Show that if a particle is in a stationary state at a given time it will always remain in a stationary state.

That statement is impossible to show, demonstrate, or prove.

A big part of the reason for the inability to show it is that the statement is ... prima facie, pro bono, and habeus corpus ... wrong, and as false as the day is long.

Although I am certainly far from what you might call a 'particle', I think I can show, using myself, that the statement is false:

As we speak, I am sitting, in a stationary state, before my computer.  It is my intention, and I fully believe it will come to pass, that in a few seconds, my fingers may move to operate one of the letter keys, or my hand may reach out to snag my coffee cup, or my entire being may arise from my place and go into the kitchen for a bowl of oatmeal.  At the INSTANT that any of these events takes place, the statement will have been proven false.

QED

a longitudinal wave moves energy that is to the direction of the wave’s movement. - true