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 :

Therefore

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.

### Like this:

Like Loading...

*Related*

Pingback: Translational invariance and conservation of momentum | Physics pages

Pingback: Finite transformations: correspondence between classical and quantum | Physics pages

Pingback: Parity transformations | Physics pages

Pingback: Rotational transformations using passive transformations | Physics pages