Basis Transformation

Well, I wrote this to remove the mystery out of basis transformations.

We have a matrix $M$ whose elements $M_{ij}$ are given to us in the $\{|\alpha_{i}\rangle\}$ basis and we have to find its elements $\tilde{M}_{ij}$ in the basis $\{|\beta_{i}\rangle\}$ basis. Further, the transformation linking the bases is unitary and given by T, where $|\beta_{i}\rangle = T|\alpha_{i}\rangle$.

Thus,

(1)
\begin{align} \tilde{M}_{ij} = \langle \beta_{i}|M|\beta_{j}\rangle = \langle \alpha_{i}|T^{\dagger}MT|\alpha_{i}\rangle \end{align}

The matrix element (i,j) of T in the $|\{\alpha_{i}\rangle\}$ basis is given by

(2)
\begin{align} T_{ij} = \langle \alpha_{i}|T|\alpha_{j}\rangle = \langle \alpha_{i}|\beta_{j}\rangle \end{align}

so it consists of old base bras and new base kets.

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