# Bra and Ket Representation in Dual Hilbert Space

• wasi-uz-zaman
In summary, kets and bras are different mathematical objects in Hilbert space. While they may appear to exist in the same space, they are actually back-to-front. This can be seen in the example of spin-1/2 states, where kets and bras are represented as column and row matrices, respectively. Therefore, we need dual Hilbert space for bras in order to properly represent them in the same space.
wasi-uz-zaman
Hi, if ket is 2+3i , than its bra is 2-3i , my question is 2+3i is in Hilbert space than 2-3i can be represented in same hilbert space, but in books it is written we need dual Hilbert space for bra?

hi wasi-uz-zaman!
wasi-uz-zaman said:
Hi, if ket is 2+3i , than its bra is 2-3i , my question is 2+3i is in Hilbert space than 2-3i can be represented in same hilbert space, but in books it is written we need dual Hilbert space for bra?

it's not the same, it's back-to-front!

you can't add a ket to a bra

it's like ordinary 3D vectors and pseudovectors (a 3D pseudovector is a cross product of two 3D vectors)

you can't add a vector to a pseudovector …

they look as if they exist in the same space, but in fact the two spaces are back-to-front

Kets are elements of a vector space, bras are linear functions defined on the kets - entirely different things.

They are, with a few caveats such as Rigged Hilbert Spaces, isomorphic via the Rietz-Fisher Theorem - but that doesn't mean they are the same.

Thanks
Bill

The easiest Hilbert space to deal with (or maybe the second easiest) is that of spin-1/2 states. Then the states are (or can be represented as) column matrices with 2 elements. The dual states are the row matrices with 2 elements. Obviously it doesn't make any sense to add a row and a column.

## 1. What is the concept of Bra and Ket representation in Dual Hilbert Space?

The concept of Bra and Ket representation in Dual Hilbert Space is a mathematical tool used to represent quantum states in quantum mechanics. It is based on the dual vector space structure of quantum mechanics, where the Hilbert space is split into two spaces - the Bra space and the Ket space. The Bra space contains the complex conjugate of the wavefunction, while the Ket space contains the original wavefunction. This representation makes it easier to perform mathematical operations on quantum states.

## 2. How is the Bra and Ket representation related to quantum measurements?

In quantum mechanics, measurements are represented by operators. The Bra and Ket representation allows us to represent these operators as matrices, making it easier to perform calculations and obtain observable quantities. The Bra and Ket notation also allows us to represent the measurement outcomes as inner products between the Bra and Ket vectors.

## 3. Can you explain the significance of the bra-ket notation in quantum mechanics?

The bra-ket notation, also known as Dirac notation, is a powerful tool in quantum mechanics that simplifies the representation of quantum states and operators. It allows for the easy calculation of inner products, expectation values, and probabilities. It is also used to represent quantum states in a basis-independent manner, making it easier to work with abstract mathematical concepts in quantum mechanics.

## 4. How is the bra-ket notation used to represent superposition in quantum mechanics?

In quantum mechanics, superposition is the principle that a quantum system can exist in multiple states simultaneously. The bra-ket notation allows us to represent these superposition states by using a linear combination of basis states. For example, a particle in a superposition state can be represented as a linear combination of two basis states, such as |0⟩ and |1⟩, where the coefficients represent the probability amplitudes of the particle being in each state.

## 5. What are the advantages of using bra-ket notation in quantum mechanics?

The bra-ket notation has several advantages in quantum mechanics. It simplifies the representation of quantum states and operators, making calculations easier and more intuitive. It also allows for the representation of superposition states and the calculation of probabilities in a basis-independent manner. Additionally, the bra-ket notation is widely used in quantum mechanics, making it easier for scientists to communicate and share their work.

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