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385lec23 - PHYS385 Lecture 23 Angular momentum Lecture 23...

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PHYS385 Lecture 23 - Angular momentum 23 - 1 ©2003 by David Boal, Simon Fraser University. Further copying or resale is strictly prohibited. Lecture 23 - Angular momentum What's important: quantum numbers of spherical harmonics ladder operators Text: Gasiorowicz Chap. 10 In deriving the solutions to the angular parts of the three dimensional Schrödinger equation, we found that two quantum numbers, and m arose naturally from the polynomial solutions of the relevant differential equations. We show now that the quantum numbers have more than just mathematical significance. Quantized angular momentum In lecture 20, we established the form for the angular momentum operator in polar coordinates as L 2 = - h 2 1 sin sin + 1 sin 2 2 2 (1) From lecture 21, we know that the spherical harmonics satisfy the equation - 1 sin sin + 1 sin 2 2 2 Y = l ( l + 1) Y (2) Combining these two equations shows that the result of applying the L 2 operator to Y ( , ) is: L 2 Y ( , ) = l ( l +1) h 2 Y ( , ). (3) In other words, angular momentum is a conserved quantity, from the eigenvalue equation, whose length is given by | L | = | L 2 | 1/2 = ( l ( l +1)) 1/2 h (4)
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