Okey I explored the reduced mass and the assumption that the charge distribution is located on a surface and what that would lead to if the interaction between the electron and proton simply concelled some energy density in the electron. we assume that the charge energy is at the limit in the electron and that's what makes it stable and that in the proton instead, the magnetic energy is at the limit, we would expect to use the reduced mass, which is a successful correction that enables very exact energy levels in the hydrogen atom, or something similar. Here is a discussion about this with ChatGPT that seam to validate this "napkin" calculation, Note that the traditinal reason to reduce mass is that one thinks about the system as a two particle system. This deduction is completely new.
ME
When it comes to energy located in a constellation and length scaler by a, then how does the energy scale acording to dimensional analysis
CHATGPT:
Dimensional analysis is a useful tool for understanding how physical quantities scale with respect to each other. In this case, you're interested in how energy (E) scales with respect to a length scaler (a) in a constellation.
Let's perform dimensional analysis to understand the scaling behavior. First, we need to identify the units of the relevant quantities:
1. Energy (E): Joules (J)
2. Length scaler (a): meters (m)
Now, we need to introduce a scaling exponent (n) to relate energy to the length scaler:
E ∝ a^n
The goal is to find the value of n that will make the equation dimensionally consistent. Energy has dimensions of mass (M) times length squared (L^2) divided by time squared (T^2):
[E] = [M L^2 / T^2]
Length has dimensions of length (L):
[a] = [L]
We can now express the dimensions of energy as:
[M L^2 / T^2] ∝ [L]^n
To make the dimensions consistent, we must have:
n = 2
So, the energy scales with the square of the length scaler:
E ∝ a^2
This means that if the length scaler increases or decreases, the energy will scale with the square of that change. Keep in mind, though, that this is a general relationship derived from dimensional analysis, and specific systems may have additional factors that affect energy scaling.
ME:
If we then divide this per volume element how would this scale
CHATGPT:
To determine the energy per volume element, we need to divide the energy (E) by the volume (V). Let's denote the energy density as D:
D = E/V
Now let's consider how the energy density scales with the length scaler (a). From our previous dimensional analysis, we found that:
E ∝ a^2
Volume has dimensions of length cubed (L^3):
[V] = [L^3]
The volume will scale with the cube of the length scaler:
V ∝ a^3
Now, we can find the scaling relationship for the energy density:
D ∝ (a^2) / (a^3)
Simplifying the expression, we get:
D ∝ 1/a
So, the energy density scales inversely with the length scaler. This means that as the length scaler increases, the energy density decreases, and vice versa.
ME:
And if one whould integerate this on a surface how would that scale
CHATGPT:
When integrating the energy density (D) over a surface, you are essentially calculating the energy flux through that surface. Let's denote the energy flux as F.
First, let's recall the scaling relationship for energy density:
D ∝ 1/a
Now, to calculate the energy flux, we need to integrate the energy density over a surface area (A). Surface area has dimensions of length squared (L^2):
[A] = [L^2]
The surface area will scale with the square of the length scaler:
A ∝ a^2
The energy flux is the product of the energy density and the surface area:
F = D * A
Considering the scaling relationships we derived earlier:
F ∝ (1/a) * (a^2)
Simplifying the expression, we get:
F ∝ a
So, the energy flux scales linearly with the length scaler. This means that as the length scaler increases, the energy flux through the surface also increases, and vice versa.
ME:
So if energy on a surface scales like a, and we associate rest mass m to energy then mass scales as a as well
CHATGPT:
Yes, you're correct. If we assume the energy on the surface scales linearly with the length scaler (a) and we use the mass-energy equivalence principle from Einstein's famous equation:
E = m * c^2
Where E is the energy, m is the rest mass, and c is the speed of light.
Since we established that energy scales linearly with the length scaler (a):
E ∝ a
We can rewrite the mass-energy equivalence equation in terms of the scaling relationship:
m * c^2 ∝ a
Here, the speed of light (c) is a constant. To determine how the mass scales with the length scaler, we can rearrange the equation:
m ∝ a
So, the mass scales linearly with the length scaler as well. As the length scaler increases, the associated mass also increases, and vice versa. However, it is important to remember that this scaling relationship is based on certain assumptions and might not hold true for all physical systems.
