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What are the types of energy that an atom or a molecule could have?

For example they have kinetic energy, could they also have other types?

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From a quantum prespective, it is helpful to categorize the energy of molecules as:

  1. electronic
  2. vibrational (bending and stretching of bonds)
  3. rotational and
  4. translational.

However, these are each inclusive of potential and kinetic energy.

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The Lagrangian $\mathcal L$ and Hamiltonian $\mathcal H$ are mathematical objects that can be used to describe the behaviour of dynamical systems.

In classical systems the Lagrangian is the Kenetic energy minus the potential energy, whereas the Hamiltonian is the the Kenetic energy plus the potential energy.

Most systems can be fully described by the ammount of energy they have from being in a given position / state (potential energy) and the ammount of energy it has by travelling at a given speed (its kenetic energy).

The potential energy is often expanded for gasses and chemical reactions into different types of potentials such as each of the chemical potential.

There also exists internal energy $E=mc^2$ which may be overlooked as usually it does not contribute to the dynamics of the system since it usually does not change, and dissappears when taking the derivative.

In the Standard model there exists an equation describing everything that we know about particle physics. Here is an image of the Lagrangian, each term being added together is an energy term, the energy that a particle can have from each of the different fields.

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