Editing Classical limit (section)

==Time-evolution of expectation values==
{{main|Ehrenfest theorem}}
One simple way to compare classical to quantum mechanics is to consider the time-evolution of the ''expected'' position and ''expected'' momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the [[Ehrenfest theorem]]. For a one-dimensional quantum particle moving in a potential <math>V</math>, the Ehrenfest theorem says<ref>{{harvnb|Hall|2013}} Section 3.7.5</ref>
:<math>m\frac{d}{dt}\langle x\rangle = \langle p\rangle;\quad \frac{d}{dt}\langle p\rangle =  -\left\langle V'(X)\right\rangle .</math>
Although the first of these equations is consistent with the classical mechanics, the second is not: If the pair <math>(\langle X\rangle,\langle P\rangle)</math> were to satisfy Newton's second law, the right-hand side of the second equation would have read
:<math>\frac{d}{dt}\langle p\rangle =-V'\left(\left\langle X\right\rangle\right)</math>.
But in most cases,
:<math>\left\langle V'(X)\right\rangle\neq V'(\left\langle X\right\rangle)</math>.
If for example, the potential <math>V</math> is cubic, then <math>V'</math> is quadratic, in which case, we are talking about the distinction between <math>\langle X^2\rangle</math> and <math>\langle X\rangle^2</math>, which differ by <math>(\Delta X)^2</math>.

An exception occurs in case when the classical equations of motion are linear, that is, when <math>V</math> is quadratic and <math>V'</math> is linear. In that special case, <math>V'\left(\left\langle X\right\rangle\right)</math> and <math>\left\langle V'(X)\right\rangle</math> do agree. In particular, for a free particle or a quantum harmonic oscillator, the expected position and expected momentum exactly follows solutions of Newton's equations.

For general systems, the best we can hope for is that the expected position and momentum will ''approximately'' follow the classical trajectories. If the wave function is highly concentrated around a point <math>x_0</math>, then <math>V'\left(\left\langle X\right\rangle\right)</math> and <math>\left\langle V'(X)\right\rangle</math> will be ''almost'' the same, since both will be approximately equal to <math>V'(x_0)</math>. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least ''for as long as'' the wave function remains highly localized in position.<ref>{{harvnb|Hall|2013}} p. 78</ref>

Now, if the initial state is very localized in position, it will be very spread out in momentum, and thus we expect that the wave function will rapidly spread out, and the connection with the classical trajectories will be lost. When the Planck constant is small, however, it is possible to have a state that is well localized in ''both'' position and momentum. The small uncertainty in momentum ensures that the particle ''remains'' well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories for a long time.