An electron is known to be somewhere inside a carbon...

An electron is known to be somewhere inside a carbon nanotube that's $370 \mathrm{~nm}$ long and $1.2 \mathrm{~nm}$ in diameter. Find the minimum uncertainties in the components of its velocity (a) along the tube and (b) perpendicular to the tube's long dimension.

Key Concepts

Quantum Confinement

Quantum confinement refers to the effect observed when a particle, such as an electron, is restricted to a very small volume of space. In systems where one or more dimensions are comparable to the de Broglie wavelength of the particle, the particle's energy levels become discretized. This confinement leads to increased uncertainties in certain components of momentum due to the limited spatial extent available for the particle’s wavefunction.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics which states that certain pairs of physical properties, such as position and momentum, cannot both be known to arbitrary precision. This principle sets a lower bound on the product of the uncertainties in these measurements. It implies that confining a particle to a smaller region in space increases the uncertainty in its momentum, which is directly related to its velocity in the case of a free particle.

Momentum-Velocity Relationship

For a non-relativistic particle like an electron, momentum and velocity are directly related by the equation p = mv, where m is the mass of the particle. This relationship allows one to convert an uncertainty in momentum, derived from the spatial confinement via the uncertainty principle, into an uncertainty in velocity. Thus, when the spatial dimensions in different directions vary greatly, this results in differing uncertainties in the velocity components along those directions.

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