Ordinary Vectors

Let’s pause for a few minutes and remind ourselves that this way of writing a quantum state vector is like the way we describe other vectors in mathematics, physics, engineering, and other sciences. For example, we use vectors to describe velocity, position, acceleration, forces, and electromagnetic fields in physics.

Let’s review the connection between an ordinary vector and its components relative to the traditional $x$ and $y$ axes. To be concrete, suppose we have a vector represented by the symbol $\vec{B}$ and, in keeping with our restriction to two-state systems, we assume that the vector “lives” in a two-dimensional space. The vector can be described as a sum of two vectors, $b_x\hat{x}$ in the $x$ direction and $b_y\hat{y}$ in the $y$ direction. $\hat{x}$ is a vector of unit length pointing in the $x$ direction with an analogous definition for $\hat{y}$. The unit vectors $\hat{x}$ and $\hat{y}$ play the role of basis vectors. We could also use two column vectors or a column vector with two entries. So, $\vec{B}$ can be written in three equivalent ways:

$$\begin{align*} \vec{B} &= b_x\hat{x} + b_y\hat{y}\\[2mm]& \Rightarrow b_x\ \begin{pmatrix} 1\\0\end{pmatrix} + b_y \begin{pmatrix} 0\\1\end{pmatrix} \\[2mm] &= \begin{pmatrix} b_x\\b_y\end{pmatrix}. \end{align*}$$

Now comes a crucial question: How do we express the length of the vector $\vec{B}$ in terms of the coefficients $b_x$ and $b_y$? The answer is given by the Pythagorean theorem. The square of the hypotenuse of a right triangle (the square of the length of the vector $\vec{B}$ in the figure below) is equal to the sum of the squares of the other two sides.