Regarding the eigenvalues of the translation operator

[otaKu]

I don't understand what the eigenvalue of a translation operator means physically. The eigenvalues of other operators like momentum and hamiltonian give us the physically measurable values I suppose. Then what exactly do we obtain by the translation eigenvalues?

I am new to the field of quantum mechanics and as far as what I understand up until now the eigenvalue of an operator give us the physically measurable value of that operator.(Please correct me if I am wrong.)
So I was reading about the eigenfunctions of translation operator(the Bloch functions I mean) for a simple 1D chain of atoms extended upto infinity by the Born von Karman boundary condition(φ(0)=φ(l)) where L is the length of the chain. The author says that the eigenvalue equation of translation operator can be written as follows:

Rφ(x) = λφ(x)​

λ being a constant eigenvalue I presume.
After a little trial and error he defines:

φ(x) = exp(2πikx)u(x)​

where k is some number and u(k) is another function with the same periodicity as the lattice(also known as cell function in solid state physics)
(The details in the exponential are for mathematical convenience in further operations and is fairly elastic)
He then goes on to verify the solution

Rφ(x) = exp(2πikR)φ(x)​

and since it is of the form as the first the formulated eigenfunction seems valid.

I cannot visualise the translational operator properly. I mean I am confused as to what this eigenvalue means. I know I still need to normalise it using the identity translation but I am still confused. In fact I don't even know what's getting me so worked up. Any sort of help or advice will be welcomed.

 

[Mr-R]

The translation operator translates the quantum state itself by a certain amount.
Say we have the translation operator $T(a)$ that translations the quantum state $|x\rangle$ which gives the eigen value $x$ when acted upon with the operator $\hat{x}$, by distance $a$ along the $x$ axis. Then:$$\hat{x}T(a)|x\rangle=(x+a)T(a)|x\rangle$$
Which implies: $$T(a)|x\rangle=|x+a\rangle$$
Makes sense? My first line is what is confusing you. I hope

 

[blue_leaf77]

I don't understand what the eigenvalue of a translation operator means physically.

Translation operator is not a Hermitian operator, therefore you should not expect its measurement is feasible.

 

[otaKu]

Thanks for the answers. I think I get it now.