Show that the angular momentum vector(L)=vector(r)×vector(p) 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 properties of axial vectors and apply them to L.
1. Rotation: Axial vectors behave differently from polar vectors (also called polar or ordinary vectors) under a coordinate system rotation. When an axial vector is rotated, it changes sign.
2. Reflection: Axial vectors also change sign when we reflect the coordinate system. If we reverse all three coordinate axes, the axial vector flips sign.
Now let's apply these properties to L = r x p:
Consider a coordinate system transformation where we rotate the coordinates by an angle of θ about an axis. Let's say this transformation is counterclockwise when viewed from positive infinity along the axis.
Under this rotation, the position vector r and momentum vector p both transform in the same way as polar vectors. The rotation matrix for the position vector is R_r(θ) and for the momentum vector is R_p(θ).
The cross product operation involves the determinant of a matrix, which, in this case, is the determinant of R_r(θ) x R_p(θ).
Now, when we rotate both r and p using the coordinate system rotation matrices, the cross product operation L = r x p also transforms according to the determinant of R_r(θ) x R_p(θ).
Since the determinant of a product of matrices is the product of the determinants, we have:
L' = r' x p' = (R_r(θ) x R_p(θ)) r x p
Now, because both r and p transform as ordinary vectors, their product r x p still transforms as an ordinary vector.
However, R_r(θ) x R_p(θ) is an axial vector because it changes sign under the rotation transformation. Therefore, L' = (R_r(θ) x R_p(θ)) (r x p) is an axial vector as the product of an axial and ordinary vector is always an axial vector.
Hence, we have proved that the angular momentum vector L = r x p is an axial vector based on the properties of axial vectors and the transformation laws under coordinate system rotations.