# Why not just complex conjugate bras and kets instead of Hermitian conjugate?

I read that one equation involving bras, kets and operators, implies another equation (its transpose conjugate), analogous to how one equation involving complex numbers implies its complex conjugate equation.

I'm not seeing the need of the transpose though. Why not just only do the conjugate? As an example:

Let's consider an inner product equation $$\langle A|B\rangle=z$$, where $$z$$ is a complex number, $$\langle A|$$ is the bra $$(a,b,c)$$, $$|B\rangle$$ is the ket $$(d,e,f)$$.

This equation basically expresses this relation between a bunch of complex numbers:

$$ad+be+cf=z$$

If we simply conjugate every complex number inside the bras and kets, we get this implied relation:

$$a^*d^*+b^*e^*+c^*f^*=z^*$$

The transpose conjugate equation, $$\langle B|A\rangle=z^*$$, also expresses the same relation as above but in a notationally flipped manner.

So what's the need of the transpose?

• The need for the transpose appears when you consider what happens when you change bases – SolubleFish Jun 4 at 6:11
• // starts humming a certain tune from Fiddler On The Roof .... – Carl Witthoft Jun 4 at 12:10
• @CarlWitthoft What's the reference? – Ryder Rude Jun 4 at 12:53
• @RyderRude the song is "Tradition," and includes the intro speech 'You may ask, why did this tradition get started? I'll tell you why - I don't know. But it's a tradition, and because of our traditions, everyone knows who he is and what God expects him to do' – Carl Witthoft Jun 4 at 13:58

$$|\psi\rangle\rightarrow \begin{pmatrix} a\\ b \end{pmatrix}$$ $$\langle \psi|\rightarrow \begin{pmatrix} a^*&b^* \end{pmatrix}$$ In this way the inner produce can be written as $$\langle \phi|\psi\rangle \rightarrow \begin{pmatrix} c^*&d^* \end{pmatrix} \begin{pmatrix} a\\ b \end{pmatrix}=ac^*+bd^*$$ $$\langle \psi|\phi\rangle \rightarrow \begin{pmatrix} a^*&b^* \end{pmatrix} \begin{pmatrix} c\\ d \end{pmatrix}=ca^*+db^* =(ac^*+bd^*)^*$$ As expected. That explains the need of taking transpose. The reason for taking conjugate is that we want $$\langle \psi|\psi\rangle$$ to be positive which represent the length of a vector.