Show that the angular momentum L=r¡Áp vector is an axial vector
To show that the angular momentum vector L = r x p is an axial vector, we need to understand the definition and properties of axial vectors.
An axial vector is a vector that changes sign under a reflection, also known as a parity inversion. In other words, if we reflect the coordinate system about a plane and reverse the direction of all the vectors, an axial vector will change its direction.
To demonstrate that the angular momentum vector L is an axial vector, we can perform the following steps:
Step 1: Define the angular momentum vector L = r x p, where r is the position vector and p is the linear momentum vector.
Step 2: Consider a reflection of the coordinate system about a plane. In this case, let's choose the x-y plane as the reflection plane.
Step 3: Under this reflection, the position vector r changes as r' = (x, y, -z), where x, y, and z are the Cartesian coordinates.
Step 4: Similarly, the linear momentum vector p changes as p' = (px, py, -pz), where px, py, and pz are the components of the momentum vector along the x, y, and z directions, respectively.
Step 5: Now we can calculate the reflected angular momentum vector L' = r' x p'.
By substituting the reflected position and momentum vectors into the definition of angular momentum, we have:
L' = (x, y, -z) x (px, py, -pz)
= (y(-pz) - (-z)(py), -[(x)(-pz) - (-z)(px)], -[(x)(py) - (y)(px)])
Simplifying this expression, we get:
L' = (zpy - ypz, -(-zpx - zx), -(-xpy - ypx))
= (-zpy + ypz, zpx + zx, xpy + ypx)
Step 6: Comparing L' to the original angular momentum vector L = (Lx, Ly, Lz), we see that the x-component and y-component have opposite signs in L' compared to L.
Therefore, under reflection, the angular momentum vector changes direction and satisfies the condition for an axial vector.
Hence, we have shown that the angular momentum vector L = r x p is an axial vector.