Can a Particle at Rest Have a De Broglie Wavelength?
The concept of a particle at rest with a de Broglie wavelength is a fascinating and often misunderstood topic in quantum mechanics. Let's explore this interesting question and the theoretical frameworks that govern it.
Theoretical Background
The de Broglie hypothesis, introduced by Louis de Broglie in 1924, states that particles, just like waves, exhibit wave-particle duality. The de Broglie wavelength for a particle is given by the equation:
λdB h / p
where λdB is the de Broglie wavelength, h is Planck's constant, and p (momentum) is given by the product of the particle's mass m and velocity v:
p mv
In the case of a particle at rest, v 0, which implies that p 0.
De Broglie Wavelength at Rest
Applying the de Broglie wavelength formula to a particle at rest, we find that:
λdB h / 0
This expression is undefined, not just because it involves division by zero, but also because it violates the principles of quantum mechanics. For a particle at rest, the wavelength becomes infinite.
In classical mechanics, an object at rest does not possess a wavelength because it is not in motion. However, in quantum mechanics, this concept is more nuanced. An infinitely long wavelength corresponds to a situation where the particle's position is indeterminate, and thus its location can be found with equal probability anywhere in the universe.
Alternative Measures
There are alternative ways to measure the properties of a particle at rest, such as the Compton wavelength. The Compton wavelength, denoted by WL, is defined as:
WL h / mc
where m is the particle mass and c is the speed of light. For an electron, the Compton wavelength is approximately:
2.426 X 10^-10 cm (about 100 times smaller than an atom)
This Compton wavelength is a finite value and provides a clear measurement even for a particle at rest, unlike the de Broglie wavelength.
Conclusion
In summary, a particle at rest cannot have a de Broglie wavelength because it would involve division by zero, which is undefined. Instead, we use alternative measures such as the Compton wavelength to describe the properties of a particle at rest. The division by zero is meaningless in this context and highlights the boundary between classical and quantum mechanics.
References:
[1] De Broglie, L. (1924). "On the Theory of Quanta". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, 177, 507-510.
[2] Planck, M. (1901). "über den Cardiff Theory Relating to Radiation and Blackbody Emission". Annalen der Physik, 4, 553-563.
[3] Compton, A. H. (1923). "A Quantum Theory of the Scattering of X-rays by Light Elements". Physical Review, 21(3), 483-493.