Harmonic oscillator, average value

Consider a one-dimensional harmonic oscillator with vibrational frequency 5x 1013 sec-1 and mass 1 x 10-23 g. (These are typical values for a diatomic molecule.) Find the average lifetime of the t>=l vibrational state. 

As we are interested in the low energy states close to the bottom of the wells, the amplitude of nuclear motion is small compared to the overall average value of the nuclear displacement. Thus the criterion for smallness comes from the small deviation qp of the displacement from the bottom of the minimum point. Ultimately, we should include nuclear motion as a part of the dynamic problem so that the parameter qp will become a dynamic variable associated with the ground harmonic oscillator state 10) in well p. However, this will not be considered further here. 

Next, first consider the average value of the coordinate of a quantum harmonic oscillator performed on a coherent state,... 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

Evaluate the average (expectation) values of potential energy and kinetic energy for the ground state of the harmonic oscillator. Comment on the relative magnitude of the.se two quantities. 

Thus, the average values of potential and kinetic energies for the harmonic oscillator are equal. This is an instance of the viiial theorem, which states that for a potential energy of the form V x) = const x, the average kinetic and potential energies are related by... 

Pio. 11-4.—The probability distribution function ( io( )]2 for the state n 10 of the harmonic oscillator. Note how closely the function approximates in its average value the probability distribution function for the classical harmonic oscillator with the same total energy, represented by the dashed curve. 

Problem 11-3. Calculate the average values of x, x1, x , and x for a harmonic oscillator in the nth stationary state. Is it true that x1 = ( ) or that = ( )> What conclusions can be drawn from these results concerning the results of a measurement of x ... 

Problem 11-4. Calculate the average values of p, and p for a harmonic oscillator in the nth stationary state and compare with the classical values for the same total energy. From the results of this and of the last problem, compute the average value of the energy W = T + V for the nth stationary state. 

Problem 64-1. Evaluate the momentum wave functions for the harmonic oscillator. Show that the average value of prx for the nth state given by the equation... 

For the ground state of the one-dimensional harmonic oscillator, find the average value of the kinetic energy and of the potential energy verify that T) = V) in this case. 

These can be considered as a perturbation, the unperturbed problem being the harmonic oscillator. Since the average value of QkQiQm is zero in any state v, the first-order perturbation energy due to the cubic terms vanishes, but the second-order energy does not. The first-order energy from the quartic terms involves the mean value of hkbnnQkQiQ,aQn, which vanishes except for two classes of terms QIQ and QjJ. The mean values of terms of the first class arc given by (see Appendix III)... 

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