ProblemSet4Hints - s ( r ) r 1 s ( r ) r 2 dr = 4 Z Z 3 / 2...

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Hints for PS4 • The radial wavefunction for the 1s orbital of Hydrogen is: R 1 s ( r ) = 2 a 3 / 2 0 e - r / a 0 (1) When we are dealing with other hydrogen-like atoms (1 electron, many protons), anywhere we have 1 / a 0 , we replace it with Z / a 0 . So the radial wavefunction for the 1s orbital in a hydrogen-like atom is: R 1 s ( r ) = 2 ( Z a 0 ) 3 / 2 e - Zr / a 0 (2) To calculate an expectation value such as, h r i 1 s , we need the full wavefunction, ψ ( r , θ , φ ) = R ( r ) Y ( θ , φ ) For the 1s orbital , the angular part of the wavefunction is very simple, Y ( θ , φ ) = 1 4 π So the complete hydrogen-like wavefunction (angular and radial part) is: ψ 1 s ( r , θ , φ ) = 1 π ( Z a 0 ) 3 / 2 e - Zr / a 0 (3) To calculate h r i 1 s we need to integrate over all values of θ and φ h r i 1 s = Z π 0 sin θ d θ Z 2 π 0 d φ Z 0 ψ 1
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Unformatted text preview: s ( r ) r 1 s ( r ) r 2 dr = 4 Z Z 3 / 2 ( a 3 ) 1 / 2 e-Zr / a r Z 3 / 2 ( a 3 ) 1 / 2 e-Zr / a r 2 dr (4) The operators, S x , S y , S z are dened as follows (see page 210 of Engel): spin up spin down 1 S x = h 2 1 S x 1 = h 2 (5) S y = i h 2 1 S x 1 =-i h 2 (6) S z = h 2 S z 1 =- h 2 1 (7) Chem 120A, Spring 2006, 1...
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This note was uploaded on 01/25/2011 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at University of California, Berkeley.

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