A Remark on the Transformations between Different Representations

In quantum mechanics, the theory about the representation of state vector (or wave function) and the operator as well as the transformation between different representations is an important subject which attracts the remarkable interesting of a lot of researchers . There are two equivalent mathematical expressions for the transformation of representations, if F̂ is an operator of representing a physical quantity, U is a linear operator, the reverse matrix of the matrix of U exists for all of the wave functions, thus, the definition of the transformation of representations for the operator F̂ and its eigenfunctions from B to A representation is:


I. Introduction
In quantum mechanics, the theory about the representation of state vector (or wave function) and the operator as well as the transformation between different representations is an important subject which attracts the remarkable interesting of a lot of researchers [1][2][3][4] . There are two equivalent mathematical expressions for the transformation of representations, if F is an operator of representing a physical quantity, U is a linear operator, the reverse matrix of the matrix of U exists for all of the wave functions, thus, the definition of the transformation of representations for the operator F and its eigenfunctions from B to A representation is [5][6] representation. It can be proven that the transformation of representation is also an unitary transformation [5][6] , thus, , multiplying the transformation formula of operator in eq. ⑴ with U from the left and with 1  U from the right, moreover, multiplying the transformation of formula of eigenfunction in eq. ⑴ with U from the left it arrives Concerning the application of the transformation of representation, the teaching materials generally employ eq.⑴ [5][6] , but there are some researchers like to employ eq.⑵ [7][8] .
The key problem for the transformation of representation is how to determine its transformation matrix, if some mistakes or puzzles occurring in the transformation result, these are usually caused by incorrect transformation matrix employed. Taking the electronic spin component operators and their eigenfunctions as an example, the paper will present some discussions and studies by use of eq.⑴.

Transformation of Operator
A typical application is the transformation of electronic spin operators ) , , With respect to the transformation of representation of operators, some people obtained the transforming matrix from the transforming formula of the eigenfunctions of one of the spin operators from z S to x S representation [ 7 ] . For example, using the matrix of x S in eq. ⑶ to solve the equations of its eigenvalues, it obtains the eigenfunctions of where the subscripts . According to the assumption of electronic spin [6] , the eigenfunctions of If using eq.⑴ to perform the transformation, thus It is easy to prove that the transformation of representation does not change the commutation relations between representation are given by [7]  

According to the commutation relation between
x Sˆ and z Ŝ , the transformation result of eq. ⑻ should be just to exchange the matrix of z S with that of x S in eq.⑶ [8] , but a negative symbol occurs in z S  of eq.⑻. The negative symbol of the transformation results in Ref. [7] occurs in the matrix of y S  , it should be a special case, because the reverse matrix of its transformation matrix is just as same as itself.

Transformation of Eigenfunctions
If similar to the transformation of representation of operator eq.⑻, using 1  U of eq. ⑺ to transform the eigenfunctions of That the reason of the mistakes occurs in eq.⑼ is due to the incorrect transformation matrix employed. The eigenfunctions of z Ŝ in the representation of itself are the totally same as eq.⑸, using eq.⑷ and eq.⑸, the transformation of representation for the eigenfunctions of z Ŝ from z S to x S representation are given by . ⑾ Therefore, with respect to the transformation of eq.⑼, we must take the 1  U of eq.⑾ as the transformation matrix, it is just the reverse matrix of the 1  U of eq.⑺.
Thus, if using eq.⑾ for the transformation of