# Classical vs. Quantum Defintion of Energy in Field Theory

• I
• LarryS
In summary, classical fields are constructed using classical harmonic oscillators, where the energy is proportional to the amplitude squared. In quantum field theory, the same classical fields are quantized and the energy is proportional to frequency. There is a mathematical connection between the classical and quantum definitions of energy, as the frequency is related to the amplitude through the spring constant. However, the quantum definition is not arbitrary and is demonstrated in nature. Classical and quantum descriptions are both modeled through Hamiltonian mechanics, with the classical version being an average of the quantum versions.
LarryS
Gold Member
Classical fields are usually constructed using a collection of classical harmonic oscillators, e.g. masses connected to springs. The energy of a classical harmonic oscillator is proportional to the amplitude squared. QFT uses quantized versions of those same classical fields. But, in the quantum domain energy is proportional to frequency, not amplitude squared. Is there any mathematical connection between this classical definition of energy and the quantum definition (frequency)? Or is the classical definition of energy simply discarded and arbitrarily replaced with the quantum definition?

The frequency of a HO is a characteristic of the system in both the classical and quantum regimes.
The energy stored in the quantum HO is proportional to "n" - the principle quantum number.
The quantization is not arbitrary - the value of h was determined from experiiment, and the quantization was demonstrated the same way.

Note: for the classical HO: ##E= m\omega^2A^2## ... so now it is in terms of both amplitude and frequency.
... for a quantized SHO, the amplitude A is related to the energy level n as: ##\frac{1}{2}kA^2 = (n+\frac{1}{2})\hbar\omega## ... that is to say, if you somehow had a physical mass on a spring that could only have quantized energy, then the amplitude would also be quantized.

None of this is arbitrary - it was not just pulled from the air.
The quantization is demonstrated in Nature.

Both the classical and quantum descriptions are modeled through hamiltonian mechanics.
The classical version is what you get on average over the quantum versions.

bhobba
Simon Bridge said:
The frequency of a HO is a characteristic of the system in both the classical and quantum regimes.
The energy stored in the quantum HO is proportional to "n" - the principle quantum number.
The quantization is not arbitrary - the value of h was determined from experiiment, and the quantization was demonstrated the same way.

Note: for the classical HO: ##E= m\omega^2A^2## ... so now it is in terms of both amplitude and frequency.
... for a quantized SHO, the amplitude A is related to the energy level n as: ##\frac{1}{2}kA^2 = (n+\frac{1}{2})\hbar\omega## ... that is to say, if you somehow had a physical mass on a spring that could only have quantized energy, then the amplitude would also be quantized.

None of this is arbitrary - it was not just pulled from the air.
The quantization is demonstrated in Nature.

Both the classical and quantum descriptions are modeled through hamiltonian mechanics.
The classical version is what you get on average over the quantum versions.

Thanks. I forgot about the "spring constant" k. That is the mathematical connection between the amplitude (classical) version of energy and the frequency (quantum) version. Also, I did not mean to imply that the quantum version of energy was completely arbitrary, only that I could not be mathematically derived from the classical/amplitude version.

I'm not being clear: classical and quantum use the same definition of energy.
 Also ... No quantum theory is derived from the classical. If it was possible to do this, then quantum theory would be a subset of classical theory and we wouldn't actually need it for what we use it for.
What you need to look for is a derivation of the classical result from the quantum theory.
Classical physics is what happens on average.

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## 1. What is the difference between classical and quantum definition of energy in field theory?

The classical definition of energy in field theory is based on classical mechanics and considers energy as a continuous, measurable quantity. On the other hand, the quantum definition of energy takes into account the principles of quantum mechanics and considers energy as discrete and quantized.

## 2. How do classical and quantum definitions of energy differ in terms of the behavior of particles?

In classical field theory, particles are seen as point-like objects with definite positions and momenta, and their energy can be calculated using classical equations. In quantum field theory, particles are described as wave-like entities with uncertain positions and momenta, and their energy is quantized according to the energy levels allowed by the system.

## 3. Can classical and quantum definitions of energy be reconciled?

There is ongoing research in the field of quantum gravity and quantum field theory to find a way to reconcile the classical and quantum definitions of energy. Some theories propose that classical mechanics can be seen as an approximation of quantum mechanics under certain conditions.

## 4. How does the concept of energy conservation differ between classical and quantum field theory?

In classical field theory, energy conservation is a fundamental principle and is always conserved. In quantum field theory, energy conservation is still a valid principle, but there are certain situations where energy can appear to be non-conserved due to the uncertainty principle and the possibility of particles spontaneously appearing and disappearing in the quantum field.

## 5. Are there any practical implications of the differences between classical and quantum definitions of energy in field theory?

Yes, the differences between classical and quantum definitions of energy have significant practical implications in fields such as particle physics and quantum computing. The ability to accurately measure and manipulate energy at the quantum level has led to the development of technologies such as quantum computers, which have the potential to revolutionize computing power. Additionally, the principles of quantum field theory are essential for understanding and predicting the behavior of particles at the subatomic level.

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