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

In summary, it seems that the sign in the exponent of the equation will change depending on the situation.
  • #1
Silviu
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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!
 
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  • #2
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
 
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  • #3
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.
 
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What is space and time translation?

Space and time translation refers to the mathematical concept of transforming a point in space-time to a new location or time. It is often used to describe the movement of objects in the universe or to make calculations in physics.

How is space and time translation related to relativity?

In the theory of relativity, space and time are considered to be interconnected and can be transformed into each other. This means that an object's position and time can change relative to an observer's frame of reference, and space and time translations are used to calculate these changes.

What are the different types of space and time translations?

There are three types of space and time translations: spatial translations, temporal translations, and Lorentz transformations. Spatial translations involve moving an object in space without changing its time coordinate, temporal translations involve changing an object's time without changing its spatial coordinates, and Lorentz transformations involve both spatial and temporal changes.

How are space and time translations used in astrophysics?

Space and time translations are used extensively in astrophysics to study the universe and understand the behavior and movement of celestial objects. They are also crucial in calculating the effects of gravity and the movement of light in the universe.

What is the importance of space and time translations in quantum mechanics?

In quantum mechanics, space and time translations are used to describe the behavior of particles and their interactions. They are also important in understanding the concept of wave-particle duality and the uncertainty principle, which states that the exact position and momentum of a particle cannot be simultaneously known.

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