Quantum oscillators harmonic and anharmonic

In accordance with Section 7.3.2, in order to solve the quantum harmonic oscillator problem, we have to write the potential energy expression in the form / x /2, to substitute it into the Schrbdinger equation (7.3.5), find the wavefunction satisfying the standard condition and then find the spectrum of the energy whether it is continuous or discrete. This particular problem can be solved in analytical form. However, while not solving the Schrodinger equations, we will give here the essence of the results. 

Firstly, the equation for the quantum harmonic oscillator shows that the energy can accept only definite values of energy equal to 

Secondly, the number of levels is not limited the quantum number v can accept any value. 

Thirdly, calculations of the quantum mechanical probability of the energy transitions showed that the quantum number can be changed in increments of 1 only (Av = 1) the transitions are allowed only between the adjacent levels. In other words, since the levels are equidistant, only one spectral line can be emitted (or absorbed) regardless of which levels the transition takes place between. The emitted/absorbed quantum energy is in any case, always fid). 

Fourthly, at T = 0 K (v = 0), the oscillations do not come to an end. The so-called zero oscillations are preserved even at absolute zero temperature. The zero point oscillation energy is equal to /iCO/2. 

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