- #1

ChrisJ

- 70

- 3

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1. Homework Statement

1. Homework Statement

Construct the matrix ##\sigma_{-} = \sigma_{x} - i\sigma_{y}## and show that the states resulting from ##\sigma_{-}## acting on the eigenstates of ##\sigma_{z} ## are also eigenstates of ##\sigma_{z} ## and comment on your result.

## Homework Equations

pauli spin matrices

## The Attempt at a Solution

I need more help with the commenting on the result and the actual physics rather than the maths here,

I constructed the matrix ##\sigma_{-} =

\left( \begin{array}{ccc}

0 & 0 \\

2 & 0 \end{array}

\right)

##

and in previous bit of question found the eigenstates of ##\sigma_{z}## to be ##

\left( \begin{array}{ccc}

1 \\

0 \end{array}\right) ## and ##

\left( \begin{array}{cc}

0 \\

1 \end{array}\right)## respectively.

So therefore ##\sigma_{-}

\left( \begin{array}{ccc}

1 \\

0 \end{array}\right)

=2

\left( \begin{array}{ccc}

0 \\

1 \end{array}\right)##

and also ##\sigma_{-}

\left( \begin{array}{ccc}

0 \\

1 \end{array}\right) =

\left( \begin{array}{ccc}

0 \\

0 \end{array}\right)##

I am pretty sure the math is correct as I ran the math past a few people who agreed but I can't see how/why that that shows they are also eigenstates of ##\sigma_{z}##. I can maybe see it mathematically with the first result, as that is explicitly an eigenstate, but the zero matrix result, I am not sure how in words I can say that it is. And what it means physically. As I said this isn't part of any coursework, just a question from a past exam paper, any help/advice is much appreciated.