Kinetic Energy, Momentum, and Their Relationship in Classical and Quantum Mechanics

Understanding the relationship between kinetic energy and momentum is crucial in both classical and quantum mechanics. This article explores how these two fundamental concepts are interconnected and under what circumstances they can or cannot exist independently.

Classical Mechanics and the Relationship Between Kinetic Energy and Momentum

In classical mechanics, momentum p and kinetic energy KE are directly related through the equation KE p2/2m. Here, m is the mass of the object and v is the velocity of the object with v p/m. From this, it becomes clear that kinetic energy is inherently tied to both mass and momentum. Without momentum, a body cannot have kinetic energy, and the product of these variables is always positive.

A common misconception arises with particles like photons, where they can have momentum without possessing mass. However, photons do not have kinetic energy in the traditional sense because their kinetic energy is defined differently: E hf, where h is Planck's constant and f is the frequency of the photon. This definition is not typically considered kinetic energy in the classical sense.

Additionally, for particles like protons and neutrons, considering spin as a form of motion can imply that kinetic energy exists even in a stationary state. Nevertheless, in a classical sense, momentum and kinetic energy cannot coexist without mass.

Momentum without Kinetic Energy: The Case of Photons

Photons are a unique case in modern physics. Despite their zero mass, photons have momentum given by p h/λ, where h is Planck's constant and λ is the wavelength. However, because photons do not have mass, they do not possess kinetic energy in the traditional sense. This relationship can be summarized using the equation for momentum: p √2mK. Therefore, photons can have momentum without kinetic energy, emphasizing the distinct nature of these particles.

Quantum Mechanics and the Expectation Value of Momentum and Kinetic Energy

In quantum mechanics, the relationship between kinetic energy and momentum is more complex and can be influenced by the particle's confinement and spatial distribution. The expectation value of kinetic energy is always nonzero if the particle is not spread over all space. Even in a state where the particle has no momentum or its momentum is not directly observable, the expectation value of kinetic energy remains non-zero.

Important to note is that every particle which is not in the zero momentum state (extended over the whole space) will give a nonzero answer when measured for momentum. Despite the momentum possibly being zero in a given reference frame, the expectation value of kinetic energy will still be non-zero due to the spatial wavefunction confinement.

Conclusion

From the classical perspective, kinetic energy and momentum are intrinsically linked, with kinetic energy being dependent on both mass and momentum. In the quantum realm, the relationship is more nuanced, with the expectation value of kinetic energy always being non-zero except in very specific cases where the particle is in a state of zero momentum spread over all space.

Whether in classical or quantum mechanics, the interplay between kinetic energy and momentum highlights their fundamental roles in understanding the behavior of particles and systems.

References

1. Feynman, R., Leighton, R., Sands, M. (1964). The Feynman Lectures on Physics. Addison-Wesley.

2. Bohm, D. (1951). A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I and II. Physical Review, 85(2), 166–193.

3. Sakurai, J. J., Napolitano, J. (2011). . Cambridge University Press.