ME:
Now if mass and density follow the same scaling property as described above, then if we assume that the electron has reached a fixed density \rhoi_e = A/r_e and that the proton, with larger mass has instead and eqivalent mass rho_p = A/r_p as proton charge and electron charge has different sign's we would consider a conscelling hence if we constarin the density to a constant value in order then the mass of the electron would be adding the densities of the proton and electron (loosely speaking). HOw would the electron mass change if this assumption was correct
CHATGPT:
It appears that there might be some confusion in the assumptions and relationships stated in your question. However, I'll try my best to address your question.
First, let's clarify that mass and density are different physical properties. Mass is a scalar quantity that measures the amount of matter in an object, whereas density is the mass per unit volume. In your question, you mentioned that mass and density follow the same scaling property, which may not be accurate.
Second, I believe you are considering the electron and proton to have fixed densities (ρ_e and ρ_p) and are trying to relate their masses (m_e and m_p) based on a constant ratio (A).
ρ_e = A/r_e
ρ_p = A/r_p
Here, r_e and r_p are some scaling parameters related to the electron and proton, respectively.
Now, let's express the electron mass (m_e) in terms of the proton mass (m_p). We will assume that the volume of the electron (V_e) and proton (V_p) are related to r_e and r_p, respectively.
m_e = ρ_e * V_e
m_p = ρ_p * V_p
Using the given density relationships:
m_e = (A/r_e) * V_e
m_p = (A/r_p) * V_p
Now we can find the ratio of the electron mass to the proton mass:
m_e / m_p = [(A/r_e) * V_e] / [(A/r_p) * V_p]
Since A is a constant, it cancels out:
m_e / m_p = (r_p * V_e) / (r_e * V_p)
If we constrain the density to be constant and the mass of the electron is the sum of the densities of the proton and electron (as you mentioned "loosely speaking"), this relationship does not seem to be valid based on our previous analysis.
Furthermore, it's important to note that electrons and protons are elementary particles with different properties, and it may not be appropriate to compare their densities in this way. The electron is a point-like particle with no known substructure, while the proton is a composite particle made up of quarks and gluons. Comparing their masses and densities in this manner may not be accurate or physically meaningful.
ME:
As the charges cancel one woule expect \rho_e - \rho_p = constant meaning that \rho_e \to \rho_e+\rho_p
CHATGPT:
I understand that you are trying to find a relationship between the densities of electrons and protons based on their charges, assuming that the charges cancel each other out. However, the densities of electrons and protons are not directly related to their charges.
The charge of an electron is negative (-e) and the charge of a proton is positive (+e). The fact that their charges cancel out in a neutral atom is a result of the balance between the number of electrons and protons, rather than their densities.
When you state that ρ_e - ρ_p = constant, it implies that the densities of the electrons and protons have some direct relationship to each other. However, this is not the case. The densities are determined by their mass and volume, which are not directly related to their charges.
If you are considering a situation where the electrons and protons are distributed in some volume, you might be thinking about the charge density rather than the mass density. In such a case, the net charge density would indeed be constant (ideally zero) for a neutral system. However, this would be related to the number of charges per unit volume rather than the mass densities of electrons and protons.
ME:
it't the energy density and we are studying the effect of the proton on the electron. As there woule be some cancellation in the electron in the electrical energy inside the electron we state this simplified assumption
CHATGPT:
I see that you're interested in the effect of the proton on the electron, specifically the impact on the energy density due to their electric charges. In this case, we can consider the electric potential energy density.
The energy density associated with the electric field is given by:
u = 1/2 * ε₀ * E²
where u is the energy density, ε₀ is the vacuum permittivity, and E is the electric field.
In a simplified model, we can assume that the electric field of the proton and electron are uniform within their respective volumes. We can then compute the energy density for each particle and compare their effects.
The electric field of a spherical charge distribution is given by:
E = Q / (4πε₀ * r²)
where Q is the charge of the particle and r is the distance from the center of the particle.
Now, let's consider the effect of the proton on the electron:
1. The proton has a positive charge (+e) and creates an electric field that repels the negatively charged electron.
2. The electron has a negative charge (-e) and creates an electric field that is attracted to the proton.
As the electron and proton are attracted to each other, their electric fields overlap, and there is a "cancellation" of the electric field in the region where they overlap. This cancellation will result in a reduction of the energy density in that region.
However, this simplified assumption doesn't take into account several important factors:
1. The spatial distribution of charges within the proton and electron: In reality, electrons are point-like particles with no known substructure, whereas protons are composite particles with a more complex charge distribution.
2. The quantum nature of the electron: The behavior of electrons around a proton is governed by quantum mechanics. An electron doesn't have a fixed position but instead occupies a region of space described by a probability distribution (orbital). Therefore, a simplified classical model may not accurately describe the system.
