Linear Algebra in Dirac Notation

[guyvsdcsniper]

Homework Statement

Convert general notation of linear operators in vector space to dirac notation

Relevant Equations

Dirac Notation

I am trying to convert the attached picture into dirac notation.
I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ>
The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS.

*Was going to type in LaTex but I can't preview my code during my intial post? is that normal?*

 

[topsquark]

Homework Statement:: Convert general notation of linear operators in vector space to dirac notation
Relevant Equations:: Dirac Notation

View attachment 313611

I am trying to convert the attached picture into dirac notation.
I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ>
The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS.

*Was going to type in LaTex but I can't preview my code during my intial post? is that normal?*

Note that $< \alpha , \beta > = \alpha ^{ \dagger } \beta$.

Hint: $< \psi , a \phi > = \psi ^{ \dagger } (a \phi) = ( \psi ^{ \dagger } a ) \phi$

The system, for some reason, occasionally flubs the LaTeX if you are writing the first LaTeX in the thread. Copy your text to the clipboard (for safety) and refresh the page. It should work after that.

-Dan

 

[PeroK]

Homework Statement:: Convert general notation of linear operators in vector space to dirac notation
Relevant Equations:: Dirac Notation

View attachment 313611

I am trying to convert the attached picture into dirac notation.
I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ>
The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS.

*Was going to type in LaTex but I can't preview my code during my intial post? is that normal?*

First, $\psi$ and $\varphi$ are vectors, which map to kets. And $a$ is an operator, with $a^{\dagger}$ its Hermitian conjugate. So, we have:
$$a\varphi \leftrightarrow a\ket{\varphi}$$$$a^{\dagger}\psi \leftrightarrow a^{\dagger}\ket{\psi}$$Now, to form the inner product in Dirac notation, we need to map the first ket to its correspondng bra:
$$a^{\dagger} \ket \psi \to \bra{\psi} a$$So, we can see that both the RHS and the LHS of the original linear algebra map to the same thing in Dirac notation:
$$\langle \psi, a\varphi \rangle \to \bra \psi a \ket \varphi$$$$\langle a^{\dagger}\psi, \varphi \rangle \to \bra \psi a \ket \varphi$$PS this is not really homework as it's just an explanation of the Dirac notation itself.