Vectors: A beginner's guide

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.

Note that a vector with two entries, or two numbers, is called a two-dimensional vector. One with three entries is called a three-dimensional vector, etc.

Examples

This is an example of a two-dimensional row vector ${\vec {s}}=(1,6).$

This is an example of a two-dimensional column vector ${\vec {u}}=\left({\begin{array}{c}7\\2\end{array}}\right).$

This is an example of a three-dimensional row vector ${\vec {v}}=(2,4,3).$

This is an example of a three-dimensional column vector ${\vec {w}}=\left({\begin{array}{c}1\\5\\4\end{array}}\right).$

Real Vectors

(If you are not familiar with vectors, you can skip this subsection.)

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 vectors (vectors with complex, i.e., imaginary numbers as entries) 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 arrays of numbers that we call vectors. 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 and the components $v_{i}\,\!$, $i=x,y,z\,\!$ are just numbers. The unit vectors are also known as basis vectors. This is because any vector in real three-dimensional space can be written in terms of these unit/basis vectors. In this vector, one can associate a point where the coordinate of the point is $(x,y,z)$. That is, a point a distance 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 x} from the origin along the 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 x} -axis, a distance 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 y} from the origin along the 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 y} -axis, and a distance 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 z} from the origin along the 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 z} -axis. In some sense, unit vectors are the basic components of any vector. Other basis vectors could be used. But this will be discussed elsewhere.

Vector Operations

To illustrate vector operations, let

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_1 \\ v_2 \end{array}\right), \mbox{ and } \vec{w} = \left(\begin{array}{c} w_1 \\ w_2 \end{array}\right). }

Also, let

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{s} = \left(s_1, s_2\right), \mbox{ and } \vec{u} = \left(u_1, u_2\right). }

Vector Addition

Vectors can be added. To do this, each element of one vector is added to the corresponding element of the other vector. In general, for a row vector, they add 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}+\vec{w} = \left(\begin{array}{c} v_1 \\ v_2 \end{array}\right)+\left(\begin{array}{c} w_1 \\ w_2 \end{array}\right) = \left(\begin{array}{c} v_1+w_1 \\ v_2+w_2 \end{array}\right). }

The addition of row vectors is similar. They are added component by component. Then

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{s}+\vec{u} = \left(s_1, s_2\right)+\left(u_1, u_2\right) = \left(s_1+u_1, s_2 + u_2\right). }

Since the way vectors are added is component to corresponding component, 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 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}_1\,\!} 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}_1\,\!} with

${\vec {v}}_{1}=\left({\begin{array}{c}1\\5\end{array}}\right),\;\;{\vec {w}}_{1}=\left({\begin{array}{c}4\\2\end{array}}\right),\,\!$

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}_1 + \vec{w}_1 = \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). \,\!}

Length or Magnitude of a Vector

Consider the 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} = \left(\begin{array}{c} v_1 \\ v_2 \end{array}\right). }

We can calculate the length of a vector the same way that one calculates the hypotenuse of a right triangle. The magnitude is the square root of the sum of the squares of the entries. For example,

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}|= \sqrt{(v_1)^2 + (v_1)^2}. }

Figure V.1: The magnitude of a vector in terms of its entries.

Vectors of Length One, Unit Vectors

Vectors of length, or 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 = 1} are quite useful. These are also called "unit vectors". As we will see, they are used as "basis vectors" and also for quantum states. If a vectors does not have length one, but it should have, then we can "normalize" it by dividing by the magnitude. When this is done, we sometimes denote the vector of length one with a "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{v}} instead of the arrow 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}} . So

${\hat {v}}={\frac {1}{|{\vec {v}}|}}{\vec {v}}={\frac {1}{|{\vec {v}}|}}\left({\begin{array}{c}v_{1}\\v_{2}\end{array}}\right).$

Then

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} |\hat{v}| &= \sqrt{\frac{\vec{v}}{|\vec{v}|}} \\ & \\ &= \sqrt{ \left(\frac{v_1}{|\vec{v}|}\right)^2 + \left(\frac{v_2}{|\vec{v}|}\right)^2 }\\ & \\ &= \frac{1}{|\vec{v}|} \sqrt{(v_1)^2 + (v_2)^2} = 1. \end{align} }

Multiplication by a Number

When a vector is multiplied by a number, each component is multiplied by that same number. For example, suppose

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_1 \\ v_2 \end{array}\right). }

Then

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 a\vec{v} = a\left(\begin{array}{c} v_1 \\ v_2 \end{array}\right)= \left(\begin{array}{c} av_1 \\ av_2 \end{array}\right). }

Notice that if $a$ is positive, then the magnitude of the vector is

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 |a\vec{v}|= \sqrt{(av_1)^2 + (av_1)^2} = \sqrt{a^2(v_1^2+v_2^2)} = a|\vec{v}|. }

So multiplying a vector by a number just changes the length, or magnitude, of the vector is 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 a} is positive. If 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 a} is negative, it changes the directions of the components and therefore of the vector itself and also changes the magnitude.

