Dot and Inner Products of State Vectors

It often turns out to be useful to express the trigonometric functions in the change of basis equations, such as this equationEq_8_10, in more abstract ways involving the basis state vectors themselves. We met up with this idea in the Orthogonality of Two-State Vectors lesson, where we introduced the dot product of two state vectors. Recall that the dot product is also called the “scalar” or “inner” product.

Let’s start with a quick review of the dot product for ordinary vectors and then switch to quantum state vectors.

The dot product is called dot because it is represented symbolically by a dot between two vectors. It gives us a number:

$$\vec{B}\cdot\vec{C} \equiv \lVert\vec{B}\rVert\lVert\vec{C}\rVert\cos\theta_{BC},$$

Here, $θ_{BC}$ is the angle between the two vectors. (See the below figure.) The double vertical bars mean the length of the vector. The above equation can be read as telling us how much of vector $\vec{B}$ lies along the direction of vector $\vec{C}$ multiplied by the length of vector $\vec{C}$ or vice versa; it treats $\vec{B}$ and $\vec{C}$ symmetrically.