A mercury atom moves in a cubical box whose edge is 1 m long. Its kinetic energy is equal to th

step 1 Conservation of energy leads half M .E. V squared, that's the initial energy equals half M .E ti

A mercury atom moves in a cubical box whose edge is 1 m...

A mercury atom moves in a cubical box whose edge is $1 \mathrm{~m}$ long. Its kinetic energy is equal to the average kinetic energy of an atom of an ideal gas at $1000 \mathrm{~K}$. If the quantum numbers $n_{x}, n_{y}$, and $n_{z}$ are all equal to $n$, calculate $n .$

Key Concepts

Quantized Energy Levels

In systems like a particle in a cubic box, the energy is not continuous but takes on specific discrete values determined by quantum numbers. These quantum numbers arise due to the boundary conditions and are used to index the different energy states, with the energy typically being proportional to the sum of the squares of these numbers.

Equipartition Theorem

In classical statistical mechanics, the equipartition theorem states that energy is distributed equally among all degrees of freedom, with each quadratic degree contributing (1/2)kT to the system’s energy. For example, an ideal gas atom in three dimensions has an average kinetic energy of (3/2)kT.

Relation Between Quantum and Classical Descriptions

By equating the quantized energy expression derived from the particle in a box model with the average kinetic energy from the equipartition theorem, one creates a bridge between quantum and classical mechanics. This approach can be used to estimate the quantum numbers corresponding to a given classical energy level.

Particle in a Box

This concept refers to a quantum mechanical model where a particle is confined within an impenetrable boundary, resulting in the quantization of energy levels. The allowed states of the particle come from solving the Schrödinger equation with boundary conditions that force the wavefunction to vanish at the edges of the box.

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