Appendix C - Vectors and Linear Algebra

Introduction

This appendix introduces some aspects of linear algebra and complex algebra which will be helpful for the course. In addition, Dirac notation is introduced and explained.

Vectors

Real Vectors

Here we review some facts about real vectors before discussing the complex analogues used in quantum mechanics.

The simple definition of a vector - an object which has magnitude and direction - is helpful to keep in mind even when dealing with complex and/or abstract vectors as we will here. In three dimensional space, a vector is often written as

${\vec {v}}=v_{x}{\hat {x}}+v_{y}{\hat {y}}+v_{z}{\hat {z}},$

where the hat (${\hat {\cdot }}\,\!$) denotes a unit vector. The components $v_{i}\,\!$, $i=x,y,z\,\!$ are just numbers. These unit vectors are also called basis vectors because any vector (in real three-dimensional space) can be written in terms of them. In some sense they are basic components of any vector. Other basis vectors could be used though. Some other common choices are those the are used for spherical coordinates and those used for cylindrical coordinates. When dealing with more abstract and/or complex vectors, it is often helpful to ask what one would do for an ordinary three-dimensional vector. For example, properties of unit vectors, dot products, etc. in three-dimensions are similar to the analogous constructions in more dimensions.

The inner product, or dot product for two real three-dimensional vectors

${\vec {v}}=v_{x}{\hat {x}}+v_{y}{\hat {y}}+v_{z}{\hat {z}},\;\;{\vec {w}}=w_{x}{\hat {x}}+w_{y}{\hat {y}}+w_{z}{\hat {z}},$

can be computed as follows

${\vec {v}}\cdot {\vec {w}}=v_{x}w_{x}+v_{y}w_{y}+v_{z}w_{z}.$

For the inner product of ${\vec {v}}\,\!$ with itself, we get the square of the magnitude of ${\vec {v}}\,\!$ denoted $|{\vec {v}}|^{2}\,\!$,

$|{\vec {v}}|^{2}={\vec {v}}\cdot {\vec {v}}=v_{x}v_{x}+v_{y}v_{y}+v_{z}v_{z}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}.$

If we want a unit vector in the direction of ${\vec {v}}\,\!$, then we divide by its magnitude to get a unit vector,

${\hat {v}}={\frac {\vec {v}}{|{\vec {v}}|}}.$

Now, of course, ${\hat {v}}\cdot {\hat {v}}=1\,\!$ as can easily be checked.

There are several ways to represent a vector. The ones we will use most often are column and row vector notations. So, for example, we could write the vector above as

${\vec {v}}=\left({\begin{array}{c}v_{x}\\v_{y}\\v_{z}\end{array}}\right).$

In this case, our unit vectors are represented by the following

${\hat {x}}=\left({\begin{array}{c}1\\0\\0\end{array}}\right),\;\;{\hat {y}}=\left({\begin{array}{c}0\\1\\0\end{array}}\right),\;\;{\hat {z}}=\left({\begin{array}{c}0\\0\\1\end{array}}\right).\,\!$

We next turn to the subject of complex vectors along with the relevant notation. We will see how to compute the inner product later, but some other defintions will be required.

Complex Vectors

For complex vectors in quantum mechanics, Dirac notation is most often used. This notation uses a $\left\vert \cdot \right\rangle \,\!$, called a ket, for a vector, so our vector ${\vec {v}}\,\!$ would be

$\left\vert v\right\rangle =\left({\begin{array}{c}v_{x}\\v_{y}\\v_{z}\end{array}}\right).$

For qubits, i.e. two-state quantum systems, will often be written as complex vectors

	${\begin{aligned}\left\vert \psi \right\rangle &=\left({\begin{array}{c}\alpha \\\beta \end{array}}\right)\\&=\alpha \left\vert 0\right\rangle +\beta \left\vert 1\right\rangle ,\end{aligned}}$	(C.1)

where

$\left\vert 0\right\rangle =\left({\begin{array}{c}1\\0\end{array}}\right),\;\;{\mbox{and}}\;\;\left\vert 1\right\rangle =\left({\begin{array}{c}0\\1\end{array}}\right)$

are the basis vectors. The two numbers $\alpha \,\!$ and $\beta \,\!$ are complex numbers, so the vector is said to be a complex vector.