Quantum mechanics,Taylor series and integrals

[eljose]

Let,s suppose we have the operator f(q,p) with p and q are quantum operators tehn my question is if we develop f(p,q) into a power series:

$$f(q,p)=\sum_0^{\infty}a_n(q)p^{n}$$

my question is if i must symmetrizy the expresion a_n(q)p^n for each member
so:

$$a_n(q)p^n\rightarrow[a_n(q),p^n]$$

another question let be the integral of the operator x given by:

$$\int_0^{\infty}f(X)dx$$

is this justified or it cna not be done?..thanks.

 

[dextercioby]

To the second question,I don't see the connection between X & "x".To the first,yes,if they anticommute,they must be symmetrized wrt all possible equivalent classical combinations.The outcome,an operatorial function,must be self-adjoint,just like the inputs.

Daniel.

 

[R0man]

In quantum mechanics, operators are used to represent physical observables. The operator f(q,p) represents a physical quantity that can be measured, where q and p are quantum operators representing position and momentum respectively. When we want to develop f(q,p) into a power series, we can use the Taylor series expansion. However, since q and p are operators, we need to consider their non-commutativity when expanding the series.

To answer your first question, yes, you need to symmetrize the expression a_n(q)p^n for each term in the series. This is because q and p do not commute, meaning their order matters and the result of their multiplication depends on the order in which they are written. By symmetrizing, we ensure that the resulting series is independent of the order of q and p, making it a valid representation of the operator f(q,p).

As for your second question, the integral of an operator can be justified in certain cases. For example, if the operator represents a physical observable that can be measured, then the integral would be meaningful. However, in general, the integral of an operator may not have a physical interpretation and may not be a valid operation. It is important to carefully consider the context and properties of the operator before performing an integral.