Lecture12 - CHEM 356, Lecture 12, Fall 2009 1 The Harmonic...

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Unformatted text preview: CHEM 356, Lecture 12, Fall 2009 1 The Harmonic Oscillator.II. The usual forms for the Hermite polynomials (the ‘well–behaved’ solutions of the Hermite DE) obey what are referred to as recurrence relations , the two most important of which are: +1 ( ) = 2 ( ) − 2 − 1 ( ) , and ′ ( ) = 2 − 1 ( ) . Let’s take a look at some of the Hermite polynomials, starting from the definition of ( ) as a power series in , which is written explicitly as ( ) = ∑ =0 . For = 0, we have ( ) = = , with arbitrary. Let us choose = 1: then ( ) = 1 [ note that we shall also define − ( ) ≡ 0 ]. We can use the first Hermite recurrence relation to generate 1 ( ) , 2 ( ) , ⋅⋅⋅ , as follows: Series solution with: 1 ( ) = 2 (1) − 0 = 2 , 1 = 2 2 ( ) = 2 (2 ) − 2(1)(1) = 4 2 − 2 = − 2, 2 = 4 3 ( ) = 2 (4 2 − 2) − 2(2)(2 ) = 8 3 − 12 , 1 = − 12, 3 = 8 4 ( ) = ⋅⋅⋅ = 16 4 − 48 2 + 12, 4 = 16, 2 = − 48, = 12 and so forth. The Hermite polynomials have the orthonormalization ∫ ∞ −∞ e − 2 ( ) ′ ( ) = √ 2 ! ′ . CHEM 356, Lecture 12, Fall 2009 2 iii) Normalization of SHO Wavefunctions. We shall require (for use in Postulate 5) that the wavefunction ( ) satisfy the normalization condition ⟨ ( ) ∣ ′ ( ) ⟩ = ′ . We have seen that the general solution to the Schr¨ odinger equation for the SHO is ( ) = ( )e − 2 / 2 , with its normalization as yet unspecified. To include the normalization, we shall therefore write ( ) as ( ) = e − 2 / 2 ( √ )...
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This note was uploaded on 02/28/2011 for the course CHEM 356 taught by Professor Prof.iaskjd during the Fall '09 term at Waterloo.

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Lecture12 - CHEM 356, Lecture 12, Fall 2009 1 The Harmonic...

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