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11_Hatom_new

# 43 substitute dimensionless radial position 2 2 1 1 1

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Unformatted text preview: s radial position ℓℓ 2 2 1 ℓℓ 1 1 2 2 2 1 ℓℓ gives 1 2 ℓℓ 1 0 8 ( or divide by ) 2 0 2 1 2 1 2 1 1 2 1 ℓℓ 1 ℓℓ 1 1 1 ℓℓ energy will come in units of (anticipating in fact 2 2 2 ℓℓ multiplied next to last term by one in form of 2/2 which is the Bohr radius 2 2 1 0 8 , solved for Let 0 4 the form of the quantum number from asymptotic solution) 0 1 0 1 differentiating by parts 0 simplify 0 6 9/30/2013 2 2 ℓℓ 1 1 1 As , this becomes 0 / 0 where only was chosen to get / So / / / / / 2 ℓℓ / 2 1 in the exponential 1 ℓℓ 2 / / / / 2 / 1 1 2 1 2 2 2 1 2 2 2 2 2 ℓℓ 2 ℓℓ ℓℓ 1 1 / 2 / 1 2 2 2 1 2 0 diff. by parts 1 1 ℓℓ 0 1 diff. by parts 1 / 0 / divide by 1 is normalizable plug this back into top eq. / 2 is asymptotic solution / 1 1 0 0 collect 0 rearranging 12 2 2 ℓℓ 1 0 ℓℓ 1 0 rearranging 1 Manipulate into recognizable form Let 1 2 /2 2 2 2 /2 , substitute , so 2 /2 /2 ℓℓ 1 1 ℓℓ 1 1 0 multiply by 2 2 2 2 2 2 2 1 2 1 1 0 2 2 2 2 1 /2 ℓℓ 1 ℓℓ 1 /2 0 0 7 9/30/2013 1 2 1 ℓℓ 1 ℓ solutions have the form ℓ ℓ 2 0 1 substitute for 1 ℓℓ 1 ℓ 0 differentiate by parts ℓ 2 ℓ ℓ ℓ 1 ℓ ℓ 1 ℓℓ ℓ 1 0 differentiate by parts ℓ ℓ ℓ ℓ ℓ ℓℓ 1 2 ℓ ℓ 1 ℓ ℓ ℓ 2ℓ 2 2ℓ 2 2ℓ 1 ℓ2 1 ℓ2 ℓ ℓ ℓ 1 1 2ℓ 1 1 ℓ ℓ 1 ℓ 0 ℓ 2ℓ 1 ℓ ℓℓ 2 2ℓ 2 1 ℓℓ 2 2ℓ ℓℓ 1 ℓ 1 ℓℓ 1 1 1 ℓ 0 1 0 collecting terms simplify 0 0 1 0 1 ℓℓ 0 write as ℓ 1 ℓ 1 ℓℓ ℓℓ ℓ divide by ℓ 1 Laguerre Polynomial Eq. 1 0 0 Associated Laguerre Eq. " This is associated Laguerre eq. where 2ℓ 1 and ℓ 1) 1 quadratically integrable solution if is zero or positive integer, so since ℓ 1 0 then ℓ ℓ 1 0,1,2, …, 1 since ℓ can equal zero, the lowest value of Principle quantum number 0 1,2,3 …...
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