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The transformation of phase translations is linked to the generator of the conjugate variable. For phase changes, this is the number operator. Hence, for a phase translation of $\text{d}\varphi$, the unitary transformation performing this shift is

$$\hat{U} = \exp\left[-i\text{d}\varphi\hat{n}\right].$$

I would like to derive a similar unitary transformation for different variables where the generator is initially unknown. To do so, it would be informative to understand how the above equation is derived. Does anyone know how to derive the form of $\hat{U}$ shown above for phase translations so that a similar approach may be taken for different observables?

NOTE

The above is a phase shifter, not a phase operator.

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  • $\begingroup$ Actually there is no hermitian phase operator (see physics.stackexchange.com/q/292633/36194 ) so your $\hat U$ cannot be unitary. $\endgroup$ – ZeroTheHero Jul 3 '17 at 14:56
  • $\begingroup$ @ZeroTheHero, see edit, it is a phase shifter. Namely $U a U^\dagger = a \exp[-i\varphi]$ $\endgroup$ – Sid Jul 3 '17 at 15:06
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We are trying to find the unitary operator $\hat U$ that shift the phase of the creation operator, i.e. $$ \hat U(\phi) \hat a \hat U^\dagger(\phi)=e^{-i \phi}\hat a.\quad\quad (1)$$

To do so, the simplest thing is to expand $\hat U(\phi)$ for small phases, $\hat U=\hat 1 -i \phi \hat A$, (the sign is arbitrary) with $\hat A^\dagger=\hat A$ to insure unitarity at lowest order in $\phi$. Expanding $(1)$ to lowest order in $\phi$, one find that $\hat A$ must be such that $$[\hat A,\hat a]=\hat a,$$ which is solved by $\hat A=\hat a^\dagger \hat a$. One can then reconstruct $\hat U$ directly, since it only involves one operator(this would not be true if $\hat A$ was the sum of two non-commuting operators).

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