What Conditions Ensure Adiabaticity in a Slowly Varying Harmonic Oscillator?

[wdlang]

suppose that the potential of a harmonic oscilltor is changed slowly

i mean, the frequency of the harmonic oscilltor \omega is varying slowly

my question is, under what conditions, the particle initially in the ground state follow the potential adiabatically?

what conditions \omega(t) sholuld satisfy?

 

[olgranpappy]

well, at the very least, it seems that the time rate of change of the frequency $d\omega / d t$ should be much less than then initial squared frequency. I.e.,
$$\frac{d\omega}{d t}<<k_0/m_0$$