# Use the uncertainty principle for momentum vs. position to e

• FlorenceC
In summary: If the electron has energy E, then how does the radius depend on the momentum?Can you relate ##\Delta x## to radius?
FlorenceC

## Homework Statement

Please see the attachment for a better picture

The energy of an electron in a hydrogen atom is: E = p^2/2m - αe2/r; where p is the momentum,
r the orbital radius, me the electron mass, e the electron charge, and α the Coulomb constant.
Use the uncertainty principle for momentum vs. position to estimate the minimum radius and the momentum corresponding to this radius.

## Homework Equations

E = p^2/2m - αe2/r

I'm not exactly sure what this means conceptually

## The Attempt at a Solution

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Is this a plug and chug? I feel like it's not that simple.
uncertainty for momentum says

delta x * delta p <= h/2π

so do I just plug in p in delta p?

#### Attachments

• Screen Shot 2016-01-28 at 8.00.26 PM.png
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FlorenceC said:
E = p^2/2m - αe2/r

I'm not exactly sure what this means conceptually
The total energy is the sum of kinetic and potential energy.

Is this a plug and chug? I feel like it's not that simple.
uncertainty for momentum says

delta x * delta p <= h/2π

so do I just plug in p in delta p?
Does that work? Did you try? How do you work the ##\Delta x##?
I would suggest understanding the relation before trying random calculations.

If the electron has energy E, then how does the radius depend on the momentum?
Can you relate ##\Delta x## to radius?

FlorenceC said:
E = p^2/2m - αe2/r

That's just the total (semiclassical) energy of the electron-nucleus system. The first bit on the left is just the kinetic energy of the electron's orbit, written in terms of momentum instead of velocity. The second bit is the electrostatic potential energy due to the attraction between the electron and the positively charged nucleus. The hydrogen nucleus is just a lone proton, so its charge is +e. Might help to rewrite the whole thing like this:
E = p^2 / 2m + a*(-e)*(+e)/r

FlorenceC said:
Is this a plug and chug? I feel like it's not that simple.

The problem is part visualization and part plugging and chugging. You can't put p's where delta p's go, and same for x. Interpreting the deltas requires some thinking. Remember that delta x and delta p are the deviations observed in measured values of x and p. So how much can x (the electron's horizontal position) vary as the electron orbits the nucleus in a circle of radius r? Likewise, how much can p (the x-component of the electron's momentum) vary in that orbit?

## 1. What is the uncertainty principle for momentum vs. position?

The uncertainty principle for momentum vs. position is a fundamental concept in quantum mechanics that states that it is impossible to simultaneously know the exact momentum and position of a particle. This means that the more precisely we know the momentum of a particle, the less precisely we can know its position, and vice versa.

## 2. How is the uncertainty principle for momentum vs. position used in experiments?

The uncertainty principle for momentum vs. position is used in experiments to understand the behavior of particles at the quantum level. It allows scientists to predict the range of possible outcomes for a particular measurement and helps to explain phenomena such as wave-particle duality.

## 3. Can the uncertainty principle for momentum vs. position be violated?

No, the uncertainty principle for momentum vs. position is a fundamental principle of quantum mechanics and cannot be violated. It is a consequence of the wave-like behavior of particles at the quantum level and the limitations of our measurement capabilities.

## 4. How does the uncertainty principle for momentum vs. position relate to Heisenberg's uncertainty principle?

The uncertainty principle for momentum vs. position is a specific case of Heisenberg's uncertainty principle, which states that there is a fundamental limit to the precision with which certain pairs of physical properties can be known simultaneously. The uncertainty principle for momentum vs. position specifically refers to the relationship between the momentum and position of a particle.

## 5. Can the uncertainty principle for momentum vs. position be applied to macroscopic objects?

No, the uncertainty principle for momentum vs. position is only applicable to particles at the quantum level. Macroscopic objects, such as everyday objects, do not exhibit the same wave-like behavior and are subject to different physical laws that do not involve uncertainty principles.

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