Is the energy difference between two energy levels unique for that particular pair of levels for a hydrogen atom ? If so how can one prove it?
As I am sure you are aware, the electrons in free atoms are in quantized energy states which means that they can only be found on a set of discrete energies. These states can be referred to as energy levels. Since these states are discrete, the difference between any two states must also have a finite value.
To answer your question as to how one could prove this, there is a simple method called atomic emission spectroscopy. You can take a tube of hydrogen gas and excite it through the use of a high voltage transformer. This addition of energy will cause the electrons to move up to higher energy states, and eventually they 'fall' back down to a lower energy level. Now, where dose that energy go? It is emitted as electromagnetic radiation. You can use a detector such as a photomultiplier to obtain rather beautiful data on the emitted spectrum from this tube. An example of such is here:
You can see that there are certain distinct lines of emission. These correspond directly to the difference in energy between the differing states. If there were not unique energy levels, the graph would have no distinct peak of intensity. Thus, it can be concluded that the energy states of an hydrogen atom are quantized.