Basically, if εi - εj << kT, then quantum mechanical behaviour will be observed: otherwise, the system will display a more-or-less continuous distribution of energy levels and behave more-or-less classically.
Read Engel and Reid
Chapter 13.2 and 13.3
Many problems in physics and chemistry are independent of time (e.g. the electron distribution or the energy of a molecule), and we can actually obtain a wave equation that does not contain t as a variable. In mathematics this procedure is known as separation of variables, and it is performed by assuming the solution ψ(x,t) can be written as the product of two functions, one of which depends only on x, and the other only on t:
ψ(x,t) = ψ(x)φ(t)
Substituting this form of ψ (x,t) into the classical wave equation results in:
Left hand side:
| ( | ∂2ψ | )t = φ(t)( | d2ψ(x) | ) |
| ∂x2 | dx2 |
Right hand side:
| 1 | ( | ∂2ψ | ) x = | 1 | ψ(x) ( | d2φ(t) | ) |
| v2 | ∂t2 | v2 | dt2 |
dividing both sides by ψ(x)φ(t) we obtain
| 1 | d2ψ(x) | = | 1 | 1 | d2&phi(t) | |||
| ψ(x) | dx2 | v2 | &phi(t) | dt2 | ||||
| function of x only | function of t only | |||||||
