Exploring Bra and Ket Notation: What is a Dual Vector?

In summary, bra and ket notation is a way of representing column and row matrices in quantum mechanics. The dot product is represented by <ψ|ψ> and |ψ><ψ| is also a product that produces a matrix. The concept of a "dual vector" is mentioned in the introduction of bra and ket notation, but can be better understood by reading the book "Quantum Mechanics" or the article "The Principles of Quantum Mechanics" by Paul Dirac.
  • #1
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This isn't a homework problem, but it's so simple that it belongs here.

Can someone please explain to me bra and ket notation? I've been consulting various books and they are all so abstract. Yesterday, my professor told me that a ket |ψ> represents a column matrix and a bra <ψ| represents a row matrix, and that's all there is to it. The (dot) product is represented by <ψ|ψ> rather than <ψ||ψ>, and |ψ><ψ| is also a product, though it produces a matrix.

Yes, this all makes sense in terms of matrices. So my first request is that someone explain this to me in modern, matrix notation.

What's this mysterious "dual vector" that all the books allude to in the introduction of bra and kets? Why isn't matrix notation, ψ for column matrix and ψT for row matrix good enough?
 
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  • #2
I would recommend you reading this:
www.users.csbsju.edu/~frioux/dirac/dirac.pdf
It helped me a lot and we got it from our QM-teacher as a supplement to the book by Brandsen and Joachain named Quantum Mechanics(Descriptive title)
 

Related to Exploring Bra and Ket Notation: What is a Dual Vector?

What is a dual vector?

A dual vector is a mathematical object that is closely related to a vector in a vector space. It is a linear functional that maps a vector to a scalar value. In other words, it takes in a vector as an input and outputs a number.

How is a dual vector different from a regular vector?

A dual vector operates on a vector to produce a scalar, while a regular vector operates on other vectors to produce a vector. Additionally, the basis for a dual vector space is the dual basis, which is the set of linear functionals that correspond to the basis vectors of the original vector space.

What is bra-ket notation?

Bra-ket notation is a notation system used in quantum mechanics to represent vectors and dual vectors. It uses the symbols (bra) and (ket) to represent the dual vector and vector, respectively.

Why is bra-ket notation useful?

Bra-ket notation is useful because it simplifies the notation for vectors and dual vectors, making it easier to manipulate and work with in mathematical expressions. It also allows for a more intuitive understanding of vector operations in quantum mechanics.

How is duality related to bra-ket notation?

Duality refers to the relationship between a vector and its dual vector. In bra-ket notation, the bra and ket symbols represent the dual vector and vector, respectively, highlighting this duality. This notation also makes it easier to identify and manipulate dual vectors in quantum mechanics.

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