In one dimensional quantum mechanics there is one to one correspondence between eigenvalues and eigenstates, there is an absence of degeneracy. So when a parameter of the Hamiltonian is varied slowly, curves can not cross but they can come quite close and then diverge from each other (Avoided Crossing). Crossings and avoided crossings of levels is commonly observed in the spectra of two or three dimensional systems. Mostly, in one dimensional systems if a parameter of the potential is varied slowly, the eigenvalues increase or decrease monotonically. For particle in an infinitely deep well of width a, {formula} decrease as function of a. For harmonic oscillator potential, E_n=(n+1/2)?w increase linearly as function of the frequency parameter w.
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