I'm trying to understand the maths of QM from Shankar's book - Principles of Quantum Mechanics: On page 21 of that book, there is a general derivation that if we have a relation:(adsbygoogle = window.adsbygoogle || []).push({});

|v'> = Ω|v>

Where Ω is a operator on |v> transfroming it into |v'>, then the matrix entries of the operator can be expressed as:

Ω_{ij}= <i|Ω|j>, where |i> and |j> are basis vectors. What the book says is essentially that the jth column of matrix Ω can be viewed as the image of the transformed jth basis vector expressed in the same basis. More explicitly, it means:

Ω =

<1|Ω|1> <1|Ω|2> ..... <1|Ω|n>

<2|Ω|1> <2|Ω|2> ..... <2|Ω|n>

<3|Ω|1> <3|Ω|2> ..... <2|Ω|1>

..............................................

..............................................

<n|Ω|1> <n|Ω|2> ..... <n|Ω|n>

Where |1>, |2> ... form the basis set. I tried to verify this but it turned out to be not correct (or so i found). For e.g. consider the 2x2 matrix example:

Ω =

2 3

4 5

If I choose basis vectors as |1> = [1 0] and |2> = [0 1] (please read them as column vectors), the I can verify that Ω_{ij}= <i|Ω|j>. However, if i choose a non-starndard but orthonormal basis such as |1> = [1 0] and |2> = [0 -1] (again, please read them as column vectors), then using Ω_{ij}= <i|Ω|j>, I get:

Ω =

2 -3

-4 5

Why is this happening ? Why am I not getting the same Ω back ? What am I missing here? because the proof in the book seems totally logical. Thanks in advance.

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# Matrix Elements as images of basis vectors

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