# Classical approximation of quantum mechanics

Consider the classical limit of quantum mechanics.

Is it okay to replace every operator, such as position operator and momentum operator, by the expectation value? In essence, is it okay to approximate that the wave-function as behaving as an eigenfunction of every operator, since a measurement (of position, say) is certain to produce that specific value?

• Not quite this simple. See arxiv.org/abs/quant-ph/0506082 – Stéphane Rollandin Apr 12 '18 at 7:14
• moreover, this replacement is not tied to approximating wavefunction as eigenfunctions, v.g. in using $\langle x\rangle$ one does not approximate wavefunctions as eigenfunctions of $\hat x$, as the latter are $\delta$-functions. – ZeroTheHero Apr 12 '18 at 13:59
• @ZeroTheHero While it is not in eigenfunctions, is it okay to replace operators with the expectation values in the classical limit? – user148792 Apr 12 '18 at 15:00
• This is a very very deep question as there are many types of classical limits. The easy answer is "not always". You might want to first look up Ehrenfest's theorem but doing as you suggest amounts to eliminating some types of quantum correlations, which might not be advisable depending on what you do. Certainly doing as you suggest is the most drastic approach. You can get a taste by reading this: arxiv.org/abs/1306.1453 – ZeroTheHero Apr 12 '18 at 15:08
• @ZeroTheHero In the book Binney and Skinner for "The Physics of Quantum Mechanics", he merely replaced the position and momentum operator with the expectation of position and expectation of momentum for a particle in a uniform magnetic field. In this case, is this valid? – user148792 Apr 12 '18 at 16:09