What Are the Uncertainties for an Electron in a Hydrogen Atom?

[Quelsita]

Consider and electron in ground state in the Hydrogen atom. Suppose the Bohr orbit lies in the x-y plane. Assume that the position of the electron along the orbit is completely unknown.
a)Estimate the uncretainties in x,p_x, y, p_y, z and p_z.
b)Repeat for and excited state n>1. With the given information, can you say anything about the unceratinty in time? Energy?


I know how to solve for part a:
$$\Delta$$x=2a₀, $$\Delta$$p_x=h/(2a₀), $$\Delta$$y=a₀, $$\Delta$$p_y=h/(a₀), $$\Delta$$z=0 since in x-y plane, $$\Delta$$p_x=$$\infty$$
but I'm a little stuck how to do part b.

any help is greatly appreciated, thanks!

 

[jbsteele]

A) For part b, the uncertainties in x, px, y, py, z and pz will remain the same as before, since the orbit of the electron is still lying in the x-y plane. The uncertainty in time and energy will depend on the excited state value of n, but since we don't know that value, there is no way to calculate any uncertainty for these two variables.

 

[thomate1]

I would like to clarify that the Heisenberg Uncertainty Principle is a fundamental principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This means that the more precisely we know the position of a particle, the less we know about its momentum, and vice versa.

In the case of an electron in ground state in the Hydrogen atom, the uncertainty in its position along the orbit (in the x-y plane) is estimated to be 2a0, where a0 is the Bohr radius. This means that the electron could be anywhere along the orbit with an uncertainty of 2a0.

Similarly, the uncertainty in its momentum along the orbit is estimated to be h/(2a0), where h is the Planck's constant. This means that the electron could have any momentum along the orbit within this uncertainty range.

For an excited state with n>1, the uncertainties in position and momentum will be different due to the larger size of the orbit. However, the uncertainty in time and energy cannot be determined with the given information. This is because the Heisenberg Uncertainty Principle also applies to the energy and time of a particle, and the uncertainty in these quantities is inversely proportional to the uncertainty in position and momentum. Therefore, without knowing the exact values of these uncertainties, we cannot say anything about the uncertainty in time and energy in this scenario.

It is important to note that the Heisenberg Uncertainty Principle is not a limitation of our measurement capabilities, but rather a fundamental property of the quantum world. It is a cornerstone of quantum mechanics and has been experimentally verified numerous times.