Which quantitiesoperators are invariant in respect to space

Which quantities/operators are invariant in respect to space reflection?
a) Kinetic Energy
b) Projection of particle’s spin on its coordinate vector
c) Mass
Please explain your answer

Solution

a) Kinetic Energy

Because

In Newton\'s theory of mechanics, given two bodies, each with mass m, starting from rest at the origin and moving along the x-axis in opposite directions, one with speed v₁ and the other with speed v₂ the total kinetic energy of the system (as calculated from an observer at the origin) is ¹₂m(v₁² + v₂²) and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the y-axis.

The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if v₁ and v₂ are interchanged.