- #1
gulfcoastfella
Gold Member
- 99
- 1
I took a semester of QM as an undergrad engineering major, and I don't recall the motivation for replacing traditional vector notation with bracket notation. Can someone enlighten me? Thank you.
gulfcoastfella said:I don't recall the motivation for replacing traditional vector notation with bracket notation. Can someone enlighten me? Thank you.
And the good news is that Dirac got it even almost mathematically rigorous. One has only to formalize it a bit. That's what's nowadays known under the name "rigged Hilbert space". Of course, you have to learn functional analysis, if you want QM mathematically rigorous. It's not necessary in all details for practitioners (physicists) of QM, but it also doesn't hurt to read a bit into it since it can help to understand better some finer details about the continuous spectra ("eigenvalues") of the self-adjoint and unitary operators, occurring in the formalism of quantum theory.dextercioby said:The brilliant invention of Paul Dirac (if I am not mistaking, this appears in the 1935 edition of his textbook) has the advantage of teaching people (such as you) Quantum Mechanics without worrying about the pesky details of Functional Analysis, which had been shown by the great John von Neumann to be the underlying mathematical theory of Quantum Mechanics.
gulfcoastfella said:I took a semester of QM as an undergrad engineering major, and I don't recall the motivation for replacing traditional vector notation with bracket notation. Can someone enlighten me? Thank you.
vanhees71 said:And the good news is that Dirac got it even almost mathematically rigorous. One has only to formalize it a bit. That's what's nowadays known under the name "rigged Hilbert space".
Strilanc said:Honestly I'm surprised that ket notation hasn't swept over all of linear algebra. It's a great tool for understanding.
For example, I always had trouble doing matrix multiplication by hand. "Is it row col or col row or... which dimensions have to match again?" But with kets that becomes downright trivial.
##|a\rangle \langle b|## is a transformation from ##a## to ##b## (or vice versa), and we can break any matrix ##M## into a sum of these single-value transformations ##M = \sum_{i,j} |i\rangle\langle j| M_{i,j}##. Correspondingly, ##\langle a | b \rangle## is a comparison (dot-product) between ##a## and ##b##. Our matrix breakdown has the property that ##\langle a | a \rangle = 1## while ##\langle a | b \rangle = 0## for ##b \neq a## so matrix multiplication is just...
lavinia said:- You use an inner product in order to eliminate the terms ##<j.k>## when ##j ≠ k##. Presumably if you change basis then you must redefine the inner product since these new bases may not be orthonormal. While this may help you to keep track of indices, it is conceptually complicated.
lavinia said:- Generally linear transformations are mappings between different vector spaces. By using only indices, your notation suggests that the ##|i>## ##|j>## ##|l>## and ##|k>## 's are the same basis in a single vector space.
lavinia said:- Once one has the proof that a linear transformation can be represented by a matrix with respect to a choice of bases, the proof that the composition of two linear transformations is the product of the two matrices falls out without reference to inner products. Inner products are an added structure imposed upon a vector space and are conceptually distinct from the idea of linear transformations themselves.
Strilanc said:Why would I, half-way through a matrix product, change the basis that I write matrices in? The same issue applies to the grid of numbers: if the numbers don't mean the same thing within each matrix, then you can't compose the two matrices via normal matrix multiplication.
Bracket notation, also known as Dirac notation, is a mathematical representation used in quantum mechanics to describe the state of a quantum system. It uses a combination of angle brackets and mathematical symbols to represent the state of a quantum system, making it easier to perform calculations and understand the behavior of quantum particles.
Bracket notation is used in quantum mechanics because it provides a concise and elegant way to represent the state of a quantum system. It also allows for simple calculations and manipulations, making it a useful tool for understanding the complex behavior of quantum particles.
Unlike traditional mathematical notation, bracket notation uses a combination of angle brackets and mathematical symbols to represent the state of a quantum system. It also allows for the representation of abstract quantities, such as quantum states and operators, which cannot be easily expressed in traditional notation.
The components of bracket notation include a ket vector, represented by |>, a bra vector, represented by < |, and a dual vector, represented by < |> These vectors are used to represent the state of a quantum system, the adjoint of a state, and the dual of a state, respectively. Operators, such as Hamiltonians and observables, are also represented using bracket notation.
Bracket notation is closely related to the principles of quantum mechanics, such as superposition and measurement. It allows for the representation of quantum states, which can be in a superposition of multiple states, and operators, which are used to measure and manipulate these states. Therefore, bracket notation is an essential tool for understanding and applying the principles of quantum mechanics.