The following working derives the energy of the hydrogen atom's ground-state Bohr orbit

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Origins of quantum theory

Reading the Spectrum, Ch The nuclear atom and the Bohr-Sommerfeld model

Problem

The following working derives the energy of the hydrogen atom's ground-state Bohr orbit. Step 1. For a circular orbit,

\dfrac{m_ev^2}{r} = \dfrac{e^2}{4\pi\epsilon_0 r^2}

, so the kinetic energy is

T = \tfrac12 m_ev^2 = \dfrac{e^2}{8\pi\epsilon_0 r}

. Step 2. The quantisation condition

m_evr=n\hbar

with

n=1

gives

r_1=a_0=0.052918\,\mathrm{nm}

. Step 3. Substituting,

T_1=\dfrac{e^2}{8\pi\epsilon_0a_0}=13.606\,\mathrm{eV}

. Step 4. "The postulates give no other form of energy in this problem besides the kinetic energy calculated above, so the total energy of the atom in its ground state is

E_1=T_1=+13.606\,\mathrm{eV}

." ((a)) Identify the step at which the reasoning goes wrong, and say what physical contribution has been left out of the total energy. ((b)) Give the correct value of

E_1

, with its sign, and check it against the relation

E_n = \tfrac12U(r_n)

Related theory

Quantum Mechanics

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