- #1

deuteron

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- 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?

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?