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lecture17 - 5 6 1 F al l 2 0 07 R i g id Ro t or p ag e 1...

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5.61 Fall 2007 Rigid Rotor page Rigid Rotations Consider the rotation of two particles at a fixed distance R from one another: m 2 m 1 ω r + r R 1 2 r 1 m r = m r center of mass (COM) 1 1 2 2 COM r 2 These two particles could be an electron and a proton (in which case we’d be looking at a hydrogen atom) or two nuclei (in which case we’d be looking at a diatomic molecule. Classically, each of these rotating bodies has an angular momentum L = I ω where ω is the angular velocity and I i is the moment of i i inertia I i = mr i 2 for the particle. Note that, in the COM, the two bodies must have the same angular frequency. The classical Hamiltonian for the particles is: L 2 L 2 1 1 1 H = 1 + 2 = m r 2 ω 2 + m r 2 ω 2 = 2 2 2 m r + m r ω 1 1 2 2 ( 1 1 2 2 ) 2 I 2 I 2 2 2 1 2 Instead of thinking of this as two rotating particles, it would be really nice if we could think of it as one effective particle rotating around the origin. We can do this if we define the effective moment of inertia as: m m I = m 1 r 1 2 + m 2 r 2 2 = µ r 0 2 µ = 1 2 m + m 1 2 where, in the second equality, we have noted that this two particle system behaves as a single particle with a reduced mass µ rotating at a distance R from the origin. Thus we have 1 2 L 2 H = I ω = 2 2 I where, in the second equality, we have defined the angular momentum for this effective particle, L = I ω . The problem is now completely reduced to a 1-body problem with mass µ . Similarly, if we have a group of objects that are held in rigid y x z θ r 0 µ φ
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5.61 Fall 2007 Rigid Rotor page positions relative to one another say the atoms in a crystal and we rotate the whole assembly with an angular velocity ω , about a given axis r then by a similar method we can reduce the collective rotation of all of the objects to the rotation of a single “effective” object with a moment of inertia I r . In this manner we can talk about rotations of a molecule (or a book or a pencil) without having to think about the movement of every single electron and quark individually. It is important to realize, however, that even for a classical system, rotations about different axes do not commute with each other . For example, Hence , one gets different answers depending on what order the rotations are Rotate 90˚ about x R x Rotate 90˚ about y R y x y z Rotate 90˚ about y R y Rotate 90˚ about x R x performed in ! Given our experience with quantum mechanics, we might define an operator R ˆ x ( R ˆ y ) that rotates around x (y). Then we would write the above ˆ ˆ ˆ ˆ experiment succinctly as: R R R R . This rather profound result has nothing x y y x to do with quantum mechanics after all there is nothing quantum mechanical about the box drawn above but has everything to do
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