When we take the conjugate of the eigenfunction - obviously that means we swap the sign of the imaginary component in math speak, but what does that represent in the physical universe...what are we manipulating with the particle's properties that some how leads to the correct value that we seek?
Nothing is being manipulated physically; it's all mathematical. The wave function is a mathematical tool/model we use to describe physical systems. The particle doesn't care how we model it or how we interpret those models.
Note that the same thing happens classically; we are just so used to the classical concepts wet don't even realize it. For example, vectors are mathematical tools used to describe things like velocity and forces, and yet there are not physical arrows moving around we objects. We can use the mathematical vector tools to model reality and describe how it behaves, but they are still mathematical. Dot products, cross products, etc. are just mathematical manipulations that the physical models tell us are physically meaningful (for good reason).
Can any one explain why that operation is required for it to work?
No one can say why it is required to work; one can only say that it does work based on experimental verification. Had we done experiments and found that these postulates of quantum mechanics didn't describe the world around us, then we would have scrapped them and moved onto something else. But there is no reason the formalism should work so well; it just does. And once you get used to quantum mechanics, relating the model to the physical becomes just like it does for classical physics.
to take a conjugate of a complex wave function feels very less connected to the actual physical phenomena compared to our classical stuff - it's hard to understand the mechanisms going on as I write out the math.
I agree; it is definitely harder to grasp. But once you are familiar with how the postulates of quantum mechanics (as well as the linear algebra) relate to the physical world, then it becomes a littler easier to see. For your particular example, we can first recognize the following two points
- Observables in QM each have an associated (Hermitian) linear operator, and their eigenvectors correspond to states of a definite value of that observable. For example, if $A$ is our operator corresponding to some observable (say, Energy), then the eigenstates $|a_i\rangle$, which satisfy $A|a_i\rangle=a_i|a_i\rangle$, are states with a definite value of $a_i$ for that observable. Two important properties of the vectors $|a_i\rangle$ are orthogonal, i.e. the inner product $\langle a_i|a_j\rangle$ is $0$ for $i\neq j$, and they are complete, i.e. $\sum_i|a_i\rangle\langle a_i|=1$
- The state vector $|\psi\rangle$ encodes all of the "information" of the particle. More specifically, we can express the state vector as a superposition of "basis vectors" $|a_i\rangle$ corresponding to states of definite $a_i$ (this could be any physical observable, such as Energy).
and the complex values $c_i$ tell us the probability of measuring value $a_i$ if we were to make a measurement of that observable on our system in state $|\psi\rangle$: $P(a_i)=|c_i|^2$. We can obtain the $c_i$ values by exploiting the orthogonality of the $|a_i\rangle$ vectors: $$\langle a_i|\psi\rangle=\sum_j\langle a_i|c_j|a_j\rangle=c_i$$
Once you grasp these two aspects of QM, then expectation values fall out. By definition, the expectation value of a random variable $X$ is given by $E[X]=\sum_iP(X=x_i)\cdot x_i$, where $x_i$ are all of the values the random variable $X$ can take. So in our case, the expectation value $\langle A\rangle$ of measurements of $A$ can be written as.
\langle A\rangle & = \sum_i|c_i|^2\cdot a_i \\
& = \sum_i|\langle a_i|\psi\rangle|^2\cdot a_i \\
& = \sum_i\langle\psi|a_i\rangle\langle a_i|\psi\rangle\cdot a_i \\
& = \sum_i\langle\psi|a_i|a_i\rangle\langle a_i|\psi\rangle \\
& = \sum_i\langle\psi|A|a_i\rangle\langle a_i|\psi\rangle \\
& = \langle\psi|A\left(\sum_i|a_i\rangle\langle a_i|\right)|\psi\rangle \\
& = \langle\psi|A|\psi\rangle