## PHY456H1F: Quantum Mechanics II. Lecture 14 (Taught by Prof J.E. Sipe). Representation of two state kets and Pauli spin matrices.

Posted by peeterjoot on October 26, 2011

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Representation of kets.

Reading: section 5.1 – section 5.9 and section 26 in [1].

We found the representations of the spin operators

How about kets? For example for

and

So, for example

Kets in

This is a “spinor”

Put

with

Use

So

**Alternatively**

In braces we have a ket in , let’s call it

then

where the direct product is implied.

We can form a ket in as

An operator which acts on alone can be promoted to , which is now an operator that acts on . We are sometimes a little cavalier in notation and leave this off, but we should remember this.

and likewise

and

Suppose we want to rotate a ket, we do this with a full angular momentum operator

(recalling that and commute)

So

## A simple example.

Suppose

where

Then

where

for

so

We are going to concentrate on the unentagled state of 2.26.

\begin{itemize}

\item

How about with

is an eigenket of with eigenvalue .

\item

is an eigenket of with eigenvalue .

\item

What is if it is an eigenket of ?

\end{itemize}

FIXME: F1: standard spherical projection picture, with projected down onto the plane at angle and at an angle from the axis.

The eigenvalues will still be since there is nothing special about the direction.

To find the eigenkets we diagonalize this, and we find representations of the eigenkets are

with eigenvalues and respectively.

So in the abstract notation, tossing the specific representation, we have

# Representation of two state kets

Every ket

for which

can be written in the form 2.34 for some and , neglecting an overall phase factor.

For any ket in , that ket is “spin up” in some direction.

FIXME: show this.

# Pauli spin matrices.

It is useful to write

where

These are the Pauli spin matrices.

## Interesting properties.

\begin{itemize}

\item

latex i < j$}\end{aligned} \hspace{\stretch{1}}(4.46)$

\item

(and cyclic permutations)

\item

\item

where

and

(note )

\item

(and cyclic permutations of the latter).

Can combine these to show that

where and are vectors (or more generally operators that commute with the matrices).

\item

\item

where

\end{itemize}

Note that any complext matrix can be written as

for any four complex numbers

where

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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