   Next: Matrices Up: linear_algebra Previous: linear_algebra

# Vectors

In quantum mechanics, we are generally interested in complex numbers. Let denote the set of all -tuples of complex numbers (a complex -space). The elements of may be called points'' or vectors'' in complex -space. The elements of the complex numbers are scalars.

A vector might be denoted by listing all its elements as . This format is called a row vector. Alternatively, the elements could be arranged as which is a column vector. Our default choice will be to write vectors as column vectors, and row vectors will result from taking (complex conjugate) transposes of vectors.

When a vector is multiplied by a scalar (a number), this is the same as multiplying each component of the vector by that scalar, e.g.,   (1)

For real numbers, the dot (or inner) product between two vectors and is given by     (2)

If and are complex, then one small complication arises. If and are given by (3)

then to take the dot product of and , we must first take the complex transpose of ,     (4)

Recall that the complex conjugate of a complex number like is (the imaginary part has its sign reversed). Any time a row vector is converted into a column vector (or vice versa), if it is complex, then one needs to take complex conjugates of each element. (5)

A vector is said to be normalized if its dot product with itself, , is 1. If this is not the case, it is always possible to normalize the vector to force the dot product to come out to 1; one merely needs to divide each component of the vector by the square root of the dot product that results before normalization. For example, the vector (6)

is not normalized, because its dot product with itself is . However, if we divide the vector by , then the resulting vector (7)

is normalized: (8)

Two vectors and are said to be orthogonal if their dot product . Two vectors are orthonormal if they are orthogonal and each one is normalized.   Next: Matrices Up: linear_algebra Previous: linear_algebra
David Sherrill 2006-08-15