Dynamic variables, wavefunctions, operators

In quantum mechanics dynamic variables are the same as in classical mechanics time (t), positional coordinates, x = (x,y,z), linear momentum p=/nv, and angular momentum M = R a p. Energy plays a leading role in QM just as it does in classical mechanics. 

In classical mechanics, dynamic differential equations are solved to obtain trajectories. In QM the behavior of bodies is described by a state function or wavefimction, y, z, t). The quantity 

The wavefimction must be single-valued and continuous, and its derivatives must also be continuous. Another important condition is  

The integral represents a summation of infinitesimal probabilities over the entire space, and therefore cannot be infinite - actually, there are ways of ensuring that its value be equal to 1, as an integral probability should (the normalization of the 

A pragmatic definition of an operator is a procedure that transforms a function into another function. Thus, 3 is a number but 3 times is an operator that transforms the function 3x -l- 1 into the function 9x -I- 3. The operator derivative with respect to x , 9/9x, transforms the function cos(x) into the new function -sin(x). When two operators act upon a given function, the result depends in general on the order of application if not, it is said that the two operators commute. 

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Postulate II For every dynamical variable, there is an associated mathematical operation. Furthermore, if there is an eigenvalue for such an operation and a particular wavefimc-tion, that eigenvalue is the result that would be obtained from measuring that dynamical variable for the system having this particular wavefunction. This explains, in part, how we can glean a picture of a quantum mechanical system from its wavefunction. An example, so far, is that the operation associated with the square of the momentum has an eigenvalue of Qi/ky for the free particle wavefunction, A(x). We extract (h/Kf from the wavefunction by applying the associated operator. 

The dynamical variables in the classical Hamiltonian are position and momentum coordinates. As long as wavefunctions are developed as functions of position coordinates, as was the case for A(x), the mathematical operation associated with a position coordinate is just multiplication by the coordinate. The operation associated with the momentum variable Pi that is conjugate to a Cartesian position variable qt is... 

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