# Understanding Dirac notation - Product of ops. is product of matrices

• ThereIam
In summary, the conversation discusses the difficulty in understanding the Dirac notation proof and how the author is demonstrating the equality of the product of two operators to the product of their corresponding matrices. The equation (Ωλ)_{ij} = <i|Ωλ|j> = <i|ΩIλ|j> = \sum(over k) <i|Ω|k><k|λ|j> = \sum (over k) Ω_{ik}λ_{kj} is also mentioned as the part the speaker is struggling with.
ThereIam

## Homework Statement

This makes intuitive sense to me, but I am getting stuck when trying to read the Dirac notation proof.

Anyway, the author (Shankar) is just demonstrating that the product of two operators is equal to the product of the matrices representing the factors.

## Homework Equations

(Ωλ)$_{ij}$ = <i|Ωλ|j> = <i|ΩIλ|j> =

(and this is the part I don't understand)

$\sum$(over k) <i|Ω|k><k|λ|j> = $\sum$ (over k) Ω$_{ik}$λ$_{kj}$

I apologize if the formatting didn't work. I'll try to fix it asap.

## The Attempt at a Solution

Could you be more precise as to what you don't understand?

## What is Dirac notation?

Dirac notation is a mathematical notation used to represent vectors and operators in quantum mechanics. It was developed by British physicist Paul Dirac and is commonly used in quantum physics and related fields.

## What is the product of operators in Dirac notation?

The product of two operators in Dirac notation is represented by the order in which they are written, with the operator on the right acting on the operator on the left. This is similar to the concept of function composition in mathematics.

## How is the product of operators different from the product of matrices?

The product of operators in Dirac notation is different from the product of matrices in that it is not commutative - the order in which the operators are multiplied matters. This is due to the non-commutative nature of quantum mechanics, where measurements of different properties of a system do not necessarily commute with each other.

## What is the significance of the product of operators in quantum mechanics?

The product of operators is important in quantum mechanics as it allows us to describe the evolution of a quantum system in time. By applying a sequence of operators, we can determine the final state of the system and make predictions about the outcomes of measurements.

## How can I understand and use Dirac notation effectively?

To understand and use Dirac notation effectively, it is important to have a strong grasp of linear algebra and complex numbers. It is also helpful to practice simplifying and manipulating expressions in Dirac notation. Working through examples and exercises can also aid in developing a deeper understanding of the notation and its applications.

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