I'm confused about bra-ket reversal, and why it requires conjugation.
$$
\langle \phi|\chi \rangle = \langle \chi|\phi \rangle ^*
$$
Or equivalently why
$$
|\chi \rangle ^* = \langle \chi |
$$
I tried to understand it with the following argument (which follows Feynman Lecture's Vol 3 Ch 5). If you have the Feynman lectures, I'm trying to understand how Feynman gets equation 5.26.
Lets say $R$ is a particular complete three-dimensional basis for the states our system can take (like for a spin-1 particle), with basis vectors $R_1,R_2,R_3$.
A state $\chi$ can be represented in this basis since the basis is complete:
$$
|\chi\rangle=\sum_{i=1,2,3}|R_i\rangle\langle R_i | \chi \rangle
$$
Now consider the quantity $$\langle \chi | \chi \rangle$$ Physically, we would say this is the complex amplitude for a system in state $\chi$ to be in state $\chi$. It is just a number (which could be complex if we haven't assumed $|\chi \rangle ^* = \langle \chi |$).
Since the probability associated with this amplitude is the norm (length) of the amplitude, we can say confidently that whatever number $\langle \chi | \chi \rangle$ is, its norm is one:
$$
\langle \chi | \chi \rangle^*\langle \chi | \chi \rangle=1
$$
In words, there is 100% probability the system is in the state it is in.
But it seems to me that only restricts it to the unit circle in the complex plane: $\langle \chi | \chi \rangle = e^{i\theta}$ where $\theta$ could be anything.
If we write $\langle \chi | \chi \rangle$ in our $R$ basis, we get
$$
\langle \chi | \chi\rangle=\sum_{i=1,2,3}\langle \chi |R_i\rangle\langle R_i | \chi \rangle = e^{i\theta}
$$
We see that each term $\langle \chi |R_i\rangle\langle R_i | \chi \rangle$ can only vary (for different choices of $R$) in ways that preserve the last equality, but it seems like requiring
$$
\langle R_i|\chi \rangle = \langle \chi|R_i \rangle ^*
$$
Is too strong a condition if $\theta$ is arbitrary. If $\theta=0$ or you needed to pick a value and stick with it, I could see why that condition may be required, but I'm not even sure about that. Or maybe I just don't see the argument.
Can anyone help with the reasoning here?
In Feynman lectures, he says effectively that because the probability for $\chi$ to be any state in $R$ is 1 (R is a complete basis)
$$
\langle R_1 | \chi \rangle \langle R_1 | \chi \rangle ^ * +
\langle R_2 | \chi \rangle \langle R_2 | \chi \rangle ^ * +
\langle R_3 | \chi \rangle \langle R_3 | \chi \rangle ^ * = 1
$$
and because the amplitude $\langle \chi | \chi \rangle = 1$…
$$
\langle \chi | R_1 \rangle \langle R_1 | \chi \rangle +
\langle \chi | R_2 \rangle \langle R_2 | \chi \rangle +
\langle \chi | R_3 \rangle \langle R_3 | \chi \rangle = 1
$$
Then because $R$ and $\chi$ can freely vary, I guess (?) each term could maybe vary independently requiring for each term that
$$
\langle \chi | R_i \rangle = \langle R_i | \chi \rangle ^*
$$
If this were not true, probability wouldn't be "conserved," and
particles would get "lost." -Feynman
but this does assume the amplitude in the second equation is $1$ (not $e^{i \theta} $ ) while it safely assumes the probability in the first is $1$.
Care to help me understand this argument?
Someone could instead prove that any choice other than $1$ would result in absurd physics. Or someone could explain that $\langle \phi|\chi \rangle = \langle \chi|\phi \rangle ^*$ results from a deeper symmetry.
Best Answer
Recall the definition of an inner product space, a vector space equipped with a map $n$ that returns a scalar for any pair of vectors. In quantum mechanics, the Hilbert space of states for any quantum system is always an inner product space with the interpretation you noted above: The inner product of a state $A$ on a state $B$, $\langle A|B\rangle$, is the probability amplitude of measuring the system to be in state $A$ given you have knowledge the system is in state $B$.
Rather than considering the set of vectors and a two vector map $n(w,v)$, we can take the perspective instead that there is a dual space. Physical state vectors are "kets" if you will, but there exists another vector space, called the dual space, which is filled with "bras". These bras form the space of linear maps from the kets to complex numbers (the scalars in quantum mechanics).
Notice that there is a one-to-one correspondence between the bras and kets since, by linearity, operators in the dual space are fully defined by their action on a complete set of basis vectors in the ket space. That means that we can have a notion of taking a ket to a bra and back, normally we denote that by $\dagger$, but also sometimes by $*$ or a bar.
And, again using the correspondence, we parameterize the space of bras by objects, $\langle x |$, with the property $\langle x | y \rangle = \delta_{xy}$. Immediately we can work out the transformation properties for an orthonormal basis of bras by considering the action of a properly normalized bra on a properly normalized ket
$$|x\rangle = a_1 |a_1\rangle + a_2 |a_2\rangle.$$
The dual vector must be given by
$$\langle x | = \bar{a}_1 \langle a_1 | + \bar{a}_2 \langle a_2| $$
if we are to have that
$$\begin{align}\langle x|x\rangle &= \bar{a_1} a_1 \langle a_1|a_1\rangle + \bar{a_2} a_2 \langle a_2|a_2\rangle + \bar{a_1} a_2 \langle a_1|a_2\rangle + \bar{a_2} a_1 \langle a_2|a_1\rangle \\ &= \bar{a_1} a_1 \langle a_1|a_1\rangle + \bar{a_2} a_2 \langle a_2|a_2\rangle + 0 + 0 \\ &= 1,\end{align}$$
which is required by orthonormality.