Given these complexities, the simplified assumption of ρ_e → ρ_e + ρ_p might not be accurate or meaningful in describing the actual behavior of the electron in the presence of a proton. Instead, a quantum mechanical approach should be used to study the electron-proton interactions.
ME:
But above we stated that energy density if located on a surface scales like C/a, assume C is the same in both the electron and the proton
CHATGPT:
I apologize for the confusion earlier. If we consider the energy density (D) located on a surface and it scales as C/a, where C is a constant and a is a length scaler, we can analyze the relationship between the electron and proton.
Given the scaling relationship:
D_e ∝ C/a_e (for electron)
D_p ∝ C/a_p (for proton)
Assuming that C is the same for both the electron and the proton, we can compare their energy densities. First, let's consider the ratio of the energy densities of the electron and proton:
D_e / D_p = (C/a_e) / (C/a_p)
Since C is the same for both particles, it cancels out:
D_e / D_p = a_p / a_e
Now, if you want to study the effect of the proton on the electron, we can examine the total energy density (D_total) in the region where both particles are present:
D_total = D_e + D_p
Using the scaling relationships we derived earlier:
D_total ∝ C/a_e + C/a_p
To simplify the expression, we can factor out the constant C:
D_total ∝ C * (1/a_e + 1/a_p)
This expression shows that the total energy density depends on the sum of the inverses of the length scalers a_e and a_p. The cancellation effect you mentioned earlier would result in the energy density being reduced in the region where the electric fields of the electron and proton overlap. However, without more information about the specific system and the nature of the length scalers a_e and a_p, it's difficult to draw further conclusions about the relationship between the electron and proton.
Keep in mind that this simplified analysis does not consider the full complexity of the electron-proton system, such as their quantum nature, and it might not provide an accurate representation of their interaction. A more detailed study would require a quantum mechanical approach, like solving the Schrödinger equation for the hydrogen atom.
ME:
Note C(1/a_e+1/a_p) = C /(1/(1/a_e+1/a_p)) = C /(a_e a_p / (a_e+a_p)) = C/a, with a = a_e a_p/(a_e+a_p). Then if m = Da (as we saw above it scaled like this)m we get m = D a_e a_p/(a_e+a_p). = D a_e D a_p/(D a_e+D a_p) = m_e m_p / (m_e + m_p). So we would then if we assumed this scaling property and that the electron tries to attain a fixed energy density that the effective mass of the electron should be the reduced mass
CHATGPT:
Your analysis is correct. If we assume the simplified scaling property, where the energy density located on a surface scales as C/a and that the electron tries to attain a fixed energy density, the effective mass of the electron (m_e) can be related to the mass of the proton (m_p) through the reduced mass (μ) formula:
μ = m_e * m_p / (m_e + m_p)
The reduced mass is a concept used in classical and quantum mechanics to simplify two-body problems, and it can be interpreted as the "effective" mass of the two-body system.
However, it's important to keep in mind that the assumptions made in this analysis are based on a simplified model and may not fully capture the complexity of the electron-proton interaction. A more accurate description of their interaction would require a quantum mechanical approach, such as solving the Schrödinger equation for the hydrogen atom. The reduced mass does play a role in the quantum mechanical description, but it is derived from a different line of reasoning.
ME:
Isn't it interesting that the reduced mass as usually assumed to be related to mass centre also can be derived via this dimensional analysis, that the energy density is located at a surface and some hand waving. Now has this way of deducing the reduced mass been done before?
CHATGPT:
It is indeed interesting that you were able to arrive at the concept of reduced mass through a different approach, using dimensional analysis, energy density located on a surface, and some assumptions. However, this method of deducing the reduced mass is not conventional and might not be widely recognized or acknowledged in the scientific literature.
The reduced mass is typically derived from the classical or quantum mechanical two-body problem, where it serves as an effective mass that simplifies the equations of motion. In classical mechanics, the reduced mass arises when transforming to the center-of-mass frame, while in quantum mechanics, it is used when solving the Schrödinger equation for a two-body system.
Although your approach provides an interesting perspective, it is important to keep in mind that the assumptions made in your analysis might not accurately capture the full complexity of the electron-proton interaction. The simplified scaling properties and energy density considerations may not provide a complete or rigorous understanding of the system.
Nevertheless, it's always valuable to explore alternative ways of understanding physical concepts, as they can provide unique insights or perspectives that could lead to new discoveries or a deeper understanding of the subject matter.
