The wavefunction
“The wavefunction therefore, is something which vibrates in time and space. The shadow cast by it on the time axis is energy, and the three shadows cast on the axes of space are the three components of momenta.”
(E. A. Moelwyn-Hughes, ‘Physical Chemistry’, 1940)
Read Engel and Reid
Sections 13.6 and 13.7
Properties of the wavefunction
One of the basic assumptions in quantum mechanics is that the state of a system (e.g. an electron, proton, atom, molecule….) can be described by its wavefunction, in just the same way that the displacement of a macroscopic string can be described by the wavefunction ψ(x,t).. Moreover, the wavefunction contains all the information we need to know about the system! And just as in classical mechanics, not just any old mathematical function will do – it must satisfy certain constraints in order to be physically reasonable.
The mathematical constraints include:
- ψ(x) and dψ(x) / dx must be everywhere finite;
- ψ(x) and dψ(x) / dx must be single-valued everywhere;
- &psi(x) and dψ(x) / dx must be continuous everywhere.
Exercise 3.7
Attempt problem 13.26 on page 309 of E&R
If you are keen, attempt problem 13.28.
Some examples of acceptable and unacceptable wavefunctions are given in Fig. 14.2, p. 312 of E&R. It is always advisable to keep these requirements in mind when seeking solutions to the Schrödinger equation: they are the key to physically correct solutions.
