# Energy of an electron at rest?

• K.Callaghan
In summary: A good proof is on the you tube channel minute physics).Your non-relativistic Hamiltonian would correspondingly equal zero from being stationary but considering... (A good proof is on the you tube channel minute physics).
K.Callaghan
An electron rest mass in kilograms is calculated from the definition of the Rydberg constant R∞:

where α is the fine structure constant and h is simply Plancks constant. Now, assuming the electron has an associated rest energy, Any idea on how this is calculated? I'm assuming using relativistic means...but of what nature? Any thoughts? Formulas? Ideas?...

Use E = mc^2 with m being the rest mass of the electron

K.Callaghan
Thank you for responding! Yes of course, of which I assume would be the easy answer. My concern is more of where this relationship has mechanical meaning... Which is to say what other formulas explain this relationship...this energy = mass light squared? Is there an equivalent? Can anyone explain in lamen terms why a particle at rest has energy other than to say it has "quantum behaviour"?

Topolfractal
Please can you explain more on what you mean by quantum behavior. Also there is another formula E^2=m^2 c^4 + p^ 2 c^2 which when p=0 reduces to e= mc^2. The proof of e= mc ^2 is,... ( A good proof is on the you tube channel minute physics).

K.Callaghan
Topolfractal said:
Please can you explain more on what you mean by quantum behavior. Also there is another formula E^2=m^2 c^4 + p^ 2 c^2 which when p=0 reduces to e= mc^2. The proof of e= mc ^2 is,... ( A good proof is on the you tube channel minute physics).
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Sure. I'm not doubting whether the proof is sound, yet rather searching for another explanation for this relationship other than mass displays energy in states of "quanta", or as a wave function... As we know an electron exhibits both paticle and wave like characteristics... Such is what we would call quantum like behaviour. However such behaviour is usually calculated using multiple systems of momenta rather than a single particle at rest which essential to my understanding still has an associated energy, frequency, and wavelength...

so you want a proof of $R_\infty = \dfrac{m_ec \alpha^2}{2h}$?

K.Callaghan
No I don't want a proof of that , but thank you for offering. Now it sounds like you are trying to prove a relativistic formula quantum mechanically. The proof though in that video has nothing to do with quantization and proves it using the relativistic Doppler effect. He equates the doppler shifted perspective and the stationary perspective from energy conservation.

K.Callaghan
My last message is to Callaghan sorry I should have quoted, and the first sentence to Chris ver

K.Callaghan
Topolfractal said:
My last message is to Callaghan sorry I should have quoted, and the first sentence to Chris ver
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Exactly. Although more specifically, when considering a rest particle with an associated E irrespective of multiple systems of momenta and of course it's relationship with C^2... And I appoloize as I am not familiar with the video you speak of ? From minute physics did you say?

Yes minute physics ,and please can you teach me how to quote people.

K.Callaghan
Topolfractal said:
Yes minute physics ,and please can you teach me how to quote people.

I'll will definately check it out ! As far as quoting someone Its my understanding that when you hit reply the previous post is quoted automatically.

Although I am online via mobile thus the reply layout may be different if your on your computer.

I'm on my iPad, please though can you teach me how to quote. It could very well still work. No for me it isn't quoted.

Instead of replying directly in the "have something to add?" Column. Tap on the last response and hit reply. It's my understanding that the quotes are highlighted automatically. Hope this helps.

K.Callaghan said:
Although I am online via mobile thus the reply layout may be different if your on your computer.
Shsjsjsjsj

Thank you

Now minute physics yay or nay ( help or not )

K.Callaghan said:
--------
Sure. I'm not doubting whether the proof is sound, yet rather searching for another explanation for this relationship other than mass displays energy in states of "quanta", or as a wave function... As we know an electron exhibits both paticle and wave like characteristics... Such is what we would call quantum like behaviour. However such behaviour is usually calculated using multiple systems of momenta rather than a single particle at rest which essential to my understanding still has an associated energy, frequency, and wavelength...
K.Callaghan said:
Thank you for responding! Yes of course, of which I assume would be the easy answer. My concern is more of where this relationship has mechanical meaning... Which is to say what other formulas explain this relationship...this energy = mass light squared? Is there an equivalent? Can anyone explain in lamen terms why a particle at rest has energy other than to say it has "quantum behaviour"?
Your quantum thinking has inspired me to wonder if e= mc^2 can be derived from quantum mechanics.
Now the non - relativistic Hamiltonian would correspondingly equal zero from being stationary but considering a relativistic free Hamiltonian instead that takes in moving particles. The formula E^ 2 = m^2c^4+ p^2 c^2 can be quantized to produce such a Hamiltonian and reduces to e= mc^2 on p= 0. This isn't a proof of course but is the foundational insight to the Dirac equation.

K.Callaghan
Now actually because traditional quantum mechanics is based from classical mechanics which doesn't contain e= mc^ 2 there is no such notion in traditional quantum mechanics and therefore it can't be derived or proven using that theory.

Topolfractal said:
Now minute physics yay or nay ( help or not )

Excluding the hamiltionain operator.

No,relativistic quantum mechanics make use of $E^2 - |\vec{p}|^2 c^2 =m^2 c^4$ formula, it doesn't prove it. This formula is taken from special relativity alone and it's a frame independent combination of the frame dependent quantities $E,\vec{p}$.

Also there is no quantization of that formula either... from that formula you obtain the classical field lagrangian for (mainly) the Klein Gordon field (and maybe you can say something about the Dirac field too)... The quantization is then applied on the fields (when you impose [anti]commutator relations and insert the ladder operators or when you write the canonical [anti]commutation relations). In both cases the fields are the ones that get the quantized...

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Topolfractal said:
Now actually because traditional quantum mechanics is based from classical mechanics which doesn't contain e= mc^ 2 there is no such notion in traditional quantum mechanics and therefore it can't be derived or proven using that theory.
Formulated appropriately is identical to what you just said.

## 1. What is the energy of an electron at rest?

The energy of an electron at rest, also known as its rest energy, is equal to its mass multiplied by the speed of light squared, as described by Einstein's famous equation E=mc^2. This results in a very small amount of energy, approximately 0.511 megaelectronvolts (MeV).

## 2. How is the energy of an electron at rest calculated?

The energy of an electron at rest can be calculated by using the formula E=mc^2, where E is the rest energy, m is the mass of the electron, and c is the speed of light. The speed of light, c, is approximately 299,792,458 meters per second.

## 3. What is the significance of the energy of an electron at rest?

The energy of an electron at rest is significant because it represents the minimum amount of energy required to create an electron-positron pair, as well as the minimum amount of energy that an electron can possess. It also plays a crucial role in understanding the behavior of particles at the atomic and subatomic level.

## 4. Can the energy of an electron at rest change?

No, the energy of an electron at rest, also known as its rest energy, remains constant regardless of its surroundings or the conditions it is in. However, the total energy of an electron can change if it is in motion, as it will have both kinetic energy and its rest energy.

## 5. How does the energy of an electron at rest relate to other forms of energy?

The energy of an electron at rest, or its rest energy, is just one form of energy. It is the minimum energy that an electron can possess, but it can also have kinetic energy from its motion, potential energy from its position in an electric field, and other forms of energy depending on its environment. The total energy of an electron is always conserved, and can be converted into different forms according to the laws of physics.

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