When the phrase “free particle” is used in quantum physics as opposed to a “boxed particle” it means that the particle is not being held by a potential energy (the box) and so is free to move. From a mathematical standpoint solving the Schrödinger equation can be seen as easier with the potential term removed but it is made equally harder by the fact the wave is now progressive (moving) while it has to be a standing (stationary) wave when in the box.
The mathematical analysis and solutions for both the time dependent and time independent Schrödinger equation are essential when trying to understand the quantum nature of a system. To apply relativity to quantum physics a full quantum field theory is required but for certain circumstances when there is no massive fluctuation in the local electric field a half measure called the semi-classical approach can be used and is often easier. Using the semi-classical method it has been recently proven that for free, coherent (when the quantum oscillation is very similar to a classical one) particles the mathematics is identical to that of a simple particle moving along the same trajectory with a centre of mass at the centre of the oscillation. It appears like the two scenarios are so similar the position of the quantum state is determined just by initial position and momentum also (with the added problem of not being able to know both) exactly the same as a tennis ball thrown. Finding these golden comparisons between the macro and quantum world help prove that he physics is not just descending into mathematical manipulation and still has real relevance to the world around us.