Change of Basis (Bra-Ket Notation)

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:

\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.

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