A vector, in classical physics, is an entity that has magnitude and direction. 'Normal' numbers such as those refering to money or things that have been counted have magnitude only. An example of a vector would be velocity, which in physics has both magnitude and direction. We sometimes are a bit careless and use the words 'speed' and 'velocity' interchangeably when speed actually has only magnitude (we call such number a 'scalar') whereas velocity has magnitude and direction.

Vectors can be added and subtracted. Visually, this is done by laying them head to tail and measuring the distance from the tail of the first vector to the head of the last vector. Mathematically, this is accomplished using unit or 'standard' vectors and multiplying them by a scalar to obtain the desired vector.

Linear momentum, L, is defined as mass (m) times velocity (v), L = mv (where the boldface indicates a vector quantity). Note that multiplication of a vector quantity by a scalar generates a new vector with the same direction but a different magnitude. This is actually an intuitive definition of linear momentum as one would easily say that, for a given velocity, an object with more mass than another object would have more momentum. A bowling ball thrown down the alley has more momentum than a ping-pong ball thrown at the same velocity.

Vector Addition/Subtraction

As mentioned above, addition of vectors can be pictured as a laying of the vectors involved head to tail and measuring the distance and direction between the tail of the first vector to the head of the last vector:

Subtraction of the second vector from the first would just involve reversing it's direction and adding them together as above.

The magnitude of the vector is simply it's length measured in some arbitrary units. This is easy to do .. you just get out your measuring stick and measure it's length. But it isn't completely specified yet; we haven't said what it's direction is. Remember, vectors have both magnitude and direction.

To specify it's direction we use unit vectors defined in our coordinate system and then multiply each by a magnitude such that when the unit vectors are added together we end up with our vector. In three-dimensional cartesion coordinates, our unit vectors are i_x, i_y and i_z

So, let's say that we have two vectors A and B. We can represent each of these:

Each of these is just a vector that is made up of the unit or standard vectors, each of which is multiplied by an appropriate scalar quantity.

The sum of the vectors A and B is:

What is the magnitude of a vector, given its expression in unit vectors? This is just a simple application of the Pythagorean theorem:

The magnitude of our added vectors, A and B is:

Vector Multiplication

There are two kinds of vector multiplication, the dot-product (or the scalar-product) and the cross-product. The dot-product gives a scalar quantity as its result and the cross-product gives a new vector quantity. For vectors A and B, the dot-product is:

where q; is the angle between the two vectors. Obviously, if the two vectors are 90 degrees apart, the result of the dot product will be zero. If this is not obvious to you please let me know at the above email.

The cross-product is a bit more complicated. The result of a cross-product operation is a new vector with a magnitude and direction. The magnitude is:

where, again, q is the angle between the two vectors. Again, it is obvious that if the angle is zero then the magnitude of the new vector, C, is zero. It is equally obvious that the magnitude of the new vector will be at a maximum when q equals 90 degrees.

That takes care of the magnitude but what about the direction of the new vector? We use the right hand rule to get the new direction. The fingers of the right hand point in the direction that A must rotate in order to approach B. The thumb will point towards the new vector, C.

The vector A rotated towards B shows us what direction the resultant vector, C, will be in. Note that, from the right hand rule, BXA will result in C pointing along the -z axis, so AXB = -BXA since, for BXA, we would rotate B towards A and C would lie along the -z axis. We say that A and B do not commute, since reversing the order of operation changes the outcome. Commutators and non-commutators are very important in angular momentum calculations, as we will see shortly.

One thing that I have just realized as I write these web pages is that you can go backwards. That is, if you come across an equation such as:

and you know that A and B are both vector quantities then you can write:

The more rigorous description of the cross-product involves some matrix or rather determinant algebra. For vectors A = A_xi_x + A_yi_y + A_zi_z and B = B_xi_x + B_yi_y + B_zi_z, the cross product, AXB is:

Working out the value of the determinant is not all that difficult ... it just requires the application of the expansion rules (see below):

The magnitude of this vector is:

We'll need this later ...

The result of an operation on an eigenfunction is a constant times the eigenfunction:

where Y_j is the eigenfunction and k_j is the eigenvalue. The set of all Y_j's is called a "basis set". If the operator is Hermitian (see below), operating on an eigenfunction will produce an eigenvalue that is real.

Examples:

If our function is sin(x) and our operator is ∂/∂x then:

Sin(x) is clearly not an eigenfunction of ∂/∂x since the original function does not appear in the result.

However if we use ∂²/∂x² as our operator then:

Sin(x) is an eigenfunction of ∂²/∂x² ... sin(x) is regenerated multiplied by the constant, -1, which is the eigenvalue.

A matrix is a rectangular array of numbers such as:

They need not be square as in the above matrix ... they can generally be an n x m array of numbers, where occasionally n = m (where n is rows and m is columns). Thus, a single row of numbers or a single column of numbers is a matrix. Two matrices are said to be equal if they have the same numbers of rows and columns and if their corresponding elements are the same.

Matrix algebra is, for the most part, pretty simple. There are some complexities with determinants that the first-time 'user' will probably have to review, but there are no insurmountable problems.

There is, of course, addition and subtraction of matrices. Two matrices can be added (or subtracted, which is the same thing) if they have the same number of rows and columns:

+ =

Matrices can be multiplied by a scalar number very easily ... just multiply each element of the matrix by the scalar number. Matrix-matrix multiplication is a bit more complicated. An n x a matrix can multiply an a x m matrix. In other words, if the number of columns of the first matrix is the same as the number of rows of the second matrix then we can do the multiplication. The way it works is that each element of a row in the first matrix mutiplies the corresponding element in a column in the second matrix. These multiplications are summed and become a new element in the product matrix:

X =
=

Operators perform some type of mathematical operation on a mathematical function. For example, one can multiply by x:

where x is simply meant to multiply into B. Note that on this web page, operators are bolded and overlined.

Other common operators would include:

An operator is called linear if:

or:

An operator is said to be hermitian if it satisfies:

Since operators are generally taken to operate on the function to the right of them as they are written, this Hermitian property is very useful for getting the value of the operation on the function to the left.

The order that the operators occur in is generally important, since a different value may result if they are switched. For example:

ie.:

whereas:

On the other hand, depending on the operators, their order may not be important:

This is the commutation property of operators. If the order of the operators is unimportant to the result of the calculation we say that they commute:

or more compactly:

In this case A and B are said to be commutators.

A hermitian operator is defined as follows: