Vectors: A beginner's guide

Vectors

Here we introduce vectors and the notation that we use for vectors. We then give some facts about real vectors before discussing the complex vectors used in quantum mechanics.

Vectors Defining and Representing

You may have heard of the definition of a vector as a quantity with both magnitude and direction. While this is true and often used in science classes, our purpose is different. So we will simply define a vector as a set of numbers that is written in an row or a column. When the vector is written as a row of numbers, it is called a row vector and when it is written as a set of numbers in a column, it is called a column vector. As we will see, these two can have the same set of numbers, but each can be used in a slightly different way.

Examples

This is an example of a row vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} = (2,4,3). }

This is an example of a column vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w} = \left(\begin{array}{c} 1 \\ 5 \\ 4 \end{array}\right).}

Real Vectors

If you are familiar with vectors, the simple definition of a vector --- an object that has magnitude and direction --- is helpful to keep in mind even when dealing with complex and/or abstract vectors as we will here. However, this is not necessary and we will see how to perform all of the operations that we need just using our arrays of numbers. 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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\cdot}\,\!} ) denotes a unit vector and the components Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_i\,\!} , $i=x,y,z\,\!$ are just numbers. The unit vectors are also known as basis vectors. The unit vectors have magnitude equal to one. (The magnitude is the size, or length of a vector.) So that if the vector ${\hat {v}}\,\!$ is a unit vector, then $v_{x}^{2}+v_{y}^{2}+v_{z}^{2}=1\,\!$

Vector Operations

Adding Vectors

When adding vectors, it is important to note that you can only add vectors if they are the same type. (No adding "apples and oranges" so to speak.) So you can add two vectors that are both column vectors and have three entries. You can't add a column vector to a row vector and you can't add a vector with two components to a vector with three components.

Example

Adding two vectors ${\vec {v}}\,\!$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w}\,\!} with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} = \left(\begin{array}{c} 1 \\ 5 \end{array}\right), \;\; \vec{w} = \left(\begin{array}{c} 4 \\ 2 \end{array}\right), \,\!}

we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} + \vec{w} = \left(\begin{array}{c} 1 \\ 5 \end{array}\right) + \left(\begin{array}{c} 4 \\ 2 \end{array}\right) = \left(\begin{array}{c} 5 \\ 7 \end{array}\right). \,\!}

Products of Vectors

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z. }

For the inner product of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\,\!} with itself, we get the square of the magnitude of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\,\!} , denoted $|{\vec {v}}|^{2}\,\!$:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y + v_zv_z=v_x^2+v_y^2+v_z^2. }

If we want a unit vector in the direction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\,\!} , we can simply divide it by its magnitude:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{v} = \frac{\vec{v}}{|\vec{v}|}. }

Now, of course, ${\hat {v}}\cdot {\hat {v}}=1\,\!$, which 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 and the relevant notation. We will see how to compute the inner product later, since some other definitions are required.

Complex Vectors

For complex vectors in quantum mechanics, Dirac notation is used most often. This notation uses a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert \cdot \right\rangle\,\!} , called a ket, for a vector. So our vector ${\vec {v}}\,\!$ would be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, complex vectors will often be used:

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}	(C.1)

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta\,\!} are complex numbers, so the vector is said to be a complex vector.