In this chapter, we introduced completeness as the requirement that an orthonormal basis (left{left|v_{i} ightangle ight}_{i}) satisfies

In this chapter, we introduced completeness as the requirement that an orthonormal basis \(\left\{\left|v_{i}\rightangle\right\}_{i}\) satisfies

\[\begin{equation*}\sum_{i}\left|v_{i}\rightangle\left\langle v_{i}\right|=\mathbb{I} . \tag{3.159}\end{equation*}\]

In this problem, we will study different aspects of this completeness relation.

(a) What if the basis is not orthogonal? Consider the vectors

\[\vec{v}_{1}=\left(\begin{array}{l}1 \tag{3.160}\\0\end{array}\right), \quad \vec{v}_{2}=\left(\begin{array}{c}e^{i \phi_{1}} \sin \theta \\e^{i \phi_{2}} \cos \theta\end{array}\right)\]

What is their outer product sum \(\left|v_{1}\rightangle\left\langle v_{1}|+| v_{2}\rightangle\left\langle v_{2}\right|\) ? For what value of \(\theta\) does it satisfy the completeness relation? Does the result depend on \(\phi_{1}\) or \(\phi_{2}\) ?

(b) In Example 2.1 and Exercise 2.3 of Chap. 2, we studied the first three Legendre polynomials as a complete basis for all quadratic functions on \(x \in[-1,1]\). There we had demonstrated orthonormality:

\[\begin{equation*}\int_{-1}^{1} d x P_{i}(x) P_{j}(x)=\delta_{i j} \tag{3.161}\end{equation*}\]

What about completeness? First, construct the "identity matrix" II formed from the outer product of the first three Legendre polynomials:

\[\begin{equation*}\mathbb{I}=P_{0}(x) P_{0}(y)+P_{1}(x) P_{1}(y)+P_{2}(x) P_{2}(y) . \tag{3.162}\end{equation*}\]

Use the form of the Legendre polynomials presented in Eq. (2.41).

(c) You should find something that does not look like an identity matrix. How can we test it? Provide an interpretation of this outer product identity matrix by integrating against an arbitrary quadratic function:

\[\begin{equation*}\int_{-1}^{1} d x\left(a x^{2}+b x+c\right) \mathbb{I} \tag{3.163}\end{equation*}\] for some constants \(a, b, c\). What should you find? What do you find?