How to find eigenvectors/eigenspinors

[mark.laidlaw19]

Homework Statement

Hi all, I have a net spin operator that the problem has asked me to find:$$S=\frac{\hbar}{2} \left(\begin{array}{cc}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{array}\right)$$ and I need to write out the matrix representation with respect to the $S_z$ spinor basis.

Homework Equations

I know that to do this, I need to find the eigenvalues and eigenvectors (or eigenspinors) of this matrix, and then rewrite them in terms of the $S_z$ basis. From the characteristic equation of this matrix, I have found the eigenvalues to be $\pm\frac{\hbar}{2}$

The Attempt at a Solution

I am pretty confident with most of this problem, however, when I try to find the eigenvectors of the spin operator above, I end up with this:

$$\left(\begin{array}{cc}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{array}\right) \left(\begin{array}{cc}a\\b\end{array}\right) = \left(\begin{array}{cc}a\\b\end{array}\right)$$
I can't seem to solve these equations for a or b in order to find the eigenspinors, as once I substitute one equation into the other, the a and the b cancel out, and I'm wondering if there is a simple step that I am missing.

Many thanks in advance

 

[unscientific]

Homework Statement

Hi all, I have a net spin operator that the problem has asked me to find:$$S=\frac{\hbar}{2} \left(\begin{array}{cc}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{array}\right)$$ and I need to write out the matrix representation with respect to the $S_z$ spinor basis.

Homework Equations

I know that to do this, I need to find the eigenvalues and eigenvectors (or eigenspinors) of this matrix, and then rewrite them in terms of the $S_z$ basis. From the characteristic equation of this matrix, I have found the eigenvalues to be $\pm\frac{\hbar}{2}$

The Attempt at a Solution

I am pretty confident with most of this problem, however, when I try to find the eigenvectors of the spin operator above, I end up with this:

$$\left(\begin{array}{cc}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{array}\right) \left(\begin{array}{cc}a\\b\end{array}\right) = \left(\begin{array}{cc}a\\b\end{array}\right)$$
I can't seem to solve these equations for a or b in order to find the eigenspinors, as once I substitute one equation into the other, the a and the b cancel out, and I'm wondering if there is a simple step that I am missing.

Many thanks in advance

From your eigenvector equation you obtain:

$$a cos(\alpha) + b sin(\alpha) = a$$
$$\frac{a}{b} = \frac{sin (\alpha)}{1 - cos(\alpha)}$$

Can you use the double angle identity to simplify this? $\sin2x = 2sinx cos x$ and $sin^2x = \frac{1- cos2x}{2}$

 

[dauto]

Homework Statement

Hi all, I have a net spin operator that the problem has asked me to find:$$S=\frac{\hbar}{2} \left(\begin{array}{cc}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{array}\right)$$ and I need to write out the matrix representation with respect to the $S_z$ spinor basis.

Homework Equations

I know that to do this, I need to find the eigenvalues and eigenvectors (or eigenspinors) of this matrix, and then rewrite them in terms of the $S_z$ basis. From the characteristic equation of this matrix, I have found the eigenvalues to be $\pm\frac{\hbar}{2}$

The Attempt at a Solution

I am pretty confident with most of this problem, however, when I try to find the eigenvectors of the spin operator above, I end up with this:

$$\left(\begin{array}{cc}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{array}\right) \left(\begin{array}{cc}a\\b\end{array}\right) = \left(\begin{array}{cc}a\\b\end{array}\right)$$
I can't seem to solve these equations for a or b in order to find the eigenspinors, as once I substitute one equation into the other, the a and the b cancel out, and I'm wondering if there is a simple step that I am missing.

Many thanks in advance

Off course it does. That's always going to be true for an eigenvalue equation. You already found the eigenvalues and subbed them into the eigenvalue equation. That means the two equations are not independent anymore. If you use both of them you just get an identity such as 1=1 (duh...) What you have to do is use only one of the equations to find a/b like unscientific did and than impose some normalization condition on the resultant eigenvector.

 

[unscientific]

Off course it does. That's always going to be true for an eigenvalue equation. You already found the eigenvalues and subbed them into the eigenvalue equation. That means the two equations are not independent anymore. If you use both of them you just get an identity such as 1=1 (duh...) What you have to do is use only one of the equations to find a/b like unscientific did and than impose some normalization condition on the resultant eigenvector.

Exactly, because both equations are exactly the same, simply rearranged.

 

[paranormal]

for any help you can provide.
Hello,

To find the eigenvectors/eigenspinors of a matrix, you need to solve the characteristic equation and then substitute the eigenvalues back into the matrix to find the corresponding eigenvectors. In this case, the characteristic equation is:

det(S-\lambda I)=0

where S is the given spin operator and I is the identity matrix. Since you have already found the eigenvalues to be \pm\frac{\hbar}{2}, you can now substitute these values back into the matrix and solve for the corresponding eigenvectors.

For the eigenvalue \frac{\hbar}{2}, the matrix equation becomes:

\left(\begin{array}{cc}\cos\alpha-\frac{\hbar}{2}&\sin\alpha\\\sin\alpha&-\cos\alpha-\frac{\hbar}{2}\end{array}\right) \left(\begin{array}{cc}a\\b\end{array}\right) = \left(\begin{array}{cc}0\\0\end{array}\right)

Solving this system of equations will give you the eigenvector for the eigenvalue \frac{\hbar}{2}. Similarly, for the eigenvalue -\frac{\hbar}{2}, the matrix equation will become:

\left(\begin{array}{cc}\cos\alpha+\frac{\hbar}{2}&\sin\alpha\\\sin\alpha&-\cos\alpha+\frac{\hbar}{2}\end{array}\right) \left(\begin{array}{cc}a\\b\end{array}\right) = \left(\begin{array}{cc}0\\0\end{array}\right)

Solving this system of equations will give you the eigenvector for the eigenvalue -\frac{\hbar}{2}. Once you have both eigenvectors, you can rewrite them in terms of the S_z basis by using the relationship S_z=\frac{\hbar}{2}(\left|\uparrow\right\rangle\left\langle\uparrow\right|-\left|\downarrow\right\rangle\left\langle\downarrow\right|) and substituting the corresponding values of a and b.

I hope this helps you in solving the problem. Good luck with your homework!