notes008b - n x = Nn Hn x exp Normaliza)on constant = 2 km...

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Ψ n ( x ) = N n H n ( α x ) exp( - α 2 x 2 2 ) Normaliza)on constant Hermite polynomial α = km 2 1 / 4 = m 1 / 2 N n = 1 2 n n ! α 2 1 / 4
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H 0 ( z ) = 1 H 1 ( z ) = 2 z H 2 ( z ) = 4 z 2 - 2 H n +1 ( z ) = 2 zH n ( z ) - 2 nH n - 1 ( z ) All higher‐order polynomials can be generated with the recursion rela,on: For example: which gives Note that this can be done on a point‐by‐point basis; a good way to write a computer program doing harmonic oscillator calcula=ons. H 3 ( z ) = 2 zH 2 ( z ) - 2(2) H 1 ( z ) H 3 ( z ) = 2 z [4 z 2 - 2] - 2(2)(2 z ) = 8 z 3 - 4 z - 8 z = 8 z 3 - 12 z
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Hermite polynomials (19 th century mathema=cs) These are “orthogonal” polynomials, but this doesn’t mean what you think it might mean. In general, orthogonal polynomials of order k (=0,1,2 …) obey equa=ons like: where n and m are the orders of the two polynomials, w(x) is the so‐called weight func=on and the thing on the right hand side is the ubiquitous (but perhaps unfamiliar) Kronecker delta. It equals zero if m and n are unequal, and equals one if m=n.
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