Basic Mathematics For Quantum Mechanics - Inner Products

What is an inner product?

Last time, we talked about vector spaces and we said that they have two binary operations: vector addition and scalar multiplication.

We can add a third operation called an inner product to a vector space to turn it into an inner product space.

An inner product is a binary operation that takes in two vectors and gives us back a scalar. We usually represent it with a dot

\[\cdot : V \times V \rightarrow S\]

Or we write it with a set of angle brackets

\[\langle x, y \rangle\]

Properties of an inner product

The inner product must obey 3 special properties

Conjugate symmetry

What this means is if we reverse the order of an inner product, we just get the complex conjugate of the original out. So

\[\langle x, y \rangle = \overline{\langle y, x \rangle}\]

In the case where the set of scalars we are working with is the real numbers, then since

\[x = \overline{x}\]

We get

\[\langle x, y \rangle = \langle y, x \rangle\]

Linearity in the first argument

Linearity in the first argument means two things. Firstly if x, y and z are all vectors then

\[\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle\]

Secondly, if s is a scalar and x and y are vectors then

\[\langle sx, y \rangle = s\langle x, y \rangle\]

Positive-definiteness

This just means that the inner product of a vector with itself is always greater than or equal to 0

\[\langle x, x \rangle \geq 0\]

Size of vectors

An inner product is often said to give geometry to a space. This is because it allows us to talk about the sizes of vectors and the angles between them.

We can denote the size of a vector v as |v| and we use the following formula:

\[|v| = \sqrt{\langle v, v \rangle}\]

Angle between two vectors

If we have two vectors x and y then the angle t between them is defined as:

\[cos(t) = \frac{\langle x, y \rangle}{|x||y|} = \frac{\langle x, y \rangle}{\sqrt{\langle x, x \rangle}\sqrt{\langle y, y \rangle}}\]

Dot product from school

In school, you might have encountered the dot product. The dot product is an example of an inner product.

Let us consider the following 2 vectors:

\[v = \begin{bmatrix}v_1 \\ v_2 \\ v_3\end{bmatrix} \in \mathbb{R}^3\] \[w = \begin{bmatrix}w_1 \\ w_2 \\ w_3\end{bmatrix} \in \mathbb{R}^3\]

Then the dot product is defined as

\[v \cdot w = v_1w_1 + v_2w_2 + v_3w_3\]

This formula meets all the criteria we outlined above and is probably the most common inner product you’ll come across. And it works in any number of dimensions we just keep multiplying the components together and adding them. So in n dimensions we have:

\[v \cdot w = v_1w_1 + v_2w_2 + ... + v_nw_n\]

Let’s look at a quick example:

\[v = \begin{bmatrix}3 \\ 4 \\ 7\end{bmatrix}\] \[w = \begin{bmatrix}5 \\ 8 \\ 1\end{bmatrix}\]

So

\[v \cdot w = 3*5 + 4*8 + 7*1 = 54\]

Dot products for complex numbers

The above dot product only works for real numbers. To extend it to complex numbers we need to make one small change.

\[v \cdot w = v_1\overline{w_1} + v_2\overline{w_2} + ... + v_n\overline{w_n}\]

All we have done is take the complex conjugate of the components of the w vector.

Let’s take a look at an example

\[v = \begin{bmatrix}3+2i \\ 4+7i\end{bmatrix}\] \[w = \begin{bmatrix}5 \\ 3+i\end{bmatrix}\]

So

\[v \cdot w = (3+2i)\overline{5} + (4+7i)\overline{(3+i)} = (3+2i)5 + (4+7i)(3-i)\] \[= 15+10i + 19+17i = 34 + 27i\]