Notes:Basics of wave functions

The lecture makes a transition from a classical wave function to the idea of a wave function applied to a quantum system, where its modulus squared is an indication of the particle density in the quantum system. It is important to connect the students' previous experience with quantum systems, namely the spin-1/2 system, with the wave function, and to motivate the utility of the wave function.

The latter point is made by describing imaging experiments where electron density can be determined and by describing quantum well systems where electron transitions between the energy eigenstates cause light emission that can be measured.

The students' previous experience with spin-1/2 systems has made them familiar with idea of projection of a quantum state vector onto an eigenstate to calculate the probability of the measurement of the particular eigenvalue. Here, the continuous limit is invoked where the position eigenstate is a delta function of position and a continuous function $\psi \left( x \right)$ is produced. ${\left| {\psi \left( x \right)} \right|^2}$ is associated with the particle density (in this course, always electron density).

The formal identification \[\left \langle x| \psi \right\rangle = \psi \left( x \right)\]

is made, but the “dot equals” notation is also invoked so that one can say “the state is represented by the wave function”: \[\left| \psi \right\rangle \doteq \psi \left( x \right) \]. This is the the same language as in the spins course, “The state is represented by the column vector”:

\[\left| \psi \right\rangle = \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{pmatrix}\]