When should I use a plus or minus sign in space and time translations?

[Silviu]

Hello! I am a bit confused about the sign in space and time translation operators acting on a state. I found it with both plus and minus sign and I am not sure which one to use when. The equations I am talking about are: $$U(t)=e^{\pm iHt/\hbar}$$ and $$T(x)=e^{\pm ixp/\hbar}$$. Is it a plus or a minus in the exponent? Thank you!

 

[bhobba]

It seems to vary a bit, but my reference, Ballentine has it as positive in accordance with what I say below.

For the detail on this and other associated matters see Chapter 3 - Ballentine - Quantum Mechanics - A Modern Development. In particular see page 66 - equation 3.4 where you see in general its positive for any such operator - time translation, space translation, rotational translation or whatever. Sometimes however the following can confuse the issue. When you speak of a translation do you move the observing apparatus or what is being observed? As an example for a position measurement if you move the measuring equipment a distance d you subtract d from everything measured. Move the thing being measured and you add d - it can be confusing.

The above does not prove the important Wigner's Theorem which is associated with it as well:
https://arxiv.org/abs/0808.0779

Thanks
Bill

 

[Charles Link]

It might help to start with a simple derivation on the wave function $\Psi(x)$: $\\$ $\Psi(x+\Delta x)=\Psi(x)+(\frac{\partial{\Psi}}{\partial{x}}) \Delta x=\Psi(x)+\frac{i}{\hbar} \hat{p} \Psi \, \Delta x$. $\\$ This operator equation can be integrated to get $\Psi(x+x_o)=e^{+\frac{i}{\hbar} x_o \hat{p} } \Psi(x)$. $\\$ If you are trying to find $\Psi(x-x_o)$, you reverse the signs. $\\$ These formulas can get complicated depending on whether the wave function is being translated or the axes. As @bhobba has mentioned, the textbooks will use different sign conventions at times.