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 numerical value $-13.6 \mathrm{eV}$.

Key Concepts

Unit Conversion (Joules to Electron Volts)

Unit conversion is a critical aspect when comparing theoretical predictions with experimental results. In atomic physics, energy is often calculated in Joules but can be more conveniently expressed in electron volts (eV) for comparison with observed values. Converting between these units is essential to validate theoretical models such as the Bohr model with the empirically determined ground state energy of hydrogen.

Bohr Model

The Bohr model is a theoretical framework for understanding the hydrogen atom, in which electrons move in fixed orbits around the nucleus with quantized energy values. This model introduces the idea that electron energies are discrete and tied to specific orbital radii, leading to a formula for the energy levels of the atom that can be compared directly to experimental data.

Quantization of Energy

Quantization of energy in atomic systems means that electrons can only occupy certain allowed energy states rather than a continuous range of energies. In the context of the Bohr model, these quantized energy levels arise from fundamental physical principles, leading to specific formulas that predict the numerical values of energies, such as the 13.6 eV for hydrogen's ground state.

Fundamental Physical Constants

Fundamental physical constants, such as the electron mass, the elementary charge, and Planck’s constant, are integral in calculating physical quantities in atomic physics. These constants, when combined in the energy expression derived from the Bohr model, yield specific numerical predictions that align with experimental observations, demonstrating the interplay between theory and measurement.

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