Energy versus uncertainties in position & momentum

[tjkubo]

Homework Statement

Is it true that if a particle has lower energy than another particle, then it also has smaller uncertainties in momentum and position?

Homework Equations

Uncertainty principle

The Attempt at a Solution

If you consider the Bohr model, lower energy states mean smaller radii, which would mean the electron in confined to a smaller space, or width, so that the uncertainty in position would be smaller. Is this correct reasoning? I don't know what to do with the uncertainty in momentum, at all. Any help will be appreciated.

 

[mark726]

Thank you for your post. Your reasoning is partially correct. According to the uncertainty principle, the product of the uncertainties in position and momentum must always be greater than or equal to a certain value determined by Planck's constant. This means that as the uncertainty in position decreases, the uncertainty in momentum must increase, and vice versa.

In the Bohr model, the energy levels are quantized, meaning that they can only have certain specific values. This also means that the momentum of the electron in each energy level is also quantized. As the energy level decreases, the momentum of the electron also decreases. This leads to a decrease in the uncertainty in momentum.

However, the uncertainty in position is not solely determined by the size of the orbit. It also depends on other factors such as the wave nature of the electron and the shape of the orbit. So, while a lower energy level may correspond to a smaller orbit and a decrease in the uncertainty in position, it does not necessarily mean that the uncertainty in position will always be smaller than that of a particle with higher energy.

In summary, the relationship between energy and uncertainties in position and momentum is more complex and cannot be generalized in a simple statement. It also depends on the specific system and its quantum properties. I hope this helps to clarify your understanding. If you have any further questions, please do not hesitate to ask.