Question

From the relation ⟨a|b⟩=⟨b|a⟩∗ it can be seen that we can get same inner product by swapping vectors in Hilbert space and taking complex conjugate of the resulting complex scalar. So it seems that ket and bra vectors are two sides of the same story. Then how bra vector ⟨a| is conceptually different from ket vector |a⟩ if both belong to the same Hilbert vector space ? To me, I see just one principal difference is that ket is column vector and bra - row vector. That's it. Are there any more radical differences

Answer 1

Bra and ket vectors are conceptually different in the context of Hilbert vector space. Ket vectors represent the state of a quantum system as column vectors, while bra vectors represent the dual space or the conjugate transpose of the ket vectors as row vectors.

Step-by-step explanation:

In the context of Hilbert vector space, bra and ket vectors are conceptually different. While they both belong to the same vector space, they represent different mathematical objects. Ket vectors |a⟩ are column vectors, representing the state of a quantum system, while bra vectors ⟨a| are row vectors, representing the dual space or the conjugate transpose of the ket vector.

The inner product between a bra vector and a ket vector, ⟨a|b⟩, can be computed by taking the complex conjugate of the ket vector and performing matrix multiplication. This inner product is a complex scalar.

In summary, the main difference between bra and ket vectors is their orientation and their role in representing the state of a quantum system and its dual space, respectively.