Taylorpotential - | x-a | required for the potential at x...

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Exercise —Phys374—Spring 2007 Prof. Ted Jacobson Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374b/ [email protected] Approximate harmonic oscillator and Taylor expansion A particle of mass m moves in a region with potential energy V ( x ) = λ ( x 2 - a 2 ) 2 , where λ > 0. 1. Sketch the graph of the potential. 2. Use the Taylor expansion to show that near x = a the potential energy has the form of a harmonic oscillator potential centered at x = a . Find the effective “spring constant” k of this oscillator. 3. Determine the condition on
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Unformatted text preview: | x-a | required for the potential at x to be well approximated by a harmonic oscillator around x = a . ( Hint : Any higher order terms in the potential must be small compared to the harmonic oscillator term.) 4. What is the frequency of small oscillations near x = a ? 5. Are there any other points x around which the potential is approxi-mately a harmonic oscillator? If so, what are they, and what is the corresponding effective spring constant?...
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This note was uploaded on 12/29/2011 for the course PHYSICS 374 taught by Professor Jacobson during the Fall '10 term at Maryland.

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