CoherentStates - Coherent States of the Simple Harmonic...

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Coherent States of the Simple Harmonic Oscillator Michael Fowler, 10/14/07 What is the Wave Function of a Swinging Pendulum? Consider a macroscopic simple harmonic oscillator, and to keep things simple assume there are no interactions with the rest of the universe. We know how to describe the motion using classical mechanics: for a given initial position and momentum, classical mechanics correctly predicts the future path, as confirmed by experiments with real (admittedly not perfect) systems. But from the Hamiltonian we could also write down Schrödinger’s equation, and from that predict the future behavior of the system. Since we already know the answer from classical mechanics and experiment, quantum mechanics must give us the same result in the limiting case of a large system. It is a worthwhile exercise to see just how this happens. Evidently, we cannot simply follow the classical method of specifying the initial position and momentum—the uncertainty principle won’t allow it. What we can do, though, is to take an initial state in which the position and momentum are specified as precisely as possible . Such a state is called a minimum uncertainty state (the details can be found in my earlier lecture on the Generalized Uncertainty Principle). In fact, the ground state of a simple harmonic oscillator is a minimum uncertainty state. This is not too surprising—it’s just a localized wave packet centered at the origin. The system is as close to rest as possible, having only zero-point motion. What is surprising is that there are excited states of the pendulum in which this ground state wave packet swings backwards and forwards indefinitely, a quantum realization of the classical system, and the wave packet is always one of minimum uncertainty. Recall that this doesn’t happen for a free particle on a line—in that case, an initial minimal uncertainty wave packet spreads out because the different momentum components move at different speeds. But for the oscillator, the potential somehow keeps the wave packet together, a minimum uncertainty wave packet at all times. These remarkable quasi-classical states are called coherent states , and were discovered by Schrodinger himself. They are important in many quasi-classical contexts, including laser radiation. Our task here is to construct and analyze these coherent states and to find how they relate to the usual energy eigenstates of the oscillator. Classical Mechanics of the Simple Harmonic Oscillator To define the notation, let us briefly recap the dynamics of the classical oscillator: the constant energy is 2 2 1 22 p Ek m =+ x or () 2 2 2, / pm x m E k ωω += = m .
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2 The classical motion is most simply described in phase space , a two-dimensional plot in the variables ( . In this space, the point ) , mxp ω ( ) , corresponding to the position and momentum of the oscillator at an instant of time moves as time progresses at constant angular speed in a clockwise direction around the circle of radius 2 mE centered at the origin.
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CoherentStates - Coherent States of the Simple Harmonic...

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