## Effect of sinusoid operators

Posted by Peeter Joot on May 23, 2010

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# Problem 3.19.

[1] problem 3.19 is

What is the effect of operating on an arbitrary function with the following two operators

On the surface with and it appears that we have just

but it this justified when the sinusoids are functions of operators? Let’s look at the first case. For some operator we have

Can we assume that these cancel for general operators? How about for our specific differential operator ? For that one we have

Since the differentials commute, so do the exponentials and we can write the slightly simpler

I’m pretty sure the commutative property of this differential operator would also allow us to say (in this case at least)

Will have to look up the combinatoric argument that allows one to write, for numbers,

If this only assumes that and commute, and not any other numeric properties then we have the supposed result 1.3. We also know of algebraic objects where this does not hold. One example is exponentials of non-commuting square matrices, and other is non-commuting bivector exponentials.

# References

[1] R. Liboff. *Introductory quantum mechanics*. 2003.

## On commutation of exponentials « Peeter Joot's Blog. said

[...] while working a Liboff problem, I wondered about what the conditions were required for exponentials to commute. In those problems [...]