A one-dimensional simple harmonic oscillator has energy levels given by En = (n+ ) hw, where w is the characteristic (angular) frequency of the oscillator and where the quantum number n can take on non-negative integral values: 0, 1,2... Suppose that such an oscillator is in thermal contact with a heat reservoir at temperature T low enough so that kT/(hw) <1. (a) Find the ratio of the probability of the oscillator being in the first excited state to the probability of its being in the ground state. (b) Assuming that only the ground state and first excited state are appreciably occupied, find the mean energy of the oscillator as a function of the temperature T. (c) For a hydrogen iodide (HI) diatomic molecule, the vibrations have a natural frequency of f = 6.69 x 10 Hz. What then is the spacing between energy levels of this quantum harmonic oscillator, hw? (d) What temperature is "low enough" here? That is. find the temperature where vour
A one-dimensional simple harmonic oscillator has energy levels given by En = (n+ ) hw, where w is the characteristic (angular) frequency of the oscillator and where the quantum number n can take on non-negative integral values: 0, 1,2... Suppose that such an oscillator is in thermal contact with a heat reservoir at temperature T low enough so that kT/(hw) <1. (a) Find the ratio of the probability of the oscillator being in the first excited state to the probability of its being in the ground state. (b) Assuming that only the ground state and first excited state are appreciably occupied, find the mean energy of the oscillator as a function of the temperature T. (c) For a hydrogen iodide (HI) diatomic molecule, the vibrations have a natural frequency of f = 6.69 x 10 Hz. What then is the spacing between energy levels of this quantum harmonic oscillator, hw? (d) What temperature is "low enough" here? That is. find the temperature where vour
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Transcribed Image Text:A one-dimensional simple harmonic oscillator has energy levels given by E, = (n+) hw,
where w is the characteristic (angular) frequency of the oscillator and where the quantum
number n can take on non-negative integral values: 0, 1,2... Suppose that such an oscillator
is in thermal contact with a heat reservoir at temperature T low enough so that kT/(hw) < 1.
(a) Find the ratio of the probability of the oscillator being in the first excited state to the
probability of its being in the ground state.
(b) Assuming that only the ground state and first excited state are appreciably occupied,
find the mean energy of the oscillator as a function of the temperature T.
(c) For a hydrogen iodide (HI) diatomic molecule, the vibrations have a natural frequency
of f = 6.69 x 1013 Hz. What then is the spacing between energy levels of this quantum
harmonic oscillator, liw?
(d) What temperature is "low enough" here? That is, find the temperature where your
result to part (a) is about 0.01 (so that it is 100 times more likely to be in the ground
state than the first excited state)?
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