Consider an harmonic oscillator with spring constant k and mass m. - Find the cl

Question

Consider an harmonic oscillator with spring constant k and mass m.
- Find the classical frequency of this oscillator as a function of the spring constant and
m.
In quantum mechanics, the spectrum of possible energies for this oscillator is given by:
En = hf (n + 1/2) where n is a natural number: n = 0, 1, 2, 3, ..., h is Planck’s constant
and f is the frequency.
- Find the ground state energy as a function of the spring constant and the mass.
- What is the di?erence of energy between the second excited state and the ground
state?
- Assume now that the oscillator carries a unit of charge, what is the wavelength of
the photon emitted as a function of k and m when the oscillator goes from the ?rst
excited state down to the ground state?
- What is the wavelength of the emitted photon of the previous question if k = 10eV /°A
2
,
where °A denotes Angstrom and eV: electron volt. Is the emitted photon in the visible
spectrum of a human eye?

Explanation / Answer

Force                          F =   -kx m ( d2x / dt2 )   = -kx       ( d2x / dt2 )   = -( k/m) x comparing this simple harmonic equation , we get   ( d2x / dt2 )   = - 2 x           since    2    =  k/m         frequency of oscillator is                     f = (1/2 ) (   k/m  ) _______________________________________________ possible energies of oscillator in quantun mechanics is                En   = h f ( n + (1/2))   for ground state energy n =    0                 Eo    =   (1/2)hf                          =   (h / 4 )(   k/m  ) second exited state energy   n = 2                  E2    =   ( 5/2) hf   energy difference                E = E2   - E0                       =   2hf                       =   ( h / ) (   k/m  ) when an unit positive charge is added the energy levels undergo perturbation effect hence   the energy levels shifted   by   electrostatic potential energy -qkx

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