1.7 Additional Points

This subsection describes a few further issues of importance for this document.

1.7.1 Dirac notation

Physicists like to write inner products such as $\langle f\vert A g\rangle$ in “Dirac notation”:

$$\langle f\vert A\vert g\rangle$$

since this conforms more closely to how you would think of it in linear algebra:

$$\begin{array}{ccc} \langle \vec f\vert & A & \vert\vec g... ...13pt} \mbox{bra} & \mbox{operator} & \mbox{ket} \end{array}$$

The various advanced ideas of linear algebra can be extended to operators in this way, but they will not be needed in this book.

In any case, $\langle f\vert A g\rangle$ and $\langle f\vert A\vert g\rangle$ mean the same thing:

$$\int_{\mbox{\scriptsize all }x} f^*(x)\,(A g(x)) {\,\rm d}x$$

If $A$ is a Hermitian operator, this book will on occasion use the additional bar to indicate that the operator has been brought to the other side to act on $f$ instead of $g$.

1.7.2 Additional independent variables

In many cases, the functions involved in an inner product may depend on more than a single variable $x$. For example, they might depend on the position $(x,y,z)$ in three dimensional space.

The rule to deal with that is to ensure that the inner product integrations are over all independent variables. For example, in three spatial dimensions:

$$\langle f \vert g\rangle = \int_{\mbox{\scriptsize all }... ...ze all }z} f^*(x,y,z) g(x,y,z) {\,\rm d}x {\rm d}y {\rm d}z$$

Note that the time $t$ is a somewhat different variable from the rest, and time is not included in the inner product integrations.