# Sakurai Quantum Mechanics - Definition of Matrix Elements [closed]

I don't understand how he represents $$X$$ as a matrix. Any help would be appreciated.

• Are you happy with WP? Commented Aug 8, 2021 at 16:04

Since your question wasn't too specific, I'll try rephrasing Sakurai's explanation.

If $$N$$ is the dimension of the Hilbert space, this means that there are $$N$$ linearly independent basis kets. Any vector in this space has N components. Likewise, any operator in this space is a square matrix with $$N^2$$ components.

The actual numbers that go into each "entry" of a vector (or matrix) depends on which basis you're using. Given a basis $$\left| a_i \right>$$ (with $$i$$ ranging from 1 to $$N$$), the components of an arbitrary vector $$\left| \psi \right>$$ in that basis are

$$c_i = \left< a_i | \psi \right>$$

(notice how there are $$N$$ different coefficients).

Much in the same way, the components (also called "matrix elements") of an arbitrary operator $$A$$ are given by

$$A_{ij} = \left< a_i | A | a_j \right>$$

(notice how there are $$N^2$$ different matrix elements).

If you assume that $$X$$ takes a (ket) vector and gives you another (ket) vector, and that's all you know about it, then a product like

$$\langle a'' | X | a' \rangle$$

makes sense, since it's just the inner product with a bra and a ket. So it is just a complex number.

If $$|a'\rangle, |a'' \rangle$$ are both basis vectors (so they could be one of the vectors $$|0\rangle, |1\rangle, ... |n \rangle$$) then there are $$n^2$$ combinations of the expression above, each one is a complex number. Those $$n^2$$ numbers are called the matrix elements of $$X$$, and those are the numbers that we use to fill in the matrix when we write it as a table of numbers.

You can check it yourself that if you have the usual basis vectors with elements 1,0,0, etc, that an inner product like this gives you the numbers that you are used to seeing as matrix elements. The reason for the definition that Shankar gives is that you can now write the matrix in another basis, just like you can do with vectors.