Finding Mutual Potential Energy of 2 1S Electrons in He

In summary, the problem is to find the mutual potential energy of two 1S electrons in helium, described by hydrogenic wave functions. The approach involves finding the average value of the potential energy operator V through an integral calculation. However, care must be taken to include details such as spin and statistics for accurate results.
  • #1
jaimon
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0
How can I find the mutual potential energy of two 1S electrons in helium? Each of the two 1S electrons is described by a hydrogenic wave function. I've been trying the addition theoremfor spherical harmonics but it seems not to work.
Thanks
 
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  • #2
This is a quantum problem of course. You begin with the wave function for the two electrons. The wave functions are identical, except one is spin up and the other is spin down. Label the two wave functions "1" and "0".

You want the average value of the potential energy operator V. I guess V is proportional to [tex]1/r = 1/|r_1-r_0|[/tex] potential, but it's been 25 years and I'm not going bet my life on it. And besides, there are several choices of constants depending on which type of E&M you prefer.

The answer is then <1,0| V |1,0>. The integral will be something like:

[tex]\int \int \psi_1^*(r_1)\;\psi_0^*(r_0) \psi_0(r_0)\psi_1(r_1) d^3r_1 d^3r_0 /|r_1 - r_0|.[/tex]

where r_1 and r_0 are 3-vectors and the integrals are over all space. In the above, I've been sloppy in ignoring the detail having to do with spin and statistics. If you want to get your work judged correct I suggest you be more careful and antisymmetrize and keep track of spin and all that.

Carl
 
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1. What is mutual potential energy?

Mutual potential energy is the energy that exists between two particles due to their mutual interactions. In the case of two 1S electrons in a helium atom, mutual potential energy describes the attractive force between the two electrons.

2. How is mutual potential energy calculated?

Mutual potential energy is calculated using Coulomb's Law, which states that the force of attraction between two charged particles is directly proportional to the product of their charges and inversely proportional to the distance between them. In the case of two 1S electrons in helium, the mutual potential energy can be calculated using the equation E = -k[(q1q2)/r], where k is a constant, q1 and q2 are the charges of the electrons, and r is the distance between them.

3. Why is it important to find the mutual potential energy of two 1S electrons in helium?

Finding the mutual potential energy of two 1S electrons in helium is important because it helps us understand the stability of the helium atom. The attractive force between the two electrons plays a crucial role in determining the overall energy of the atom and its behavior in various chemical reactions.

4. What factors affect the mutual potential energy of two 1S electrons in helium?

The mutual potential energy of two 1S electrons in helium is affected by the distance between the electrons and the charges of the electrons. The closer the electrons are, the stronger the attractive force will be, resulting in a higher mutual potential energy. Similarly, if the charges of the electrons are larger, the mutual potential energy will also be higher.

5. How does the mutual potential energy of two 1S electrons in helium compare to other atoms?

The mutual potential energy of two 1S electrons in helium is relatively low compared to other atoms. This is because helium has a closed electron shell, making it more stable and less likely to interact with other atoms. In contrast, in atoms with unpaired electrons, the mutual potential energy between the electrons can be significantly higher due to the repulsive forces between the electrons.

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