Dependence of potential on only the *difference* between two variables

In summary, the paper discusses a diatomic molecule in two dimensions with a fixed central nucleus at the origin. The locations of the nuclei are described by (r_1, \theta_1) and (r_2, \theta_2) and the molecule is subject to a potential V(r_1, r_2, \theta_1, \theta_2). The paper argues that if the potential remains the same when rotating the molecule, it must be true that V(r_1, r_2, \theta_1, \theta_2) = V'(r_1, r_2, \theta_2 - \theta_1), meaning the potential only depends on the difference between \theta
  • #1
AxiomOfChoice
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I'm reading a paper that considers a diatomic molecule living in two dimensions in which the central nucleus is fixed at the origin. Ignore the electrons for the time being. Let [itex](r_1,\theta_1)[/itex] and [itex](r_2,\theta_2)[/itex] describe the locations of the nuclei, and let the molecule be subject to a potential [itex]V(r_1,r_2,\theta_1,\theta_2)[/itex]. The paper claims that, if the potential is the same when we rotate the entire molecule, we must have
[tex]
V(r_1,r_2,\theta_1,\theta_2) = V'(r_1,r_2,\theta_2-\theta_1);
[/tex]
i.e., the potential only depends on the difference between [itex]\theta_1[/itex] and [itex]\theta_2[/itex]. So the potential really only depends on three variables: [itex]r_1[/itex], [itex]r_2[/itex], and [itex]\phi[/itex], where [itex]\phi = \theta_2 - \theta_1[/itex]. Can someone please explain why this is? I don't see it.
 
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  • #2
what are the angle representative of?
 
  • #3


This statement is based on the concept of symmetry in physical systems. In this case, the paper is considering a diatomic molecule in two dimensions with a fixed central nucleus at the origin. This system has rotational symmetry, meaning that if the entire molecule is rotated, the physical properties of the system remain unchanged.

Now, let's consider the potential V(r_1,r_2,\theta_1,\theta_2) of this system. Since the molecule is rotationally symmetric, the potential should also remain the same after a rotation. However, if we rotate the molecule, the values of \theta_1 and \theta_2 will change, but the difference between them (\phi) will remain the same. This means that the potential should only depend on the difference between \theta_1 and \theta_2, and not on their individual values.

In other words, the potential V(r_1,r_2,\theta_1,\theta_2) can be rewritten as V(r_1,r_2,\phi). This is because the potential does not change when the molecule is rotated, and the only variable that changes during rotation is \phi. Therefore, the potential only depends on three variables: r_1, r_2, and \phi.

In summary, the statement in the paper is based on the rotational symmetry of the diatomic molecule, and the fact that the potential should remain unchanged after a rotation. This leads to the conclusion that the potential only depends on the difference between \theta_1 and \theta_2, and not on their individual values.
 

1. What does it mean for potential to depend on the difference between two variables?

When potential depends only on the difference between two variables, it means that the value of potential at a certain point is determined by the difference between the values of the two variables at that point. This relationship is known as a potential difference or voltage difference.

2. Why is it important to consider the difference between two variables when studying potential?

Understanding the dependence of potential on the difference between two variables is crucial in many scientific fields, particularly in physics and engineering. This relationship allows us to calculate the potential at a point based on the values of two variables, and it also helps us to analyze and predict the behavior of electric and magnetic fields.

3. How is the dependence of potential on the difference between two variables mathematically represented?

The relationship between potential and the difference between two variables is expressed by the equation V = kΔx, where V represents potential, k is a constant, and Δx is the difference between the two variables. This equation is known as the potential difference formula and is used to calculate potential in various scenarios.

4. Can potential depend on the difference between more than two variables?

Yes, potential can depend on the difference between multiple variables. In such cases, the potential difference formula becomes V = k(Δx1 + Δx2 + ... + Δxn), where n represents the number of variables. This equation is commonly used in circuits and other complex systems where potential depends on multiple factors.

5. How does understanding the dependence of potential on the difference between two variables help in practical applications?

Knowledge of this relationship is essential in many practical applications, such as designing electronic circuits, calculating electric and magnetic fields, and analyzing the behavior of charged particles. It also allows us to measure potential differences and use them to perform useful tasks, such as powering devices and transmitting signals.

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