Motivation behind the Operator-Formalism in QM

[deuteron]

TL;DR Summary

What is the motivation behind the eigenvalue equations corresponding the the observables in QM?

I have a problem understanding the motivation behind why all observables are represented via a hermitian operator.
I understand that from the eigenvalue equation
$$ \hat A\ket{\psi} = A_i\ket{\psi}$$
after requiring that the eigenvalues be real, the operator $\hat A$ needs to be hermitian.
However, I do not understand the motivation behind the eigenvalue equation in the first place, from where do we come to this? Why do we require, that applying an operator on an eigenstate would correspond physically to the measurement of the corresponding eigenvalue?
As far as I have understood, the steps on creating the mathematical formalism are:
- observe experimentally that the measurement of a particle gives discrete values for the same measurement
- deduce that the particle must be in the superposition of the states $\ket{\phi_i}$ corresponding to the measured values
$$ \ket{\Psi} = \displaystyle\sum_i c_i\ket{\phi_i}$$
- observe that we have a value-state pair
- (this is the step I don't understand)
- create the eigenvalue equation where the operator applied to the eigenstate gives the measured value times the state

Is there a physical motivation behind the eigenvalue equation, or is there another set of axioms, from which the eigenvalue equation can be derived mathematically?

 

[PeterDonis]

I do not understand the motivation behind the eigenvalue equation in the first place, from where do we come to this?

If $\hat{A}$ represents an observable (and in principle any Hermitian operator can represent an observable), then a state $\ket{\psi}$ that satisfies the eigenvalue equation for $\hat{A}$ will have a definite value for that observable, whereas a state that doesn't satisfy the eigenvalue equation won't. Knowing which states have definite values for observables is very valuable in analyzing problems in QM.

 

[anuttarasammyak]

However, I do not understand the motivation behind the eigenvalue equation in the first place, from where do we come to this? Why do we require, that applying an operator on an eigenstate would correspond physically to the measurement of the corresponding eigenvalue?

Let me say what I think for MOTIVATION you say. In CM we feel no necessity to distinguish between physical state and physical value. So the state is (a gourp of ) numbers, e,g, coordinates, energy value, x,y,z-components of momentum, etc. In QM we know the states are vectors which is used to express priciple of superposition. Then how do we get physical value from the state vectors ? Usually in mathematics by making inner products with itself we get number scalar from vector.
$$<\xi'|\xi'>$$
But here there's no information what physical value we measure, x, p, E? So this plain self inner product has no physical meaning. we make it 1.
$$<\xi'|\xi'>=1$$
In statistical physics we use distribution function for getting mean value, variance, etc. of any physical variables by multipling them, e.g. x,p,E to distribution function and make whole integration. In analgy we try. First for measurement of physical variable $\xi$
$$\xi<\xi'|\xi'>=\xi$$
this is physical qunatitiy measurement itsellf with no information of states so meaningless. We must sandwith $\xi$. Say $|\xi'>$ is state vector which correspond to the value $\xi'$ for measuring $\xi$, we expect
$$<\xi'|\xi|\xi'>=\xi'=\xi'<\xi'|\xi'>=<\xi'|\xi'|\xi'>$$
Comparing the most LHS and the most RHS we expect the relation
$$\xi|\xi'>=\xi'|\xi'>$$
Now we know mathematically $\xi$ is an operator which transform one vector state to another.

 

[deuteron]

Let me say what I think for MOTIVATION you say. In CM we feel no necessity to distinguish between physical state and physical value. So the state is (a gourp of ) numbers, e,g, coordinates, energy value, x,y,z-components of momentum, etc. In QM we know the states are vectors which is used to express priciple of superposition. Then how do we get physical value from the state vectors ? Usually in mathematics by making inner products with itself we get number scalar from vector.
$$<\xi'|\xi'>$$
But here there's no information what physical value we measure, x, p, E? So this plain self inner product has no physical meaning. we make it 1.
$$<\xi'|\xi'>=1$$
In statistical physics we use distribution function for getting mean value, variance, etc. of any physical variables by multipling them, e.g. x,p,E to distribution function and make whole integration. In analgy we try. First for measurement of physical variable $\xi$
$$\xi<\xi'|\xi'>=\xi$$
this is physical qunatitiy measurement itsellf with no information of states so meaningless. We must sandwith $\xi$. Say $|\xi'>$ is state vector which correspond to the value $\xi'$ for measuring $\xi$, we expect
$$<\xi'|\xi|\xi'>=\xi'=\xi'<\xi'|\xi'>=<\xi'|\xi'|\xi'>$$
Comparing the most LHS and the most RHS we expect the relation
$$\xi|\xi'>=\xi'|\xi'>$$
Now we know mathematically $\xi$ is an operator which transform one vector state to another.

thank you! this was exactly what I was trying to understand!