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Lecture22Notes - Chem 120A Spring 2006 READING Angular...

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Chem 120A Angular Momentum 03/13/06 Spring 2006 Lecture 22 READING: Engel (CS): Chapters 7.4-7.5 and 9; Appendices A.4.2 and A.6 Additional optional reading: McQuarrie and Simon, Physical Chemistry , pg. 191-206 Spherical polar coordinates and angular momentum Last week we examined the internal Hamiltonian for the hydrogen atom. In atomic units, the Hamiltonian is ˆ H = - 2 r 2 - 1 r , (1) where r is the relative coordinate which defines the distance between the nucleus and the electron. The kinetic energy term now depends on 2 r , which is just the gradient. In Cartesian coordinates this is: 2 r = 2 ∂x 2 + 2 ∂y 2 + 2 2 z . (2) The form of the interaction potential, however, makes it inconvenient to work in Cartesian coor- dinates since r = p x 2 + y 2 + z 2 , which makes the differential equation given by the Schr¨ odinger equation inseparable. However, if we consider the Hamiltonian in spherical coordinates, we are able to separate the differential equation into a radial part and an angular part. We can redefine each of our Cartesian coordinates in spherical coordinates (Figure 1): x = r cos θ sin φ 0 θ π y = r sin θ sin φ 0 φ 2 π z = r cos θ 0 r ≤ ∞ . and the volume element in these coordinates is Z d 3 r Z -∞ dx Z -∞ dy Z -∞ dz = Z 0 r 2 dr Z π 0 sin θdθ Z 2 π 0 dφ. (3) Using these definitions and Eq. 2, one derives the gradient in spherical polar coordinates, 2 r = 2 ∂r 2 + 2 r ∂r + 1 r 2 2 ∂θ 2 + cot θ ∂θ + 1 sin 2 θ 2 ∂φ 2 . (4) Here, the first two terms are the radial part of the kinetic energy, while the last three terms are the angular part. This is just the mathematical definition of 2 r in spherical coordinates. We can define the angular part as the angular momentum operator, ˆ L , which we will return to later. Chem 120A, Spring 2006, Lecture 22 1
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Figure 1: Spherical polar coordinates. Dirac notation and angular momentum Remember from classical physics that the angular momentum is ~ L = ~ r × ~ p. We can similarly define the angular momentum operator in quantum mechanics as ˆ L = (ˆ y ˆ p z - ˆ z ˆ p y ) ~ i + (ˆ z ˆ p x - ˆ x ˆ p z ) ~ j + (ˆ x ˆ p y - ˆ y ˆ p x ) ~ k (5) There are also operators for each component of the angular momentum, as defined by Eq. 5, ˆ L x = ˆ y ˆ p z - ˆ z ˆ p y ˆ L y = ˆ z ˆ p x - ˆ x ˆ p z ˆ L z = ˆ x ˆ p y - ˆ y ˆ p x , (6) and ˆ L 2 = ˆ L 2 x + ˆ L 2 y + ˆ L 2
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