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lecture_13_07b

# lecture_13_07b - Hydrogen atom in QM What does it mean to...

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05/07/09 Physics 13 - Fall 07 - G.R. Goldstein 1 Hydrogen atom in QM What does it mean to have solutions? How to interpret solutions? - η 2 2 m 2 x 2 + 2 y 2 + 2 z 2 y ( x , y , z ) + U ( x , y , z ) y ( x , y , z ) = Ey ( x , y , z ) Coulomb potential: U ( r ) = - k EM e 2 r with r = x 2 + y 2 + z 2 can change variables to ( r , J , j ) so y ( r , J , j ) = R(r)Q( J )F ( j ) 3-dimensional stationary Schrödinger equation 3 variables 3 Quantum Numbers for bound state solutions n = radial ( r ) quantum no., l = orbital ( θ ) q. no.,

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05/07/09 Physics 13 - Fall 07 - G.R. Goldstein 2 Radial equation ψ ( r , J , j ) = R(r)Q( J )F ( j ) Using factorized form and substituting into the equation simpler radial equation - η 2 2 m d 2 dr 2 + 2 r d dr - l ( l +1) r 2 R ( r ) - k EM e 2 r R ( r ) = ER ( r ) where l is an integer determined by the angular dependence and l= 0,1,2, … will all label distinct solutions
05/07/09 Physics 13 - Fall 07 - G.R. Goldstein 3 Quantum numbers 3 quantum numbers are related n = 1,2,3,… l = 0,1,2, … n -1 n values for each n m l = -l, -l+1, … , +l-1, +l 2l+1 values for each l n = principal q. no. for Coulomb case l = orbital q.no. for any central m l = magnetic q.no. force L = l ( l +1) h Orbital angular momentum L Z = m l h for z defined along some r B or arbitrary

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05/07/09 Physics 13 - Fall 07 - G.R. Goldstein 4 Angular momentum Then for given L (and l ) L Z L = cos( q L ) = m l h l ( l +1) h = m l l ( l +1) quantized directions! example: l =1 L 2 h and m l = - 1,0,1 so cos( q L ) = - 1 2 ,0, +1 2 z +1 0 -1
05/07/09 Physics 13 - Fall 07 - G.R. Goldstein 5 Energy level diagram l = 0 1 2 3 4 5 n= E=0 E =-13.6eV E =-3.4eV E =-1.5eV n=1 n=2 n=3 n=4 n=5 4s 4p 4d 4f (note that vertical scale is distorted) Note the degeneracies.

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lecture_13_07b - Hydrogen atom in QM What does it mean to...

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