Seemingly simple quantum mechanics/linear algebra problem

[Void123]

Homework Statement

For some reason, I am making a trivial mistake somewhere. I just need to find the eigenvalues and normalized eigenvectors of the following matrix:

H = (1/sqrt(2))*Matrix(Row 1 Row 2 Row 3)

Row 1 = [0 -i 0]
Row 2 = [i 3 3]
Row 3 = [0 3 0]

(Sorry, I don't know the proper latex code for a matrix)

If someone can work this out and just provide the solutions so I can compare, I would really appreciate it.

Homework Equations

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The Attempt at a Solution

I tried.

 

[jdwood983]

Use the normal eigenvalue equation:

<br /> \det(H-\lambda\mathbb{I})=0<br />

Solve for \lambda (you should have 3 values since you have a 3\times3 matrix).

For the eigenvectors,

<br /> \left(\begin{array}{ccc}0&amp;-i&amp;0 \\ i&amp;3&amp;3 \\ 0&amp;3&amp;0\end{array}\right)\left(\begin{array}{c}x\\y\\z\end{array}\right)=\lambda\left(\begin{array}{c}x\\y\\z\end{array}\right)<br />

where you have to do this as many times as you have \lambda values. That should help you a lot

PS: to write matrices in LaTeX, use the command \begin{array} & \end{array}. The following produced the matrix above:

\left(\begin{array}{ccc}0&-i&0 \\ i&3&3 \\ 0&3&0\end{array}\right)

 

[gabbagabbahey]

Row 1 = [0 -i 0]
Row 2 = [i 3 3]
Row 3 = [0 3 0]

(Sorry, I don't know the proper latex code for a matrix)

Click on the \LaTeX image below to see the code that generated it

H=\frac{1}{\sqrt{2}}\begin{pmatrix}0 &amp; -i &amp; 0 \\ i &amp; 3 &amp; 3 \\ 0 &amp; 3 &amp; 0 \end{pmatrix}

If someone can work this out and just provide the solutions so I can compare, I would really appreciate it.

You've been here long enough to know that isn't what we do here.

I tried.

Then show us!

 

[jdwood983]

I forgot the \frac{1}{\sqrt{2}} constant in front of my matrix for the eigenvectors, it should read:


<br /> \frac{1}{\sqrt{2}}<br /> \left(\begin{array}{ccc}0&amp;-i&amp;0 \\ i&amp;3&amp;3 \\ 0&amp;3&amp;0\end{array}\right)\left(\begin{array}{c}x\\y\ \z\end{array}\right)=\lambda\left(\begin{array}{c} x\\y\\z\end{array}\right)<br /> <br />

 

[Void123]

Thanks guys.

For my eigenvalues, I got \lambda = 0, -2, 5

For the last two eigenvalues, when I set up the linear equations to solve I get something of the sort cy = cy for one of them. What does that mean for y? When the authors of my textbook do an example like this and run into the same situation, they postulate that y = 1. But how is that when the variable cancels out on both sides?

Thanks again.

 

[gabbagabbahey]

Thanks guys.

For my eigenvalues, I got \lambda = 0, -2, 5

Try it again without forgetting the factor of 1/\sqrt{2}...

 

[jdwood983]

For the last two eigenvalues, when I set up the linear equations to solve I get something of the sort cy = cy for one of them. What does that mean for y? When the authors of my textbook do an example like this and run into the same situation, they postulate that y = 1. But how is that when the variable cancels out on both sides?

Thanks again.

Doesn't that answer your question? If the authors are allowed to set a canceled variable to be 1, doesn't that mean you can too? Usually you let the canceled variable be 1 for simplicity, but you can make it a billion if you really wanted, it'll just make your later calculations a bit more difficult.

But as Gabba suggested, you should add in that 1/\sqrt{2} factor to find your eigenvalues before you turn in your assignment.