Minimum momentum of electron in a hydrogen atom

In summary, the uncertainty principle states that the momentum of an electron in a hydrogen atom is smaller than the minimum possible momentum. This minimum possible momentum corresponds to the minimum energy.
  • #1
fishinsea
5
0

Homework Statement



The energy of an electron in a hydrogen atom is: [itex]E = p^2/2m_e - \alpha e^2/r[/itex]; where [itex]p[/itex] is the momentum, [itex]m_e[/itex] is the electron charge magnitude, and [itex]\alpha[/itex] the coulomb constant. Use the uncertainty principle to estimate the minimum momentum in terms of [itex]m_e, a, e, \hbar[/itex].

Homework Equations



[itex]\Delta p \Delta r = \hbar/2[/itex]

The Attempt at a Solution



The answer sheet set [itex]dE/dp = 0[/itex] to find [itex]r[/itex], and solved for [itex]p[/itex] using the uncertainty principle, but I'm confused why [itex]dE/dP[/itex] would give you minimum momentum, and whether the minimum momentum corresponds to minimum energy. All the questions I've seen that are related to this confines the electron to a certain radius. If [itex]r \to \infty[/itex], shouldn't both potential and kinetic energy (thus momentum) go to 0?
 
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  • #2
The ground state has minimum energy. You can find this by varying the radius or the momentum. Both will lead to a rough estimate of the momentum uncertainty in the ground state.

r -> infinity won't give the ground state.
 
  • #3
But why would minimum energy imply minimum momentum? E = KE + PE, if you increase r, the potential energy increases (becomes less negative), allowing for a lower kinetic energy. I'm also tempted to think of this as a gravity/orbit problem, where the velocity would decreases the further away you move away from the rotation center.
 
  • #4
It does not give the minimal momentum. But you can use it to calculate the minimal momentum uncertainty in the ground state. The momentum uncertainty is smaller for higher energy levels.
 
  • #5
So the momentum uncertainty (and thus momentum) does not occur at ground state? As r -> infinity, would the momentum -> 0?
 
  • #6
fishinsea said:
So the momentum uncertainty (and thus momentum) does not occur at ground state?
You mean the minimal? Then yes, otherwise no.

fishinsea said:
As r -> infinity, would the momentum -> 0?
Yes.
 
  • #7
Okay that cleared up the question mostly, thank you!
 

1. What is the minimum momentum of an electron in a hydrogen atom?

The minimum momentum of an electron in a hydrogen atom is known as the Bohr momentum and is equal to 5.29 x 10^-25 kg*m/s.

2. How is the minimum momentum of an electron in a hydrogen atom calculated?

The minimum momentum of an electron in a hydrogen atom is calculated using the Bohr model, which takes into account the mass of the electron, the reduced Planck's constant, and the Bohr radius.

3. Why is the minimum momentum of an electron in a hydrogen atom important?

The minimum momentum of an electron in a hydrogen atom is important because it helps us understand the behavior and properties of electrons in atoms, and is a key factor in determining the energy levels and spectral lines of hydrogen.

4. Can the minimum momentum of an electron in a hydrogen atom change?

Yes, the minimum momentum of an electron in a hydrogen atom can change if the atom absorbs or emits energy, causing the electron to move to a different energy level. This results in a change in the electron's momentum.

5. What is the significance of the minimum momentum of an electron in a hydrogen atom in chemistry?

The minimum momentum of an electron in a hydrogen atom is significant in chemistry because it helps us understand the arrangement and behavior of electrons in atoms, which is crucial in understanding chemical bonding and reactions.

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