Bibliography A

Consider the right-handed Cartesian coordinate system shown in Figure 4.4. Picture two regular vectors in this system, oriented along the first two basis vectors:

(4.102) (4.103) By “regular,” we mean that the components of these vectors obey Equation 4.25 when we transform the coordinates.

The cross product between and can be formed using the determinant,

= A,B,&, (4.104)

iet

Figure 4.4 The Right-Handed System

3

A x B

t

2

Figure 4.5 Vectors in the Right-Handed System

or, equivalently, using the Levi-Civita symbol:

(4.105) The resulting vector is shown in Figure 4.5. Notice how the direction of X B is given by the standard right-hand rule. If you point the fingers of your hand in the direction of

x

and then curl them to point along B, your thumb will point in the direction of the cross product. Keep in mind that the cross product is not commutative. If the order of the operation is reversed, that is if you form B X

A,

the result points in exactly the opposite direction.

Now consider the left-handed system shown in Figure 4.6, with coordinates and basis vectors marked with primes to distinguish them from the coordinates and basis vectors of the right-handed system. This system results from a simple inversion of the I-axis of the unprimed system. It can also be looked at as a reflection of the right- handed system about its x2x3-plane. The equations relating the primed and unprimed coordinates are

(4.106)

3'

Figure 4.6 The Left-Handed System

so that the transformation matrix becomes -1 0 0

0 0 1

[ a ] =

[

^{0 1}

^{o j}

The regular vectors

A

and B in the primed coordinate system are simply

= -A, 6;

= B, 6;.

(4.107)

(4.108) (4.109) We just wrote these results down because they were obvious. Remember, formally, they can be obtained by applying [a] to the components of the unprimed vectors. The matrix multiplication gives

(4.1 10) 0 0 1

and

-1 0 0

[:I]

⁼

^[

^{0 0 01 01 1 [ l J =}

[;,I

It is important to remember that the vectors are the same physical objects in both coordinate systems. They are just being expressed in terms of different components and different basis vectors.

Now form the cross product of

A

^and B in the left-handed system. To do this we will use the same determinant relation:

(4.111)

6;

c;

6;

A X B = -A, 0 0 = -A, B,C&, (4.112)

I

^{O B, O}

Let

or in terms of the Levi-Civita symbol:

A

X B ⁼ A{ BI 6; E i j k = A ; B; 6; €123 = - A , B, 6:. (4.113) The vectors

A

and B and the cross product X B are shown in Figure 4.7 for the left-handed coordinate system. Notice now, the right-hand rule used earlier no longer works to find the direction of the cross product. If we define the cross product using the determinant in Equation 4.112, then we must use a left-hand rule if we are in a left-handed coordinate system.

There is something peculiar here. Compare Figures 4.7 and 4.5 notice that while A and B point in the same directions in the two systems, their cross product does not!

-

2'

I

Figure 4.7 Vectors in The Left Handed System

By changing the handedness of the coordinate system, we have managed to change the vector X B.

Let's look at this cross product from another point of view. If the unprimed components of the quantity X

B,

given in Equation 4.104, are transformed into the primed system using the [a] matrix, as one would do for regular vector components, we obtain

[ ^{y ]}

^{[At,,] =} [ A t , ] (4.1 14)

Combining these components with the appropriate basis vectors gives for the cross product vector

-1 0 0

A, B ,

$4.

^{(4.1 15)}

This result disagrees with Equation 4.1 12 by a minus sign. To get around this difficulty, a quantity formed by the cross product of two regular vectors is called apsedovector.

Pseudovectors are also commonly called axial vectors, while regular vectors are called polar vectors. If

v

is a regular vector it transforms according to Equation 4.25.

However, if

v

is a pseudovector, its components transform according to

V: = b i d e t Viari- ^(4.1 16) In this way Equation 4.1 14 becomes

(A x B);

-1 0 0

( A x B);

0 0 1 A,B, - ' 4 3 "

[ ( A X E 1 : ]

^{= -}

[

^{0 1}

.] [ : ]

⁼

[ : ]

giving

(4.117)

X B = -A, B, $;, (4.1 18)

in agreement with Equations 4.1 12 and 4.1 13.

To summarize, if

v

is a regular vector its components transform as

v:

^{= vi a,i.}

If instead it is a pseudovector, it components transform as

(4.1 19)

(4.120) If the handedness of two orthonormal coordinate systems is the same, a transformation between them will have = 1 and vectors and pseudovectors will both transform normally. If the systems have opposite handedness, la/& = -1 and vectors will transform normally but pseudovectors will flip direction. A vector generated by a cross product of two regular vectors is actually a pseudovector.

It is tempting to think that all this balderdash is somehow a subtle sign error embedded in the definition of the cross product. In some cases, this is correct. For example, when we define the direction of the magnetic field vector, which turns out to be a pseudovector, we have implicitly made an arbitrary choice of handedness that must be treated consistently. Another example is the angular momentum vector, which is defined using a cross product. While you could argue that the “pseudoness”

of these two examples is just a problem with their definition, there are cases where you cannot simply explain this property away. It is possible to design situations where an experiment and its mirror image do not produce results which are simply the mirror images of each other. In fact, the Nobel Prize was won by Lee and Yang for analyzing these counterintuitive violations of purity conservation. The classic experiment was first performed by Wu, who showed this effect with the emission of beta particles from Cobalt-60, under the influence of the weak interaction.

In document Open design and medical products (Page 120-136)