A bead of mass 5.00 g slides freely on a wire 20.0 cm long. Treating this system as a particle i

step 1 Okay, so we have a bead that slides freely on a wire. We're going to treat the system as a parti

A bead of mass 5.00 g slides freely on a wire 20.0 cm long....

A bead of mass $5.00 \mathrm{g}$ slides freely on a wire $20.0 \mathrm{cm}$ long. Treating this system as a particle in a one-dimensional box, calculate the value of $n$ corresponding to the state of the bead if it is moving at a speed of $0.100 \mathrm{nm}$ per year (that is, apparently at rest).

A bead of mass $5.00 \mathrm{g}$ slides freely on a wire $20.0 \mathrm{cm}$ long. Treating this system as a particle in a one-dimensional box, calculate the value of $n$ corresponding to the state of the bead if it is moving at a speed of
$0.100 \mathrm{nm}$ per year (that is, apparently at rest).

Key Concepts

Particle in a One-Dimensional Box

This concept involves the quantum mechanical model where a particle is confined to move in a one-dimensional region with infinitely high potential barriers at the ends. The restrictions imposed by the boundaries quantize the allowed energy levels and momenta of the particle, meaning that only certain discrete states (or quantum numbers) are possible.

Energy Quantization in Confined Systems

In a confined system like a particle in a one-dimensional box, the energy levels are not continuous but are given by specific formulas that depend on the quantum number and the dimensions of the confinement. This quantization arises due to the wave nature of particles and the boundary conditions imposed on their wavefunctions.

de Broglie Wavelength and Momentum Quantization

The de Broglie relation connects the momentum of a particle with its wavelength, implying that the confined particle’s wavelength must fit an integral number of half-wavelengths into the length of the box. This leads to the quantization of momentum, where the momentum is directly related to the quantum number and fundamental constants such as Planck’s constant.

Determination of Quantum Number (n)

The quantum number 'n' identifies the specific state of the particle within the quantized energy spectrum. It is determined by relating the classical observable (such as speed or momentum) to the quantized expressions derived from the particle-in-a-box model. In problems like this, matching the given speed to the corresponding quantum mechanical expression helps deduce the value of 'n'.

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