General theorem on the Schrödinger equation.

It is unnecessary, nowadays, to point out the importance of the Schrodinger equation to disciplines of science and technology. In principle, one can understand any natural phenomenon if one can solve the corresponding Schrodinger equations. Unfortunately, the Schrodinger equation can only be analytically solved in a few special cases. Therefore, it is very important to know the properties of the equation. There are many theorems and properties [1,2] which help us to understand various aspects of the Schrodinger equation. One of them concerns the ground state of a particle in two different potential wells: It is well known that if one well V, is less than or equal to the second well V2 everywhere in space, the ground-state energy in the well V, will not always be higher than that in the well V2. This property can be easily shown using the variational theorem, i.e., energy expectation of a trial state is always greater than or equal to the ground state of the system. However, to the best of our knowledge, there is nothing in the literature comparing levels other than the ground state. It is the purpose of this letter to show that a similar property is also true for the excited states. Let us consider the energy spectrum of a particle moving in a potential well, V(r). According to quantum mechanics, the spectrum well be governed by the timeindependent Schrodinger equation