# Matrix Representation of an Operator (from Sakurai)

• I
• jaurandt
In summary, the operator equation is:$$\sigma_z = 1 \cdot |\!\uparrow_z \rangle \langle \uparrow_z \!| + 0 \cdot |\!\uparrow_z \rangle \langle \downarrow_z\!| + 0 \cdot |\!\downarrow_z \rangle \langle \uparrow_z\!| + (-1) \cdot |\!\downarrow_z \rangle \langle \downarrow_z\!|.$$
jaurandt
Look, I am sorry for not being able to post any LaTeX. But I am stuck at a place where I feel I should not be stuck.

I can not figure out how to correctly do this. I can't seem to recreate the Pauli matrices with that form using the 3 2-dimensional bases representing x, y, and z spin up/down.

Does anyone have any advice on this?

Just to be clear, you're trying to project the spin matrices in an eigenspin bases right? (say in the z-direction) You only need the spin up and spin down vectors in anyone direction to form a complete basis.
For ##\sigma_z## this is trivial in its eigenbasis:
$$\sigma_z = \sum_s \sum_r |s\rangle\langle s|\sigma_z|r\rangle\langle r|$$
$$\sigma_z = |\uparrow \rangle \langle \uparrow | - | \downarrow \rangle \langle \downarrow |$$

For ##\sigma_x## and ##\sigma_y## you could use the raising and lowering operators to make the same decomposition.

aaroman, jaurandt and vanhees71
HomogenousCow said:
Just to be clear, you're trying to project the spin matrices in an eigenspin bases right? (say in the z-direction) You only need the spin up and spin down vectors in anyone direction to form a complete basis.
For ##\sigma_z## this is trivial in its eigenbasis:
$$\sigma_z = \sum_s \sum_r |s\rangle\langle s|\sigma_z|r\rangle\langle r|$$
$$\sigma_z = |\uparrow \rangle \langle \uparrow | - | \downarrow \rangle \langle \downarrow |$$

For ##\sigma_x## and ##\sigma_y## you could use the raising and lowering operators to make the same decomposition.

Can you please give the same example, but with the ##\sigma_y## operator? What I'm trying to say is that I don't understand how

$$\sigma_z = \sum_s \sum_r |s\rangle\langle s|\sigma_z|r\rangle\langle r|$$

Reveals the entries of the matrix...

The numbers $$\left\langle a'\right|X\left|a''\right\rangle$$
are the entries of the matrix. For the z Pauli matrix we have $$\left\langle \uparrow\right|\sigma_{z}\left|\uparrow\right\rangle =1; \left\langle \downarrow\right|\sigma_{z}\left|\uparrow\right\rangle =0; \left\langle \uparrow\right|\sigma_{z}\left|\downarrow\right\rangle =0; \left\langle \downarrow\right|\sigma_{z}\left|\downarrow\right\rangle =-1$$

jaurandt
You don't even have to deal with matrices if you don't want to, all that's happening is we're rewriting the operator by inserting some identity operators. Since $$I = \sum_s |s\rangle\langle s|,$$ we can just stick one in front of and behind an operator to rewrite it in terms of the operator basis ##|s \rangle \langle r|##,
$$A = I A I = \sum_s \sum_r |s\rangle\langle s| A |r\rangle\langle r| = \sum_s \sum_r A_{sr} |s\rangle \langle r|,$$ where ##A_{sr}## are the matrix elements.

aaroman and jaurandt
andresB said:
The numbers $$\left\langle a'\right|X\left|a''\right\rangle$$
are the entries of the matrix. For the z Pauli matrix we have $$\left\langle \uparrow\right|\sigma_{z}\left|\uparrow\right\rangle =1; \left\langle \downarrow\right|\sigma_{z}\left|\uparrow\right\rangle =0; \left\langle \uparrow\right|\sigma_{z}\left|\downarrow\right\rangle =0; \left\langle \downarrow\right|\sigma_{z}\left|\downarrow\right\rangle =-1$$

So then what happens to the rest of the construct if you just pull out

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

What happened to the summation and what becomes of $$\left|a'\right\rangle\left\langle a''\right|$$

jaurandt said:
So then what happens to the rest of the construct if you just pull out

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

What happened to the summation and what becomes of $$\left|a'\right\rangle\left\langle a''\right|$$
Put everything in the formula and you have the representation of the operator in that basis of vectors. HomogenousCow already showed how the z Pauli operator looks like written in terms of its own set of eigenvectors.

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jaurandt
The object ##|a' \rangle \langle a''|## is an operator. Every entry in a representation matrix is tied to such an operator.

Consider this rewrite of the third Pauli matrix:
$$\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} = 1 \cdot \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix} + 0 \cdot \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} + 0 \cdot \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix} + (-1) \cdot \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix}$$
This is what corresponds to the operator equation
##\sigma_z = 1 \cdot |\!\uparrow_z \rangle \langle \uparrow_z \!| + 0 \cdot |\!\uparrow_z \rangle \langle \downarrow_z\!| + 0 \cdot |\!\downarrow_z \rangle \langle \uparrow_z\!| + (-1) \cdot |\!\downarrow_z \rangle \langle \downarrow_z\!|.##

(Note that the symbol ##\sigma_z## is ofen used to symbolize both the operator and its matrix representation in the z-basis. This is a sloppy but very common notation.)

Last edited:
jaurandt

## 1. What is the Matrix Representation of an Operator?

The Matrix Representation of an Operator is a way of representing a linear transformation between two vector spaces using a matrix. It allows us to perform calculations and make predictions about the behavior of the system.

## 2. Why is the Matrix Representation of an Operator important?

The Matrix Representation of an Operator is important because it allows us to solve problems in quantum mechanics and make predictions about the behavior of a system. It also makes it easier to perform calculations and manipulate equations.

## 3. How is the Matrix Representation of an Operator related to quantum mechanics?

The Matrix Representation of an Operator is closely related to quantum mechanics because it is used to describe the behavior of quantum systems. In quantum mechanics, operators are used to represent physical quantities such as position, momentum, and energy. These operators can then be represented by matrices, which allows us to make predictions about the behavior of the system.

## 4. What are the properties of the Matrix Representation of an Operator?

The Matrix Representation of an Operator has several important properties, including linearity, unitarity, and hermiticity. Linearity means that the transformation described by the matrix is linear, while unitarity means that the matrix is invertible and its inverse is equal to its conjugate transpose. Hermiticity means that the matrix is equal to its conjugate transpose, which is important in quantum mechanics as it ensures that physical observables are real numbers.

## 5. How is the Matrix Representation of an Operator calculated?

The Matrix Representation of an Operator is calculated by finding the matrix elements, which are the coefficients that relate the input vector to the output vector. These matrix elements can be found by applying the operator to a set of basis vectors and then writing down the resulting vectors in terms of the original basis. The resulting matrix will then be the Matrix Representation of the Operator.

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