SimpleHarmOsc (1)

SimpleHarmOsc (1) - MichaelFowler , 1 kx2 isan 2...

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The Simple Harmonic Oscillator Michael Fowler 11/13/06 Einstein’s Solution of the Specific Heat Puzzle The simple harmonic oscillator, a nonrelativistic particle in a potential 2 1 2 , kx is an excellent model for a wide range of systems in nature. In fact, not long after Planck’s discovery that the black body radiation spectrum could be explained by assuming energy to be exchanged in quanta, Einstein applied the same principle to the simple harmonic oscillator, thereby solving a long­standing puzzle in solid state physics—the mysterious drop in specific heat of all solids at low temperatures. Classical thermodynamics, a very successful theory in many ways, predicted no such drop—with the standard equipartition of energy, kT in each mode (potential plus kinetic), the specific heat should remain more or less constant as the temperature was lowered (assuming no phase change). To explain the anomalous low temperature behavior, Einstein assumed each atom to be an independent (quantum) simple harmonic oscillator, and, just as for black body radiation, he assumed the oscillators could only absorb or emit energy in quanta . Consequently, at low enough temperatures there is rarely sufficient energy in the ambient thermal excitations to excite the oscillators, and they freeze out, just as blue oscillators do in low temperature black body radiation. Einstein’s picture was later somewhat refined—the basic set of oscillators was taken to be standing sound wave oscillations in the solid rather than individual atoms (making the picture even more like black body radiation in a cavity) but the main conclusion—the drop off in specific heat at low temperatures—was not affected. The Classical Simple Harmonic Oscillator The classical equation of motion for a one­dimensional simple harmonic oscillator with a particle of mass m attached to a spring having spring constant k is 2 2 . d x m kx dt = - The solution is 0 sin( ), , k x x t m w d = + = and the momentum p = mv has time dependence 0 cos( ). p mx t = + The total energy 2 2 2 2 (1/ 2 )( ) m p m x E + = is clearly constant in time.
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2 It is often useful to picture the time­development of a system in phase space , in this case a two­dimensional plot with position on the x ­axis, momentum on the y ­axis. Actually, to have ( ) , x y coordinates with the same dimensions, we use ( ) , . m x p w It is evident from the above expression for the total energy that in these variables the point representing the system in phase space moves clockwise around a circle of radius 2 mE centered at the origin. Note that in the classical problem we could choose any point ( ) , , m x p place the system there and it would then move in a circle about the origin. In the quantum problem, on the other hand, we cannot specify the initial coordinates ( ) , m x p precisely, because of the uncertainly principle. The best we can do is to place the system initially in a small cell in phase space, of size / 2 x p h . In fact, we shall find that in quantum mechanics phase space is always divided into cells of essentially this size for each pair of variables.
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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SimpleHarmOsc (1) - MichaelFowler , 1 kx2 isan 2...

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