Inner Products for Complex Vectors

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Section 3.7 Inner Products for Complex Vectors

Column matrices play a special role in physics, where they are interpreted as vectors or, in quantum mechanics, states. To remind us of this uniqueness they have their own special notation; introduced by Dirac, called bra-ket notation. In bra-ket notation, a column matrix, called a ket, can be written

\begin{equation} \left|v\right> \doteq \begin{pmatrix} v_x\\ v_y\\ v_z \end{pmatrix}\text{.}\tag{3.7.1} \end{equation}

The Hermitian adjoint of this vector is called a bra

\begin{equation} \left\lt v\right| \doteq \left(\left|v\right>\right)^\dagger =\begin{pmatrix} v_x^*\amp v_y^*\amp v_z^* \end{pmatrix}\text{.}\tag{3.7.2} \end{equation}

Notice that we are not assuming that the components of the vector are real.

Now we want to generalize the concept of dot product from Section 3.6 to this case where the components of the vector or state are not necessarily real. This generalization is technically called an inner product although many scientists will still casually use the term dot product. If we take \(\left|v\right>\) to be a 3-vector with components \(v_x\text{,}\) \(v_y\text{,}\) \(v_z\) as above, then the inner product of this vector with itself is called a braket

\begin{equation} \left\lt v|v\right> = \begin{pmatrix} v_x^*\amp v_y^*\amp v_z^* \end{pmatrix} \begin{pmatrix} v_x\\ v_y\\ v_z \end{pmatrix} = \left|v_x\right|^2+\left|v_y\right|^2+\left|v_z\right|^2\tag{3.7.3} \end{equation}

which is a positive real number. We will call the square root of this positive real number the magnitude (or, more casually, the length) of the vector, even when the vector is abstract and there is no length that we would be able to measure with a ruler. Similarly, we will say that two vectors are orthogonal (or, more casually, perpendicular) whenever their inner product is zero, even if there is no angle that we could measure with a protractor.

In physics, you will encounter many other abstract spaces that have the same algebraic properties as vectors and dot products. A list of these abstract properties and some examples can be found in Section 14.1 and Section 14.2.