Use uncertainty principle to obtain the result of Bohr's Model

In summary, to find the minimum energy of a hydrogen atom using the uncertainty principle, we take the uncertainty of the position to be approximately equal to the distance from the nucleus, and approximate the momentum as Δp. Treating the atom as a 1-D system, we calculate the total energy and rearrange the terms to obtain a quadratic equation. To obtain the correct result, we need to replace Δr with (1/2)Δr, but the reason for this is not clear.
  • #1
HAMJOOP
32
0
Problem
Find the minimum energy of the hydrogen atom by using uncertainty principle

a. Take the uncertainty of the position Δr of the electron to be approximately equal to r
b. Approximate the momentum p of the electron as Δp
c. Treat the atom as a 1-D system


My step

1. Δr Δp ≥ h/4(pi)
Δp ≥ h/4(pi)r

2. Total energy E = (p^2 /2m) - ke^2 /r

≥ (h^2/8(pi)^2 m r^2) - (ke^2)/r

3. rearranging the term

(Emin)r^2 + (ke^2)r - (h^2 /8(pi)^2 m ) = 0

Require Δ = 0 for the quadratic equation

I obtain E = -54.7 eV ≠ -13.6 eV


If I replace Δr by (1/2)Δr, I can obtain the correct result. But I don't know why.
 
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  • #2
HAMJOOP said:
1. Δr Δp ≥ h/4(pi)
Δp ≥ h/4(pi)r

2. Total energy E = (p^2 /2m) - ke^2 /r

≥ (h^2/8(pi)^2 m r^2) - (ke^2)/r
you have p=h/4(pi)r, but then you seem to say p^2 /2m = h^2/8(pi)^2 m r^2 there is a mistake in this step
 
  • #3
woah there! you can't shift Δr to the RHS in case there is a > sign,right? if Δr.Δp=h/4π, ⇒Δp=h/4πr
 

1. What is the uncertainty principle?

The uncertainty principle, also known as Heisenberg's uncertainty principle, states that it is impossible to know both the exact position and momentum of a particle at the same time. This means that the more precisely we know the position of a particle, the less we know about its momentum, and vice versa.

2. How does the uncertainty principle relate to Bohr's model?

In Bohr's model, electrons are described as having specific orbits around the nucleus. However, according to the uncertainty principle, it is impossible to know the exact position and momentum of an electron simultaneously. Therefore, Bohr's model cannot fully explain the behavior of electrons in an atom.

3. Can the uncertainty principle be used to obtain the result of Bohr's model?

Yes, the uncertainty principle can be used to obtain the result of Bohr's model by considering the limitations it imposes on our ability to know the position and momentum of electrons. This allows us to better understand the behavior of electrons in an atom and refine Bohr's model.

4. How does the uncertainty principle affect our understanding of the atom?

The uncertainty principle challenges the classical understanding of the atom, which assumes that particles have definite positions and momentums. Instead, it suggests that particles, such as electrons, behave more like waves with a range of possible positions and momentums. This has led to the development of quantum mechanics, which provides a more accurate description of atomic behavior.

5. Are there any practical applications of the uncertainty principle?

Yes, the uncertainty principle has many practical applications in various fields, such as quantum computing and cryptography, where it is used to manipulate and control particles at the quantum level. It also plays a crucial role in modern technologies, such as transistors and lasers.

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