Question

1) What about the probability density? 1 Add file 2) What is the space translation of point particles? 1 Add file

Answer 1

1)

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2)

A very important symmetry in physics is translational invariance: If it does not matter where you choose the origin of space, the physics must be invariant. Since moving the origin is a translation, we find that translations are a symmetry of most microscopic systems.

For a classical point particle, whose state of motion is specified by and , the way to represent a change of origin from to is by the transformation while remains unchanged. We conclude that is also unchanged. We shall require the laws of physics to be invariant under the transformation for a translationally invariant problem. We use the fact that the object describing the dynamics of a point particle completely is the Hamiltonian, For a translationally invariant particle, the Hamiltonian must thus be the same, independent of translation. The kinetic energy is invariant since momentum is unchanged, but the external potential must satisfy i.e., the potential is a constant, which we may well choose zero. Thus we can have no external forces when we have translational invariance