In studying mechanical systems in general and ones with angular momentum in particular, it is valuable to have a good handle on vectors. Any physical property which requires the specification of both magnitude and direction requires a vector (for instance, velocity, momentum, or angular momentum). This is in contrast to scalars which only need the specification of their magnitude (for instance, temperature or pressure).
To work with vectors, a vector space is set up. A very useful such space is often formed as a Cartesian coordinate system in 3 dimensional space. This is useful since it correlates to the space in which we live. However, higher dimensional spaces are also useful - even infinite dimensional spaces - in quantum theory. It is important to appreciate that the usage of the term "space" refers to the domain in which a family of vectors operates. Our usual understanding of space ("The Final Frontier" variety) is only 1 example of an infinitude of spaces (the vector space variety). With spaces of a different dimension (NOT the science fiction "Dimension X" variety), it only describes the number of basis vectors needed to describe the space. Our universe needs three vectors to describe a position in space whereas our universe needs an infinitude of vectors (though we usually get away with 60 or 70) to describe a molecule. But the algebra we need is the same for a 3 dimensional space or for a 70 dimensional space, so we usually teach in 3 dimensions for that is easiest on our minds.
Here is such a space. The unit vectors i, j, and k are chosen so that through the addition of multiples of themselves with each other, the three of them can describe all vectors possible in the space. An arbitrary vector V, is described by specifying the amounts of i, j, and k which, when summed together, make V.The components of V, Vx, Vy, and Vz, are these multiples. To specify V, it is sufficient to specify its three components (Vx, Vy, Vz). Hence, a three dimensional vector is an ordered set of three numbers. A seventy dimensional vector is an ordered set of 70 numbers.
Two vectors, A and B, are equal if and only if each of their components are equal: Ax = Bx; Ay = By; Az = Bz. It is interesting to observe that 1 vector equation (A = B) is equivalent to three scalar equations. This brevity is a nice aspect to vector algebra.
When we add vectors, we add each of their components separately. By this it is clear that in order to add two vectors, they must have the same dimension - otherwise the operation is undefined. When we visualize this in space, we imagine moving the start point of one vector to the endpoint of the other vector. The sum vector is the resultant.
In this two dimensional case, we have mathematically A + B = (4,7) + (4,1) = (8,8). We should also remember that multiplication of a vector by scalar is multiplication of each component by the same scalar, namely cA = (cAx, cAy, cAz)
The length of a vector is called its magnitude and is usually denoted by |A|. The directionality of the vector is lost for this quantity so it is a scalar.
One way of combining two vectors is through an operation called the dot product. It is written as
This last form can be seen clearly when we consider the dot product of the unit vectors i, j, and k. Because they are fixed at 90 degrees from each other we have
This property of these three vectors (the dot product with themselves produces 1 and the dot product with each other produces 0) is what defines them as being a set of orthogonal, normalized vectors or orthonormal, for short. This is an extremely important property throughout quantum mechanics. In a 2 or 3 dimensional space, we can characterize this condition as having the vectors placed at right-angles to each other. The same concept holds in higher order spaces, but we are unable to visualize "right angles" in a 70 dimensional space!
With this information it is clear to see that this provides a quick route to the magnitude of a vector, namely to take the dot product of a vector with itself.
There is another way to combine vectors. This is the cross product but in this case the result is another vector, rather than a scalar as with the dot product. It is a little more complicated mathematically to remember the form of the operation so we cast it in the form of a determinant (is that any easier to remember??)
Take a look at the order of the subscripts in the result and you will see a cyclical appearance of each one. Learn to appreciate the order in this for it will appear time and again. There are a couple of other properties worth noting here.
Graphically, the concept to remember is that the cross product produces a vector which is perpendicular to both vectors making up the argument of the product. This means it is orthogonal to both (though the two argument vectors need not be orthogonal to each other). When the two original vectors are orthogonal to each other, the cross product vector has the greatest magnitude (it is at its longest). As the two vectors are rotated in towards each other, the resultant vector shortens until it disappears when the two overlap. This same happens when the two initial vectors rotate away from each other, the resultant disappearing when the point opposite each other. Click here for a short animation of this concept.
Another important vector operator is called del. When del operates upon a scalar function, it produces a vector which is the gradient of that function. The gradient functions points in the direction of the greatest spatial rate of change of the function. For instance, if you are standing onthe top of a hill, the gradient of the function describing the shape of the hill is in the direction of the steepest descent. In 3 dimensional space, the operator del is defined as
Related to this is the del-squared operator. When squaring this operator, we remember the dot product results for the unit vectors and can quickly note that, in 3 dimensions, del-squared is
The del and del-squared operators are very important in quantum mechanics. For instance, del shows up in the definition of the linear momentum operator, while del-squared appears in the definition of the Hamiltonian operator.