I have looked at the time development of the creation and annihilation operators for a mode of the quantized free electromagnetic field. It was assumed that the equations of motion for these operators were given by the usual prescription in the Heisenberg picture, for operators which do not include the time, t, explicitly. That is, that their time rate of change is proportional to their commutators with the Hamiltonian. See the attachment equations (1) and (2).
The solutions to these equations are easy to write down, IF , it is assumed that the 'a' operators have behaviours, under the appropriate differential and integral calculus operations, that is analogous to the behaviour of functions of a variable, t, under differentiation and integration. See the attachment, where the solutions of equations (1) and (2) or equivalently of equations (3) and (4), are expressed as in equations (5) and (6).
This suggests that there is some sort of isomorphism at play here between
1) some set of linear operators, together with the appropriately defined operations of differential and integral calculus of these operators, the Frechet derivative of an operator may come to mind here, and
2) some set of functions of t, together with the relevant operations of differential and integral calculus.
Could someone explain "this" isomorphism in detail?
Perhaps someone could specify the precise isomorphism involved, or perhaps could give a reference to solving equations involving operator valued functions in quantum optics, or a more general reference?
Apart from this first paragraph there is no mention of 'isomorphism' or 'Frechet derivative' in this answer. The answer concentrates upon finding an acceptable solution to a "fleshed out" version of equation (3), for (3) see the attachment to the question posed. This fleshed out version of equation (3), is equation (1), of the pdf attached to this post.
An acceptable solution to a fleshed out version of equation (4), may be found similarly.
This answer is essentially the same as my first answer but I have introduced an abstract notation for coefficients such as Cn. Please see attached pdf.
Please find various latex sources attached, in the zip file.
I eloquently refer to the file mentioned in the link below.
http://tuvalu.santafe.edu/~moore/esa-talk.pdf
In the question, one thing I asked for was "a reference to solving equations involving operator valued functions". I have not found anything that explains what goes on in any detail, however I have found a reference that seems to use the idea that 'operators' behave as 'functions'.
This reference is packaged in the attached pdf, where various links are given to MIT web pages. The reference given is at the web address entitled 'Quantum Dynamics (PDF)'.
It appears to use some isomorphism between ' functions + differential equations for functions' and 'operators + differential equations for operators'. See getting to (3.40) from (3.39) and getting to (3.53) from (3.52).
In this answer I correct the earlier proof that I gave. I mistakenly did not use a general enough solution to a certain differential equation, at one stage of the proof, this has been corrected. An initial condition at t=0 is then used to pick out appropriate solutions.
At t=0, Heisenberg picture operators and Schrodinger picture operators for a dynamical variable are equivalent, this is the basis of the initial condition.
Please find the answers pdf and corresponding latex source, attached.
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