It is true that E=Ex+Ey+Ez is valid for both classical physics and quantum mechanics.
What is surprising is that this formula is the least understood and the least justified.
The formula (E=Ex+Ey+Ez) you mentioned typically represents the idea that the total energy E of a system in three-dimensional space can be decomposed into the energy contributions along each of the three spatial dimensions—x, y, and z. This decomposition is an expression of the principle that physical quantities like energy in different directions can be treated independently and then summed to get the total.
In classical physics, this formulation aligns with the concept of energy conservation and the additive nature of scalar quantities like energy. For example, if you have a moving object, its total kinetic energy can be considered as the sum of kinetic energies due to motion in each of the three perpendicular directions (x, y, and z axes). This is because the kinetic energy, a scalar quantity, is dependent on the square of the velocity vector components in each of these directions.
In quantum mechanics, the situation is subtly different but the formula still holds under specific conditions. Quantum mechanics often deals with probabilities and wave functions, but when it comes to observable quantities like energy, these also add up linearly across independent directions. When the Hamiltonian (the energy operator in quantum mechanics) of a system is separable (i.e., can be written as a sum of Hamiltonians for each dimension), then the total energy of the system is indeed the sum of the energies associated with each dimension.
This ability to sum energies linearly across dimensions, even in quantum mechanics, is due to the fundamental principles underlying the physics in both classical and quantum contexts. The energy in each dimension behaves independently of the others when the forces or influences acting in those dimensions do not interact or interfere with one another. For example, in a quantum system like a hydrogen atom, the Hamiltonian can be broken down into radial and angular parts (which correspond to separations in different coordinate dimensions), and the total energy is the sum of these parts when no external fields or interactions complicate this separation.
The justification for this formula in both realms hinges on the independence of motion or states in different dimensions and the linear, scalar nature of energy as a physical quantity. This is a powerful concept because it simplifies the analysis of complex systems, whether you're looking at a macroscopic object moving through space or an electron's behavior in an atom. I hope this somehow answered your question, which if I may add, is a great one! :D
To my understanding,, there is no single one-dimension formula exist to describe three-dimension of nature.
On the other hand, science has no explanation for energy. What is energy?
to continue, there is no evidence of collapsing CP, or modern physics in to quantum. The last 100 years is hard evidence.
Again, every formula is flat static that does not recognize temperature, pressure, color, taste, smells....simultaneously, while nature in three-dimension that it is changing does.
unfortunately we accustomed to think that our math can describe nature, but it does not.
Dear Javad Fardaei ,
''On the other hand, science has no explanation for energy. What is energy?''
# As simple as it may sound ''energy is just information (including temperature, pressure, color, taste, smells etc... all being measurable consequences of motion)''.
''unfortunately we accustomed to think that our math can describe nature, but it does not.''
# In real-time/imaginary-time (in the visible/invisible universe), a 100 % accurate Information would simply mean a 100% choerence between math and the described physical system (nature)... although we can't commonly define and measure stuffs like consciousness, still the effectiveness of math must not be confused with humans' failures to achieve 100% accuracies in measurements and definitions.
''To my understanding, there is no single one-dimension formula exist to describe three-dimensions of nature.''
# I agree 100%, and one thing I'm sure about:
Like bits, such a formula (to describe three or even more dimensions of nature) is dimensionless, and that's why dear Ismail Abbas ''even if E=Ex+Ey+Ez isn't wrong'' it still can't clearly be justified since it is incomplete to be called total.