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**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:

**R**φ(x) = λφ(x)

After a little trial and error he defines:

φ(x) = exp(2πikx)u(x)

(The details in the exponential are for mathematical convenience in further operations and is fairly elastic)

He then goes on to verify the solution

**R**φ(x) = exp(2πikR)φ(x)

**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.**