- #1
cire
A is an operator, in the expression <m|A|n>|m><n|, can I insert the matrix element <m|A|n> between the |m> and <n| like:
|m><m|A|n><n|?
|m><m|A|n><n|?
n>Tom Mattson said:Remember that <m|A|n> is just a complex number, so it commutes with everything you can imagine, including bras and kets. So placing it between |m> and <n| doesn't result in an illegal expression, but it is a strange way of writing it.
It's almost like writing the expression "x2y" instead of "2xy" in algebra. There's nothing wrong with either expression, but how often do you see coefficients sandwiched inbetween variables?
I thought <m|A|n> could be written as [itex] A_{mn} [/itex] which constitutes a matrix and thus will not commute with everything.
Operator algebra is a branch of mathematics that deals with the study of operators, which are mathematical objects that transform one mathematical object into another. It is a powerful tool used in various fields of mathematics and physics, such as functional analysis, quantum mechanics, and differential equations.
There are several types of operators in operator algebra, including linear operators, self-adjoint operators, unitary operators, and normal operators. Linear operators are those that preserve linearity, self-adjoint operators are those that are equal to their own adjoint, unitary operators are those that preserve the inner product of vectors, and normal operators are those that commute with their adjoint.
Operator algebra is essential in quantum mechanics because it provides a mathematical framework for understanding the behavior of quantum systems. Operators in quantum mechanics represent physical observables, and their algebraic properties play a crucial role in the predictions and interpretations of quantum phenomena.
Operator algebra is used in signal processing to analyze signals and systems. It allows for the representation of signals and systems as operators, which can then be manipulated using algebraic techniques to perform operations such as filtering, modulation, and demodulation.
Operator algebra has various applications in different fields of mathematics and science. Some examples include quantum mechanics, signal processing, control theory, differential equations, and functional analysis. It is also used in areas such as data analysis, machine learning, and image processing.