonglization: f,= x' i= 0,123,... for x=—1to+{ 1 1 No =< folfo >= Lg): = 2 so (Pa = 75 ' 1 N1 =< f1|f1 > —l<¢olf1 >|2=<x|x>-|<7§|x>|’ 1 1x 2 x3 x21 2 _ _J-1X2dX—[I_17§'dXJ —§[1"[2 2 —§'—0-2’3 —1 (p1 = _JIL=1(f1_ < (palf1 > (p0) = fix- [IL—ffdx)—J—§—] = J2): N2 =< lefz > _I< ‘Polfz >|2"l< (P1l-f2 >I2 =< XZIXZ >--|<1%2|x2 >124< \lgxlx? >l: ...
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