By directly substituting the values of the fundamental...

By directly substituting the values of the fundamental constants, show that the ground state energy for hydrogen in the Bohr model $E_{1}=-m_{\mathrm{e}} k^{2} e^{4} /\left(2 \hbar^{2}\right)$ has the nu- $-$ merical value $-13.6 \mathrm{eV}$.

Key Concepts

Bohr Model

The Bohr model of the hydrogen atom is a foundational concept in atomic physics that introduces the idea of quantized electron orbits. In this model, an electron moves in circular orbits around the nucleus with specific allowed radii and energies. The quantization arises from the condition that the electron’s angular momentum is an integer multiple of the reduced Planck constant, leading to discrete energy levels. This model provides a simple explanation for the spectral lines of hydrogen and forms the basis for more sophisticated quantum mechanical treatments of atoms.

Fundamental Physical Constants

Using physical constants accurately is essential in deriving meaningful numerical results in physics. Constants such as the electron mass (m_e), the elementary charge (e), the reduced Planck's constant (?), and Coulomb's constant (k) appear in the formula for the ground state energy in the Bohr model. Direct substitution of these constants, measured in SI units, into the energy expression demonstrates the bridge between theoretical derivations and experimentally observed values, thereby validating the model.

Unit Conversion (Joules to Electron Volts)

The energy calculated from the Bohr model is initially obtained in joules, which is the standard unit of energy in the International System of Units. However, for atomic-scale energies, the electron volt (eV) is a more practical unit. Converting between joules and electron volts is a key step in comparing theoretical predictions from the Bohr model with observed spectral data. This conversion factor helps in expressing the ground state energy value in a familiar form for atomic and molecular physicists.

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