For $ n=3 $: $ E_3 = -13.6 / 3^2 = -13.6 / 9 \approx -1.511 $ eV.
["E3 Ground State Energy: Understanding the Calculated Value of $ -1.511 $ eV for $ n = 3 $", "In quantum mechanics, calculating the energy levels of an electron in a hydrogen-like atom is fundamental to understanding atomic structure. For the hydrogen atom (and hydrogen-like ions where only one electron is present), the energy of an electron in the $ n^{th} $ principal quantum state is given by the formula:", "[\nE_n = -\frac{13.6 \ \ ext{eV}}{n^2}\n]", "This equation stemms directly from the Bohr model of the atom, where $ n $ is the principal quantum number, $ E_n $ represents the electron’s kinetic and potential energy in bound states, and the constant $ 13.6 $ eV is the ionization energy of hydrogen — the energy required to remove an electron from the ground state.", "## What is $ E_3 $?", "When $ n = 3 $, the electron resides in the third energy level. Applying the formula:", "[\nE_3 = -\frac{13.6}{3^2} = -\frac{13.6}{9} \approx -1.511 \ \ ext{eV}\n]", "This negative value signifies that the electron is bound to the nucleus with a kinetic energy of $ 1.511 $ eV relative to complete ionization. Even though the sign is negative (indicating a bound state), the magnitude — approximately $ 1.511 $ electronvolts — quantifies how tightly the electron is held.", "## Why This Value Matters", "The energy $ E_3 \approx -1.511 $ eV plays a key role in multiple areas:", "- Atomic Spectra: Transitions of electrons between energy levels result in photon emissions at characteristic wavelengths. The energy difference between $ n=3 $ and lower levels defines spectral lines within the Balmer or Lyman series.\n- Chemical Bonding: The binding energy reflects how energetically favorable forming ions or molecules is.\n- Quantum Education: This calculation exemplifies how analytic formulas model quantum phenomena, enabling students and researchers to predict atomic properties accurately.", "## Direct Calculation and Precision", "Breaking down the calculation:", "[\nE_3 = -\frac{13.6}{9} = -1.511111\ldots \ \ ext{eV}\n]", "Rounded to three decimal places, $ E_3 \approx -1.511 $ eV. This precision supports high-accuracy modeling in atomic physics simulations and educational demonstrations.", "## Summary", "For $ n = 3 $, the ground-state energy of an electron in a hydrogen-like atom is:", "[\n\boxed{E_3 = -\frac{13.6}{9} \approx -1.511 \ \ ext{eV}}\n]", "This elegant result not only underscores the predictive power of fundamental quantum theory but also serves as a reference point in concurrent exploration of atomic behavior and chemical bonding. Understanding values like $ E_3 $ helps deepen insights into the microscopic world governed by quantum mechanics.", "---", "Keywords: $ E_3 $, $ n = 3 $, $ -13.6 / 9 $ eV, hydrogen energy levels, ground state energy, atomic physics, quantum mechanics, ionization energy, electron binding energy."]
