In quantum mechanics, operators are mathematical entities that represent physical observables, such as position, momentum, and spin. These operators act on quantum states to produce new states or to extract information about the system. In the formalism of quantum mechanics, operators are typically represented by matrices.
Matrix multiplication is used in quantum mechanics because it provides a convenient and efficient way to describe the action of operators on quantum states. When an operator acts on a quantum state, it transforms the state into a new state. This transformation can be represented by a linear transformation, which can be conveniently represented using matrices.
Matrix multiplication allows us to combine the effects of multiple operators by multiplying their corresponding matrices. For example, if we have two operators A and B, their matrix representations are denoted as matrices A and B, respectively. The composite operator representing their successive action, denoted as AB, corresponds to the matrix multiplication of matrices A and B.
Using matrix multiplication for operator composition has several advantages. First, it allows for the concise representation of complex quantum systems involving multiple observables and states. Second, it enables the formulation of quantum mechanical equations in a compact and elegant way. Finally, matrix operations have well-defined algebraic properties, such as associativity and distributivity, which make them mathematically tractable and amenable to analysis.
It's worth noting that while matrix multiplication is a powerful tool for representing and manipulating operators in quantum mechanics, there are alternative mathematical formalisms, such as Dirac's bra-ket notation and the framework of Hilbert spaces, that provide a more abstract and concise representation of quantum states and operators. However, even in these alternative formalisms, matrix representations are often used for practical calculations and numerical simulations.