Sample_Problem_Math_Chpt_D

Sample_Problem_Math_Chpt_D - integrals. This is not true in...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Chem 370 February 20, 2008 Sample Problem from Math Chapter D The H atom 2 2p orbital is a wavefunction with the following form in Cartesian coordinates ( 29 ( 29 ( 29 ( 29 1 2 2 2 2 1 2 2 2 2 2p 2p 1 2 2 2 2 z z x y z z N x y z exp 2 x y z + + ψ = + + - + + r (1) and the following much simpler form in spherical polar coordinates ( 29 r 2 2p 2p z z N re cos . - ψ = θ r (2) Our problem is to choose 2p z N so that ( 29 2p z ψ r is normalized. This may be done by choosing 2p z N as ( 29 ( 29 z 1 2 2p 2p 2p 2p 2p z z z z N N N - = ψ ψ 6 dr r r . (3) Using the Cartesian form for h dr dx dy dz h - - - I = dr (4) and Eqs. (1) and (3) yields an intractable integral form for 2p z N . However using the spherical polar form for h dr 2 2 0 0 0 r dr sin d d h π π = θ θ φ dr (5) and Eqs. (2) and (3) yields a much simpler integral for 2p z N z 1 2 2p N I - = . (6) where } 3 2 1 2 4 r 2 0 0 0 I r e dr sin cos d d h π π - = θ θ θ φ 647 48 6 4 47 4 48 . (7) Note from Eq. (7) the spherical polar form for I reduces to a product of three-one-dimensional
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: integrals. This is not true in Cartesian coordinates. Let us perform the three integrals in Eq. (7). Integral 1. 2 1 I d 2 = = . (8a) Integral 2. 2 2 I sin cos d . = Letting y cos = and thus 2 dy sin d , I = - becomes 1 2 2 1 2 I y dy 3-= = . (8b) Integral 3 We need the result at the bottom of text page 244, which is n ax n 1 n! x e dx a h-+ = . Using this relation we have that 4 r 3 I r e dr 4! 24 h-= = = . (8c) Thus comparing Eq. (7) and (8) gives that 1 2 3 I I I I 32 = = . (9) Thus from Eq. (6) T ....
View Full Document

This note was uploaded on 06/27/2008 for the course CHEM 370 taught by Professor Oldsleepyman during the Fall '08 term at Purdue University.

Page1 / 2

Sample_Problem_Math_Chpt_D - integrals. This is not true in...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online