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Angular momentum

1. Definitions

1. Moment: Moment of force (often just moment) = torque he moment may be thought of as a measure of the tendency of the force to cause rotation about an imaginary axis through a point. 2. Momentum: power residing in a moving object Momentum is sometimes referred to as linear momentum to distinguish it from the related subject of angular momentum. Like energy and linear momentum, angular momentum in an isolated system is conserved. The angular momentum (L) is defined as the cross product of the position vector (r) and the linear momentum vector (p) of a paticle of mass m (m) and velocity (v); that is : p = m v, and L = r x p In quantum mechanics angular momentum is quantized. We dfined p as: P = - i ∂/∂r Therefore: L = - i r x ∂/∂r Using Cartesian coordinates, yields: We have: Lx = - i (y ∂/∂z - z ∂/∂y) Ly = - i (z ∂/∂x - x ∂/∂z) Lz = - i (x ∂/∂y - y ∂/∂x)

2. Commutation relations

Different components of the angular momentum do not commute with another while all of The components Lx, Ly, and Lz of L commute with the square L2 of L. But the components do not commute with each other. L2 = Lx2 + Ly2 + Lz2 [L2, Lx] = [L2, Ly] = [L2, Lz] = 0 [Lx, Ly] = iLz [Ly, Lz] = iLx [Lz, Lx] = iLy

3. Expression of L2 in spherical coordinates:

Using the related spherical coordinates: ∂/∂x = sin θ cos φ (∂/∂r) + cos φcos& theta;/r (∂/∂θ) - sin φ/ρ (∂/∂φ) ∂/∂y = sin θ sin φ (∂/∂r) + sin φ cosθ/r (∂/∂θ) + cos φ/r sinθ (∂/∂φ) ∂/∂z = cos θ (∂/∂r) - sinθ/r (∂/∂θ) We have: Lx = i [sin φ ∂/∂θ + (cosφ/tanθ) ∂/∂φ] Ly = i [- cos φ ∂/∂θ + (sinφ/tanθ) ∂/∂φ] Lz = - i ∂/∂φ We obtain: L2 = -2[(1/sinθ) ∂/∂θ (sin θ ∂/∂θ) + (1/sin2θ) ∂2/∂φ2 ] That's what we have assumed at the equation (1.7) in: when we separated the Schrodinger equation, and have written A2 = L2/2

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