Consider a one-dimensional harmonic oscillator a. Using the raising and lowering operators introduced in class, calculate (mlx|n),(m|p|n), (m|{x, p}|n) (ma2n), and (m |p2|n) b. The virial theorem in quantum mechanics says (in one dimension) that the expectation of twice the kinetic energy operator, p2/2m, of a particle is equal to the expectation value (r,where V is the potential energy operator. Show this is true for the harmonic oscillator, when the expectation is taken with respect to any energy eigenket. In this special case, it also demonstrates the expected equal division between kinetic and potential energy
Consider a one-dimensional harmonic oscillator a. Using the raising and lowering operators introduced in class, calculate (mlx|n),(m|p|n), (m|{x, p}|n) (ma2n), and (m |p2|n) b. The virial theorem in quantum mechanics says (in one dimension) that the expectation of twice the kinetic energy operator, p2/2m, of a particle is equal to the expectation value (r,where V is the potential energy operator. Show this is true for the harmonic oscillator, when the expectation is taken with respect to any energy eigenket. In this special case, it also demonstrates the expected equal division between kinetic and potential energy
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Transcribed Image Text:Consider a one-dimensional harmonic oscillator
a. Using the raising and lowering operators introduced in class, calculate (mlx|n),(m|p|n), (m|{x, p}|n)
(ma2n), and (m |p2|n)
b. The virial theorem in quantum mechanics says (in one dimension) that the expectation of twice
the kinetic energy operator, p2/2m, of a particle is equal to the expectation value (r,where V
is the potential energy operator. Show this is true for the harmonic oscillator, when the expectation
is taken with respect to any energy eigenket. In this special case, it also demonstrates the expected
equal division between kinetic and potential energy
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