Discrete and canonical quantum variables

Abstract

A connection betweend discrete, Cartesian and angular quantum variables is obtained and discussed. I Introduction -Schwinger's approachQuantum mechanics, and in particular the language of quantum mechanics, is deeply entangled with the way physics is done on its many branches. Therefore, the interpretation of the formalism might have a crucial role on areas where both theoretical and experimental efforts are highly developed. Nuclear physics, with its need to take into account so many different effects -which are better understood mostly in languages of different theories -is a sound example of the above remark. Motivated by this basic idea, we shall here explore some aspects of the formal structure of quantum mechanics, in a commented overview of the work recently published on [1,2]. In the beginning of the sixties Schwinger had developed a program of treating quantum degrees of freedom characterized by a finite number of states, therefore with no classical counterpart. We start considering quantum systems described on Hilbert spaces of finite dimension. Let the set {|u k } denote the eigenstates of an arbitrary Hermitian operator acting on the space of interest. So, the states {|u k } might represent a multiplet in a closed shell, eigenstates of the quantized axis of spin, isospin, and so on. The index k is an integer which runs from 0 to N − 1.We define an operator V as:Where we use the notation convention |u j ≡ |u j(modN ) . Two integers a e b are said to be congruent modulo m when m divides the difference a − b to a integer l, so a = b + ml, Applying V s , where s is an integer, on a given |u j ,and we see that the action of integer powers of V shifts the index of the state, and in particular, on |u 0 it results on |u N −1 , 'connecting' the extremes of the set. Choosing s = N above and using the fact that j − N = j (modN ) we conclude that V is a cyclic operator, that is, V N =1. Let us look for the eigenstates (and associated eigenvalues) of the operator V,FromFor the eigenstates of V we obtain, once we decompose |v j in the {|u k } basis,where 1 √ N is an assumed real normalization factor. We can now construct, similarly to V , an operator acting on {|v j } as U |v j = |v j+s ,where modulo N notation is still assumed. As V lowers the index of the state |u j , U raises the index of |v j . Choosing the integer s = N we see that also U N =1. Finally, we can see that our initial states are precisely the eigenstates of U,