Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 11.
We’ve seen that the translation operator in quantum mechanics can be derived by considering the translation to be an active transformation, that is, a transformation where the state vectors, rather than the operators, get transformed according to
Using this approach, we found that
so that the momentum is the generator of the transformation.
We can also derive using a passive transformation, where the state vectors remain the same but the operators are transformed according to
This is equivalent to an active transformation since
As before we start by taking
where is some Hermitian operator, so that . Plugging this into 3 we get, keeping only terms up to order :
Since we see that
The extra is there because any function of alone commutes with , so
We can eliminate by considering 4.
Thus we must have , which means that must be a function of alone. This means that the most general form for is , but there’s nothing to be gained by adding some non-zero constant to , so we can take . Thus we end up with the same form 2 that we got from the active transformation.
Translational invariance is the condition that the Hamiltonian is unaltered by a translation. In the passive representation this is stated by the condition
Since translation is unitary, we can apply a theorem that is valid for any operator which can be expanded in powers of and . For any unitary operator , we have
This follows because for a unitary operator so we can insert the product anywhere we like. In particular, we can insert it between each pair of factors in every term of the power series expansion of , for example
For 21 this means that
As before, this leads to the condition
which means that is conserved, according to Ehrenfest’s theorem.