2.3 The Dot, oops, INNER Product

The dot product of vectors is an important tool. It makes it possible to find the length of a vector, by multiplying the vector by itself and taking the square root. It is also used to check if two vectors are orthogonal: if their dot product is zero, they are. In this subsection, the dot product is defined for complex vectors and functions.

The usual dot product of two vectors and can be found by multiplying components with the same index together and summing that:

(The emphatic equal, , is commonly used to indicate “is by definition equal” or is always equal.) Figure 2.6 shows multiplied components using equal colors.

Note the use of numeric subscripts, , , and rather than , , and ; it means the same thing. Numeric subscripts allow the three term sum above to be written more compactly as:

The is called the summation symbol.

The length of a vector , indicated by or simply by , is normally computed as

However, this does not work correctly for complex vectors. The difficulty is that terms of the form are no longer necessarily positive numbers. For example, 1.

Therefore, it is necessary to use a generalized “inner product” for complex vectors, which puts a complex conjugate on the first vector:

 (2.7)

If the vector is real, the complex conjugate does nothing, and the inner product is the same as the dot product . Otherwise, in the inner product and are no longer interchangeable; the conjugates are only on the first factor, . Interchanging and changes the inner product’s value into its complex conjugate.

The length of a nonzero vector is now always a positive number:

 (2.8)

Physicists take the inner product bracket verbally apart as

and refer to vectors as bras and kets.

The inner product of functions is defined in exactly the same way as for vectors, by multiplying values at the same -​position together and summing. But since there are infinitely many values, the sum becomes an integral:

 (2.9)

Figure 2.7 shows multiplied function values using equal colors:

The equivalent of the length of a vector is in the case of a function called its “norm:”

 (2.10)

The double bars are used to avoid confusion with the absolute value of the function.

A vector or function is called “normalized” if its length or norm is one:

 (2.11)

(“iff” should really be read as if and only if.)

Two vectors, or two functions, and , are by definition orthogonal if their inner product is zero:

 (2.12)

Sets of vectors or functions that are all

• mutually orthogonal, and
• normalized
occur a lot in quantum mechanics. Such sets should be called “orthonormal”, though the less precise term orthogonal is often used instead. This document will refer to them correctly as being orthonormal.

So, a set of functions or vectors is orthonormal if

and

Key Points
For complex vectors and functions, the normal dot product becomes the inner product.

To take an inner product of vectors,
• take complex conjugates of the components of the first vector;
• multiply corresponding components of the two vectors together;
• sum these products.

To take an inner product of functions,
• take the complex conjugate of the first function;
• multiply the two functions;
• integrate the product function.

To find the length of a vector, take the inner product of the vector with itself, and then a square root.

To find the norm of a function, take the inner product of the function with itself, and then a square root.

A pair of vectors, or a pair of functions, is orthogonal if their inner product is zero.

A set of vectors forms an orthonormal set if every one is orthogonal to all the rest, and every one is of unit length.

A set of functions forms an orthonormal set if every one is orthogonal to all the rest, and every one is of unit norm.

2.3 Review Questions
1.

Find the following inner product of the two vectors:

2.

Find the length of the vector

3.

Find the inner product of the functions and on the interval 0 1.

4.

Show that the functions and are orthogonal on the interval 0 .

5.

Verify that is not a normalized function on the interval 0 , and normalize it by dividing by its norm.

6.

Verify that the most general multiple of that is normalized on the interval 0 is where is any arbitrary real number. So, using the Euler formula, the following multiples of are all normalized: , (for 0), , (for ), and , (for ​2).

7.

Show that the functions and are an orthonormal set on the interval 0 1.