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Section 5.7 Change of Basis (Bra-Ket Notation)
Definition 5.4 . The Completeness Relation.
If we have an orthonormal basis \(\{|v\rangle,|w\rangle,\dots\}\text{,}\) the definition of orthonormality tells us that
\begin{align}
\langle v | v \rangle = 1 \amp = \langle w | w \rangle = \dots ~,\notag\\
\langle v | w \rangle \amp = 0 = \dots ~\text{.}\tag{5.7.1}
\end{align}
Then, we can build projection operators (see
Section 5.6 ) from this orthonormal basis
\begin{equation}
|v \rangle \langle v|,\qquad |w \rangle \langle w|, \qquad \dots ~\text{.}\tag{5.7.2}
\end{equation}
If we add up all these projections, we obtain the identity matrix, that is,
\begin{equation}
I = |v \rangle \langle v| + |w \rangle \langle w| + \dots ~\text{.}\tag{5.7.3}
\end{equation}
Equation
(5.7.3) is called the
completeness relation .
Question 5.5 .
Check that (5.7.3) is indeed the identity matrix by having it act on the arbitrary vector \(\alpha |v \rangle + \beta |w \rangle + \dots ~\text{.}\) Using the Identity Matrix to Change Basis.
If you know a vector \(|\psi\rangle = \alpha |v_i\rangle + \beta |w_i\rangle + \dots ~\) in some initial basis
\begin{equation}
\left\{|v_i\rangle , |w_i\rangle, \dots \right\}\tag{5.7.4}
\end{equation}
and want to find it in some final basis
\begin{equation}
\left\{|v_f\rangle , |w_f\rangle, \dots \right\}\text{,}\tag{5.7.5}
\end{equation}
then just use the strategy
multiply by one in a complicated form , that is, in this case, act on the vector with the identity matrix in the form
(5.7.3) . The general calculation goes like this:
Don’t be intimidated by the sea of algebra. Just FOIL like mad, rearrange terms, and keep track of which terms are vectors and which are scalars.
\begin{align}
|\psi\rangle \amp = I |\psi\rangle \amp\notag\\
\amp = \left(|v_f \rangle \langle v_f| + |w_f \rangle \langle w_f|
+ \dots\right)
\left(\alpha |v_i\rangle + \beta |w_i\rangle + \dots\right)\notag\\
\amp = |v_f\rangle\left(\alpha \langle v_f|v_i\rangle
+ \beta \langle v_f|w_i\rangle +\dots\right)\notag\\
\amp \qquad + |w_f\rangle\left(\alpha \langle w_f|v_i\rangle
+ \beta \langle w_f|w_i\rangle +\dots\right)\notag\\
\amp \qquad +\dots\tag{5.7.6}
\end{align}
Of course, this method is only useful if you already know, or can quickly find, the change of basis elements, e.g. \(\langle v_f|v_i\rangle\text{,}\) between the initial and final basis vectors.