Dot Products or Inner Products

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

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_1 \\ v_2 \end{array}\right), \mbox{ and } \vec{w} = \left(\begin{array}{c} w_1 \\ w_2 \end{array}\right) }

or

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{s} = \left(s_1, s_2\right), \mbox{ and } \vec{u} = \left(u_1, u_2\right). }

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_1w_1 + v_2w_2, \mbox{ and } \vec{s}\cdot\vec{u} = s_1u_1 + s_2u_2. }

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 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\,\!} :

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_1v_1 + v_2v_2 = v_1^2+v_2^2. }

If we want a unit vector in the direction of ${\vec {v}}\,\!$, 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, 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}\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_1 \\ v_2 \end{array}\right). }

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

${\hat {e_{1}}}=\left({\begin{array}{c}1\\0\end{array}}\right),\;\;{\hat {e_{2}}}=\left({\begin{array}{c}0\\1\end{array}}\right).\,\!$

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 e_1\,\!} 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 e_2\,\!} are called basis vectors. Notice they are unit vectors.

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

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

$\left\vert v\right\rangle =\left({\begin{array}{c}v_{1}\\v_{2}\end{array}}\right).$

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

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

where

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 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 \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. Recall from Chapter 1 that quantum states are represented by complex vectors, such as the quantum particle being in one of two different wells.

The Complex Conjugate of a Vector

To take the complex conjugate of a vector, each element of the vector has to be complex conjugated. So,

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

The Transpose of a Vector

Let us reconsider the vector from above,

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_1 \\ v_2 \end{array}\right). }

To transpose this vector, the vector will be turned into a row vector. It is the almost the same vector, just made into a row vector. The transpose is denoted with a superscript 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 T} :

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} \vec{v}^T &= \left(\begin{array}{c} v_1 \\ v_2 \end{array}\right)^T \\ &= \left( \begin{array}{cc} v_1, & v_2 \end{array}\right). \end{align}}

The Hermitian Conjugate, or "Dagger", of a Vector

The "dagger" of a vector, which is also called the hermitian conjugate, is the transpose and complex conjugate. This is denoted by a "dagger" superscript $^{\dagger }$

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} (\vec{v}^*)^T &= \left(\begin{array}{c} v_1^* \\ v_2^* \end{array}\right)^T \\ &= \left( \begin{array}{cc} v_1^*, & v_2^* \end{array}\right). \end{align}}

In the case of complex vectors, the following notation is 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)^\dagger &= \left\langle \psi \right\vert \\ &=\left(\begin{array}{c} \alpha \\ \beta \end{array}\right)^\dagger \\ &= \left(\begin{array}{cc} \alpha^*,& \beta^* \end{array}\right). \end{align}}	(C.1)

Complex Inner Product

To calculate the inner product of two complex, two-dimensional vectors, the complex conjugate of one is multiplied, term by term with the second. So if vectors 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}} 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}} are complex vectors, the inner product, or dot product, is calculated by

${\vec {v}}\cdot {\vec {w}}=v_{1}^{*}w_{1}+v_{2}^{*}w_{2}.$

Let us define a 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 \left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta \end{array}\right). }

Then, recalling the notation above, 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 \left\vert\phi\right\rangle} 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 \left\vert\psi\right\rangle} is

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 \phi \right\rangle)^\dagger (\left\vert\psi\right\rangle) &= \left\langle\phi\mid\psi\right\rangle \\ &= \left(\begin{array}{cc} \gamma^*,& \delta^* \end{array}\right)\left(\begin{array}{c} \alpha \\ \beta \end{array}\right) \\ &= \gamma^*\alpha + \delta^*\beta. \end{align}}

In quantum mechanics, this is one of the most useful relations for calculating a variety of quantities of physical interest. For examples, when projective quantum measurements are made; to normalize, or renormalize vectors; and to calculate expectation values, these products are used.

Exercises

1. Vectors
1. About the author
2. Foreword to the first edition
3. Foreword to the second edition
2. Matrices
1. About the author
2. Foreword to the first edition
3. Foreword to the second edition
3. Dirac Notation (bras and kets)
1. About the author
2. Foreword to the first edition
3. Foreword to the second edition
4. Transformations
1. About the author
2. Foreword to the first edition
3. Foreword to the second edition
5. Eigenvalues and Eigenvectors
1. About the author
2. Foreword to the first edition
3. Foreword to the second edition
6. Tensor Products
1. About the author
2. Foreword to the first edition
3. Foreword to the second edition