# Representing spin operators in alternate basis

• pondzo
In summary, the conversation discusses finding the matrix representation of the ##\hat{S}_x,\hat{S}_y,\hat{S}_z## and ##\hat{S}^2## operators in the ##S_x## basis and the difficulties in determining the eigenvectors and eigenvalues in this basis. The conversation also considers the use of similarity transformations and the preference for finding the matrix representation using a different method.

## Homework Statement

I want to find the matrix representation of the ##\hat{S}_x,\hat{S}_y,\hat{S}_z## and ##\hat{S}^2## operators in the ##S_x## basis (is it more correct to say the ##x## basis, ##S_x## basis or the ##\hat{S}_x## basis?).

## Homework Equations

$$\hat{S}^2|s,m_s\rangle=s(s+1)\hbar^2|s,m_s\rangle$$
$$\hat{S}_z|s,m_s\rangle=m_s\hbar|s,m_s\rangle$$
$$\hat{S}_x|s,m_s\rangle=\frac{1}{2}(\hat{S}_++\hat{S}_-)|s,m_s\rangle$$
$$\hat{S}_y|s,m_s\rangle=\frac{1}{2i}(\hat{S}_+-\hat{S}_-)|s,m_s\rangle$$
$$\hat{S}_{\pm}|s,m_s\rangle=\sqrt{s(s+1)-m_s(m_s \pm 1)}\hbar|s,m_s \pm 1\rangle$$

3. The Attempt at a Solution

Finding these in the ##S_z## basis is simple enough. All I need to do is investigate how the basis vectors ##\{|\frac{1}{2},\frac{1}{2}\rangle \equiv |\alpha\rangle, |\frac{1}{2},\frac{-1}{2}\rangle\equiv |\beta\rangle\}## transform under the action of the operator i wish to represent. Then the columns of the matrix become the image of the basis vectors under the operation. However, I'm not sure what the eigenvectors (basis vectors) of ##S_x## are. On top of that, the actions of the operators i have supplied would no longer apply in a different basis, would they? I would prefer to do it via this method rather than using a similarity transform if possible. Thanks.

Would I do it by defining the operators as follows?

##\hat{S}^2|s,m_s\rangle=s(s+1)\hbar^2|s,m_s\rangle##
##\hat{S}_x|s,m_s\rangle=m_s\hbar|s,m_s\rangle##
##\hat{S}_+=\hat{S}_y+i\hat{S}_z##
##\hat{S}_-=\hat{S}_y-i\hat{S}_z##
Which both imply that:
##\hat{S}_y=\frac{1}{2}(\hat{S}_++\hat{S}_-)##
##\hat{S}_z=\frac{1}{2}(\hat{S}_+-\hat{S}_-)##

And with the basis vectors of ##S_y##, ##|\alpha\rangle## and ##|\beta\rangle## defined as before?

pondzo said:
the basis vectors ##\{|\frac{1}{2},\frac{1}{2}\rangle \equiv |\alpha\rangle, |\frac{1}{2},\frac{-1}{2}\rangle\equiv |\beta\rangle\}## t
Which eigenvectors are they, ##S_x##, ##S_y##, or ##S_z##?
pondzo said:
However, I'm not sure what the eigenvectors (basis vectors) of SxSxS_x are.
Find out ##S_x## in matrix form in the basis of the eigenvectors of ##S_z## (this is the most common form found in any literature) then find its eigenvalues as well as its eigenvectors (in the basis of the eigenvectors of ##S_z##).
pondzo said:
On top of that, the actions of the operators i have supplied would no longer apply in a different basis, would they?
The operator equations you wrote above are the action on an eigenvector of ##S_z##. Moreover, the forms of those equation are in the basis-free, operator forms, therefore no matter which basis you chose the form of the above equations will not change. However, if you redefine the notation ##|s,m_s\rangle## to be the eigenvector of ##S_x##, then those equations will indeed look different.
pondzo said:
I would prefer to do it via this method rather than using a similarity transform if possible. Thanks.
It's not clear to me which method you were referring to. To be honest, in matrix form I know no other way to transform a matrix from one basis into another one unless using the usual similarity transformation.

Last edited:
This belongs in the advanced physics forum IMO.

## 1. What are spin operators?

Spin operators are mathematical representations of the intrinsic angular momentum of a quantum particle. They are used to describe the spin state of a particle, which is a quantum property that determines its behavior in a magnetic field.

## 2. What is an alternate basis for representing spin operators?

An alternate basis for representing spin operators is the Pauli spin matrices, which are a set of 2x2 matrices that can be used to describe the spin state of a particle in terms of its projection onto different axes.

## 3. How are spin operators represented in an alternate basis?

To represent spin operators in an alternate basis, the original spin operators are transformed using the corresponding basis transformation matrix. This results in a new set of spin operators that describe the spin state of the particle in the alternate basis.

## 4. Why is representing spin operators in an alternate basis useful?

Representing spin operators in an alternate basis can be useful for simplifying calculations and gaining a better understanding of the spin state of a particle. It can also provide insight into the symmetries and properties of the system being studied.

## 5. Are there any limitations to representing spin operators in an alternate basis?

One limitation is that the alternate basis must be orthogonal and complete in order for the spin operators to be represented accurately. Also, different alternate bases may be more suitable for different systems, so it may not always be clear which basis is the most appropriate to use.