Q17Engineering Physics

The "particle in a 1D box" (or infinite potential well) is a foundational model in quantum mechanics. It describes a particle perfectly free to move back and forth along a single line segment of length $L$, but strictly bounded by impenetrable, infinitely high potential walls at either end. Solving the Schrödinger equation for this system reveals that the particle cannot possess arbitrary energy; its kinetic energy must be strictly quantized into discrete, specific levels.

1. The Quantum Energy Formula

The rigorously derived formula for the allowed energy levels ($E_n$) of a particle of mass $m$ confined within a 1D box of width $L$ is:

Where: - $n$ is the principal quantum number ($n=1,2,\mathrm{3...}$). Crucially, $n$ cannot be zero. The lowest possible energy state ($n=1$) is called the ground state, and its non-zero energy is known as zero-point energy. - $h$ is Planck's universally constant ($6.626\times 10^{-34}\text{ Jcdots}$). - $m$ is the rest mass of the confined particle (for an electron, $m=9.11\times 10^{-31}\text{ kg}$). - $L$ is the physical width of the well.

2. Data Preparation and Unit Conversion

We are asked to find the ground state energy, so we set $n=1$. The width of the box is given in Angstroms, which must be converted to standard SI meters to maintain dimensional consistency.

• Quantum State ($n$) = 1
• Width ($L$) = $1\text{ A˚}=1.0\times 10^{-10}\text{ m}$
• Planck's Constant ($h$) = $6.626\times 10^{-34}\text{ Jcdots}$
• Electron Mass ($m$) = $9.11\times 10^{-31}\text{ kg}$

3. Calculation in Joules

Substitute the constants into the energy formula:

Evaluate the numerator (squaring Planck's constant):

Evaluate the denominator:

Perform the division to find the energy in standard Joules:

4. Final Conversion to Electron-Volts (eV)

In atomic physics, energies are almost exclusively expressed in electron-volts (eV). We convert Joules to eV by dividing by the elementary charge ($1.602\times 10^{-19}\text{ J/eV}$):

This incredibly high ground state energy (compared to the $13.6\text{ eV}$ ionization energy of hydrogen) illustrates the immense quantum pressure generated when forcing a particle into severe spatial confinement.