Section 14.7 Expanding a Vector in a Basis
The Method: Finding an Expansion of a Vector in an Orthonormal Basis.
If you have an
orthonormal basis 14.3 for an
inner product space 14.2, i.e. the set
\(\left\{ |n\rangle\right\}\text{,}\) where
\(n\) stands for a set of labels for the basis, you can always express any vector
\(|\psi\rangle\) as a linear combination of the basis vectors,
\begin{equation}
|\psi\rangle = \sum_n c_n|n\rangle\tag{14.7.1}
\end{equation}
It is straightforward to use the inner product to find the values of the unknown expansion coefficients
\(c_n\text{;}\) take the inner product of both sides of
(14.7.1) with an arbitrary basis vector
\(|n'\rangle \text{.}\) (Don’t forget to convert the ket
\(|n'\rangle \) to a bra!)
\begin{align}
\langle n'|\psi\rangle \amp
= \langle n'|\left(\sum_n c_n|n\rangle\right)\tag{14.7.2}\\
\amp = \sum_n c_n
\cancelto{\delta_{n', n}}{\langle n'|n\rangle}\tag{14.7.3}\\
\amp = \sum_n c_n \delta_{n', n}\tag{14.7.4}\\
\amp = c_{n'} \text{,}\tag{14.7.5}
\end{align}
where the orthonormality of the basis was used to simplify the second line,
(14.7.3).
This process is so ubiquitous in physics and other applied settings that many notations exist, which can obscure the simple process of expansion. The physical examples below are intended to help you recognize this process in other notations.
Example 14.9. Velocity Vectors.
For the familiar case of velocity vectors in 3-dimensional space, we have the orthonormal basis \(\{\hat{x}, \hat{y}, \hat{z}\}\text{.}\) In this notation, the velocity vectors are denoted with arrows, the basis vectors are denoted with hats, and the coefficients in the expansion (usually called the components of the vector) are labeled with the same base symbol as the vector.
\begin{equation}
\vec{v} = v_x \hat{x} + v_y \hat{y} + v_z \hat{z}\tag{14.7.6}
\end{equation}
The inner product is just the ordinary dot product and the calculation
(14.7.5) of the
\(x\)-component translated to this notation becomes
\begin{align}
\hat{x}\cdot\vec{v}
\amp = \hat{x}\cdot(v_x \hat{x}
+ v_y \hat{y} + v_z \hat{z})\tag{14.7.7}\\
\amp = v_x \cancelto{1}{\hat{x}\cdot\hat{x}}
+ v_y \cancelto{0}{\hat{x}\cdot\hat{y}}
+ v_z \cancelto{0}{\hat{x}\cdot\hat{z}}\tag{14.7.8}\\
\amp = v_x \text{.}\tag{14.7.9}
\end{align}
The \(y\)- and \(z\)-components are calculated analogously.
Note 14.10.
If you take a snapshot of a wave on a rope that has fixed endpoints, you can decompose the shape \(f(x)\) as a linear combination of waves with specific wave numbers. (A mathematically equivalent problem is for the shape of the initial state of a wavefunction in a quantum infinite square well).
Question 14.11. Try it yourself: Snapshot of a Wave on a Rope with Fixed Endpoints (also isomorphically, Snapshot of the Wave Function for a Quantum Infinite Square Well).
Rewrite the calculation
(14.7.5) for the wave on a rope
\(f(x)\text{.}\) The first hint gives the formula for the orthonormal basis functions, second hint gives the inner product in this notation.
Hint 1.
The orthonormal basis is given by
\begin{equation*}
\sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}
\end{equation*}
for \(n\) any positive integer. Notice that this set includes sine functions where both full periods and half periods exactly fit between the endpoints of the rope.
Hint 2.
The inner product is given by
\begin{equation}
\int_0^L \left(\sqrt{\frac{2}{L}} \sin \frac{n'\pi x}{L}\right)^*
\left(\sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\right)\, dx
=\delta_{n', n}\text{.}\tag{14.7.10}
\end{equation}
Answer.
\begin{equation}
I\doteq \begin{pmatrix}
\delta_{11}\amp \delta_{12}\amp \delta_{13}\\
\delta_{21}\amp \delta_{22}\amp \delta_{23}\\
\delta_{31}\amp \delta_{32}\amp \delta_{33}
\end{pmatrix}\tag{14.7.11}
\end{equation}
