Understanding Particle Energies in Ohanian's Book: A Potential Problem Explored

[vin300]

I saw today in Ohanian's book:
"It is convenient to separate particle energies into kinetic and rest mass energies:
M=∑m +(∑T/c^2 +U/c^2)
The term in the paranthesis is negative for a bound system,hence the total mass is less than the sum of the particle rest masses"
I thought the rest masses were indeed less.If it is otherwise, there's a problem with the Energy momentum four vector

 

[Dale]

I saw today in Ohanian's book:
"It is convenient to separate particle energies into kinetic and rest mass energies:
M=?m +(?T/c^2 +U/c^2)
The term in the paranthesis is negative for a bound system,hence the total mass is less than the sum of the particle rest masses"

This is correct. The "missing" mass is called the "mass deficit" and that mass/c^2 is called the "binding energy". Note, the mass deficit only appears when the binding energy has been removed from the system, usually either through radiation or through doing work on another system.

I thought the rest masses were indeed less.If it is otherwise, there's a problem with the Energy momentum four vector

The problem goes away if you include the four-momentum of the binding energy which left the system.

 

[Entropia]

You are correct in your understanding that the rest masses of particles are indeed less than the total mass of a bound system. This is known as mass defect and is a fundamental concept in nuclear physics. It arises from the release of energy during the formation of a bound system, such as a nucleus. This energy is released in the form of binding energy, which contributes to the negative term in the equation you mentioned.

The concept of mass defect is crucial for understanding the stability and energy balance of atomic nuclei. It also plays a significant role in nuclear reactions and the production of energy in nuclear power plants.

As for the energy-momentum four vector, it is a mathematical representation of the energy and momentum of a particle in special relativity. The negative term in the equation you mentioned represents the energy contribution to the total mass, while the positive term (Σm) represents the rest masses of the particles. Thus, there is no problem with the energy-momentum four vector, as it accurately reflects the energy and mass relationship of bound systems.

Overall, understanding the separation of particle energies into kinetic and rest mass energies is crucial for understanding the behavior of particles in bound systems. It also highlights the intricate balance between energy and mass in the universe.