lecture12_umn

# lecture12_umn - Lecture 12 Polar Coordinates Angular...

This preview shows pages 1–6. Sign up to view the full content.

Lecture 12 Polar Coordinates Angular Momentum Eigenfunctions The Initial Spherical Harmonics October 7, 2009 McQuarrie and Simon Chapter 6 pp 191-230

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

View Full Document
Polar coordinates Today we will study angular momentum eigenfunctions . Transform the coordinate system to spherical polar coordinates : angular momentum intrinsically involves rotational motion , so it is useful to work in a coordinate system natural to rotation . Spherical polar coordinate system : a point in space is defined by its distance from the origin, r , its angle from the z axis, θ , and its angle from the x axis when the coordinate vector is projected into the xz plane, φ . We need the relation between these variables and x , y , and z , and their differential operators.
Transformation (12-1) x = r sin " cos # y = r sin sin z = r cos \$ x = sin cos r + cos cos r \$" % sin r sin \$# y = sin sin r + cos sin r + cos r sin z = cos r % sin r L 2 , L z , L + and L - in spherical polar coordinates by substitution.

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

View Full Document
L + and L - in spherical polar coordinates (12-2) The effect of L + operating on the angular momentum eigenfunction having the maximal value of the z component of the angular momentum is to annihilate it: (12-3) Unknown eigenfunctions: Y ( θ , φ ) ll . First subscript : total angular momentum squared. Second subscript : z component of the angular momentum. Since Y is maximal for the latter value, both subscripts are l . L + = h cos " + i sin ( ) # #" + i cos sin #\$ % ( ) * L + Y l , l , ( ) = h cos + i sin ( ) \$ \$" + i cos sin \$# % ( ) * Y l , l , ( ) = 0
Since (cos θ + i sin θ ) is never equal to zero, we can divide both sides by that quantity (12-4) Assume we may separate Y into a part dependent on θ and a part dependent on φ (12-5) " "# + i cos # sin "\$ % ( ) * Y l , l , \$ ( ) = 0 + i cos sin ( ) , \$ ( ) = , \$ ( ) "+# ( ) + i cos sin + # ( ) ",\$ ( ) h

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## lecture12_umn - Lecture 12 Polar Coordinates Angular...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online