I am trying to solve an applied physics problem with the expectation value of an operator associated with the problem. However, I have too many unknowns and as a step in moving forward, I needed to compute for a function associated with the problem.
I am not quite sure how to address the problem and I would really appreciate help in terms of direction. I would also appreciate information on relevant texts to consult. Kindly refer to the attachment for few explanations.
I think you should first clarify what you mean the variable x. Since you integrate over it, does not depend on x. Presumably in Ix x is a component, but is that the same as the x in f(x)? Certainly you can not derive (2) from (1) the way the integral is written. If it is an indefinite integral, is not an expectation value.
Suppose I make it simpler : A = \int_a^b dx f(x). Then A is a number and its derivative with respect to x would be 0. If you mean something like A(x) = \int_0^x dx' f(x'). the derivative of A(x) would indeed be f(x). If it is an indefinite integral it just shows that you could write it either as an integral or as a differential equation.
Thank you Professor Van der Zwan. I really appreciate your kind response. I have made clarifications in the attachment (A new file)!
The remarks I made earlier still apply. However, I looked at the references (of which I only have Slichter at hand) to try to understand what you mean. I found an expression in Slichter, eq. 2.12), similar to yours. Note, however, that there is no x in the function Ψ(t) only t, and that the integration variable τ in that case is symbolic as explained in the footnote on p 14, to denote the inner product of the wave functions.
The wave functions are determined as eigenfunctions, or linear combination of eigenfunctions, of the Hamiltonian of the system, not from eq. (2.12)!
Thank you Professor. I have just sent the other books into your inbox. Yes you are right, I have been having challenges with how to place the integration variable τ. The problem I wish to apply the solution to would be meaningless if the integral has infinite upper limit; it has to be finite. I have already obtained analytical function for the expectation value (which can be easily differentiated or integrated as the situation requires); the challenge (which concerns the evaluation of any existing commutation relations; as shown in Principles of nuclear magnetism by A. Abragam, 1961, page 23) now is how to handle the right hand side of equation (6) in order to obtain f(x) in terms of the analytical function. I am really sorry if I am not asking the right questions!
I don't think I understand your question.
Again, in Slichter's τ is not an integration variable in the case of spin variables, it is a symbolic notation for the inner product, just as for instance if you use Dirac notation. The commutation relations on p. 23 of Abragam are used to show what the time evolution is of the operator Ix. Maybe I don't understand you properly, but the eigenfunctions of the Hamiltonian in a spin system have no position dependence. If you mean the parameter t as in Slichter's expression, the time dependence is governed by the Heisenberg equation, of which Abragam basically gives the solution (Slichter also, on p. 14, but somewhat more convoluted).
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