# Adjoint ket vectors in Dirac notation?

In books on quantum physics you often see things like $\left(A|\Psi\rangle\right)^\dagger=\langle\Psi|A^\dagger$. However, $|\Psi\rangle:=\Psi\in\mathcal{H}$ is a vector and $\langle\Psi|:=\Psi^\ast\in\mathcal{H}^\ast$ is its dual counterpart, while $A$ is an operator, i.e. $A\in Aut(\mathcal{H})$. From what I understand, the property 'adjoint' is only defined with respect to operators.

Let $A,B:\mathcal{H}\rightarrow\mathcal{H}$ be operators. Then $B=A^\dagger$, iff $\langle B\Psi,\Phi\rangle=\langle\Psi,A\Phi\rangle$ for all $\Psi,\Phi\in\mathcal{H}$.

Then how is $|\Psi\rangle^\dagger$ defined and justified as notation and why do we have $|\Psi\rangle^\dagger=\langle\Psi|$?

• Have you seen this? Jan 13, 2017 at 19:34
• No, but it is of great help to me. Thanks a bunch! Jan 13, 2017 at 19:35
• Glad to hear that! Jan 13, 2017 at 19:36
• You can view a bra as a linear functional that takes a vector in a hilbert space and gives you a number. There are bras that take a basis ket and spit out a number one for example...but for all the other basis kets, they spit zero. These make a dual basis in a dual space of these linear functionals. So this dual space is also a vector space. Jan 13, 2017 at 21:59
• Rigorously, let $\Psi \in H$ with $H$ a Hilbert space. Then $H^\star= \mathrm{Hom}(H,\mathbb{C})$ and $\Psi^\dagger \in H^\star$. In physics then we often dress the elements of $H$ and $H^\star$ as kets and bras without much gain actually. Sep 29, 2019 at 20:13

A vector is an arrow in spatial dimensions, it is not related to states in a hilbert space of the quantum mechanics. "Adjoint" is defined only in relation to the inner product of the states, not the operators. What you make us of when writing things like $|\Psi\rangle^\dagger=\langle\Psi|$ is the inner product of your complex state space, which relates complex conjugacy to the symmetries of said inner product !
• No, that's wrong. A vector space is concisely defined in mathematics: link. The analogy to our space in reality makes sense for the vector space $\mathbb{R}^3$ with the standard scalar product and the Euklidean norm. In quantum physics the span of a vector in a Hilbert space, i.e. a one-dimensional subspace of the Hilbert space is called a state. Jan 13, 2017 at 19:59