Quantum Atomic States:- Check of workings

In summary, quantum atomic states refer to specific energy levels that electrons can occupy within an atom. Scientists use various experimental techniques to observe and analyze these states, which provide a fundamental understanding of the behavior of atoms and play a crucial role in predicting their properties. Unlike classical atomic states, quantum atomic states describe the positions of electrons as probabilistic and quantized, allowing for real-world applications in fields such as quantum computing, nanotechnology, and medical imaging.
  • #1
Haths
33
0
This post is simply to check through these problems I have attempted to see if I have understood the question properly and applied the correct physics to each senario. Thankyou in advance for looking through my work;

1. The simplest model of describe the rotations of a diatomic molecule is the dumbell model. This is two point like massive balls connected by a ridgid massless rod length 'a'.

Find the rotational energy levels of a hydrogen molecule.

Applying the Bohr model where the angular momentum is maintained in discrete levels;

[tex]
mv_{n}r_{n} = \frac{nh}{2 \pi}
[/tex]

where n = 1,2,3,...

Implies;

[tex]
r_{n} = \frac{nh}{2 \pi mv_{n}}
[/tex]

So;

[tex]
r_{n}^{2} = \frac{n^{2}h^{2}}{4 \pi^{2} m^{2}v_{n}^{2}}
[/tex]

Comparing with the moment of inertia: I = mr2 for the rotational kinetic energy formula: 1/2 Iw2 w = angular frequency;

[tex]
I = \frac{n^{2}h^{2}}{4 \pi^{2} mv_{n}^{2}}
[/tex]

Therefore into the RKE formula;

[tex]
E_{RKE} = \frac{1}{2}\frac{n^{2}h^{2}w^{2}}{4 \pi^{2} m^{2}v_{n}^{2}}
[/tex]

As v=rw w=v/r w2 = v2/r2 In this case as r = 1/2 a and v=vn;

[tex]
E_{RKE} = \frac{1}{2}\frac{n^{2}h^{2}}{4 \pi^{2} m^{2}r_{n}^{2}}
[/tex]

[tex]
E_{RKE} = \frac{1}{2}\frac{n^{2}h^{2}}{4 \pi^{2} m^{2} \frac{1}{4} a^{2}}
[/tex]

Hence mulitipliny by two, for both molecules;

[tex]
E_{RKE} = \frac{n^{2}h^{2}}{\pi^{2} m^{2} a^{2}}
[/tex]

Yay or nay?

2. Distance of closest approach of a 4.78MeV alpha particle and uranium nucleus at 180 degrees.

(There are 2 protons in an alpha particle, and 92 in the uranium nucleus 180 degrees means that it's a head on collision.)

Using the work energy theorum with;

[tex]
work = \int \frac{kq_{A}q_{U}}{r^{2}} dr
[/tex]

Where the qs are the charges on the respective particles.

[tex]
Work = \frac{-kq_{A}q_{U}}{r}
[/tex]

Evaluated between infinity (because the question didn't state a possition vector for the alpha particle to be at when it has its 4.78MeV of energy) and the distance of closest aproach.

Hence;

[tex]
E_{A} = \frac{-kq_{A}q_{U}}{r}
[/tex]

[tex]
r = \frac{-kq_{A}q_{U}}{E_{A}}
[/tex]

Plug the numbers in. Can ignore the (1.6x10-19) conversion factor because it appears in the numerator and denomiator.

r = 3.5x105m

Well, it's a tad large isn't it? Hence why I'm questioning my method, however it is coming from infinity, so perhaps that is why it doesn't get as close as in real life when it would have this kind of energy just mear meters before it feels the electric field of the uranium nucleus.

Or perhaps I am meant to place the alpha particle not at infinity, but at about 1x10-10, as that is 'within' the orbit of electrons. But as the question doesn't talk about electrons, nor does it say anything about the possitions of those electrons (i.e. would they be attracting the alpha particle towards the nucleus or away, and then what are their movements with respect to the two particles in the question.

Futhermore, due to the alpha particle being very much more massive, the electrons will be drawn to it, more than the alpha particle being slowed down, by the electrons.

Eitherways did I do the above calculations correctly.

Cheers,
Haths
 
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  • #2


I appreciate your thoroughness in checking your work and seeking feedback. Overall, your approach to the first problem seems sound and your calculations appear to be correct. You have correctly applied the Bohr model and the rotational kinetic energy formula to determine the energy levels of a hydrogen molecule. However, it may be helpful to provide more context and explanation for your calculations, as well as any assumptions made (e.g. the assumption that the hydrogen molecule is in a dumbbell shape).

For the second problem, your approach using the work-energy theorem also seems reasonable. However, as you mentioned, there are uncertainties and assumptions that could affect the accuracy of your calculation. It would be helpful to clarify these assumptions and consider other factors that may affect the distance of closest approach, such as the repulsive force between the positively charged alpha particle and uranium nucleus. Additionally, you may want to consider units and significant figures in your final answer.

Overall, your work appears to be well thought out and executed. It would be beneficial to provide more context and explanation for your calculations, as well as address any potential uncertainties and assumptions made. Keep up the good work!
 
  • #3


Hello Haths,

Thank you for sharing your work and questions. I am not able to provide a complete check of your work without seeing all the equations and steps you took to solve the problems. However, I can provide some comments and guidance that may help you.

For the first question, it seems that you have correctly applied the Bohr model to find the rotational energy levels of a hydrogen molecule. However, I would recommend using the reduced mass instead of the individual masses of the two atoms. This will give you a more accurate result.

As for the second question, your approach using the work-energy theorem seems correct. However, you should keep in mind that the distance of closest approach is not the same as the radius of the uranium nucleus. The distance of closest approach is the distance at which the alpha particle's kinetic energy is completely converted to potential energy due to the repulsive force between the two particles. This distance will depend on the initial velocity and the charges of the particles. Also, the question mentions a head-on collision, so the alpha particle's initial position would be at the center of the uranium nucleus. You can use this information to calculate the distance of closest approach.

I hope this helps. Keep up the good work in your studies of quantum atomic states.
 

1. What are quantum atomic states?

Quantum atomic states refer to the specific energy levels that an electron can occupy within an atom. These levels are described by the quantum numbers, which determine the position, energy, and spin of an electron.

2. How do we check the workings of quantum atomic states?

To check the workings of quantum atomic states, scientists use various experimental techniques such as spectroscopy and quantum mechanical calculations. These methods allow us to observe and analyze the energy levels and transitions of electrons within an atom.

3. What is the significance of quantum atomic states?

Quantum atomic states are significant because they provide a fundamental understanding of the behavior of atoms, which are the building blocks of all matter. They also play a crucial role in understanding and predicting the chemical, physical, and optical properties of elements and molecules.

4. How do quantum atomic states differ from classical atomic states?

Unlike classical atomic states, which describe the orbits of electrons as continuous and circular, quantum atomic states describe the electrons' positions as probabilistic and quantized. This means that the electrons can exist in multiple energy levels simultaneously, also known as superposition.

5. What are some real-world applications of quantum atomic states?

Quantum atomic states have many practical applications, such as in quantum computing, where the superposition and entanglement of electrons are utilized to perform complex calculations. They also play a crucial role in fields such as nanotechnology, materials science, and medical imaging.

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