Lect31_DipoleMoments - Lecture 31 The Hydrogen Atom 2...

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Lecture 31: The Hydrogen Atom 2: Dipole Moments Phy851 Fall 2009
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The interaction between a hydrogen atom and an electric field is given to leading order by the Electric Dipole approximation: The dipole moment of a pure dipole: Vector quantity Points from - to +. – Magnitude is charge _ distance For Hydrogen atom this gives: Electric Dipole Approximation ( ) + = r r q d r r r ) ( CM E r E D V r r = `Semi-Classical’ Approx: •Electric field is classical •COM motion is classical - + r r r r + R e D r r = -R r
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Dipole Moment Operator The electric dipole moment is an operator in H (R) , which means that its value depends on the state of the relative motion: Choosing the z-axis along the electric field direction gives: Expanding onto energy eigenstates gives: ) ( | | CM E r E Z e V = ( ) ' ' ' ' ' ' ; 1 1 ' 1 0 1 ' 0 ' ' ' ' m n V m n V m n m n E n n n n m m E l l l l l l l l l l ∑∑∑∑ ∑ ∑ = = = = = = = R e D r r = ) ( CM E r E D V r r = ) ( CM E r E R e V r r = ( ) ) ( ' ' ' ; ' ' ' ; CM m n m n m n m n E r E d V l l l l =
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Dipole-Moment Matrix Elements Separate radial and angular Hilbert spaces: SELECTION RULES: Arfken, 3 rd ed., 12.213 The important thing to remember is that Electric Dipole Forbidden Transitions ( ) + +
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