AcademyAtomic Quantum Structure
Academy
Hydrogen Atom
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
The hydrogen atom is the central example of a three-dimensional bound quantum system. Solving the Schrodinger equation in a Coulomb potential gives quantized energy levels and orbital quantum numbers.
The energy depends on the principal quantum number \(n\), while angular behavior is described by \(\ell\) and \(m_\ell\).
Notation
\(n\)
principal quantum number
1
\(\ell\)
orbital angular momentum quantum number
1
\(m_\ell\)
magnetic quantum number
1
\(E_n\)
hydrogen energy level
\(\mathrm{eV}\)
\(a_0\)
Bohr radius
\(\mathrm{m}\)
\(L\)
orbital angular momentum magnitude
\(\mathrm{J\,s}\)
Method
Derivation 1: Use the Coulomb potential
The electron is bound by the proton's electric potential energy.
Potential energy
\[U(r)=-\frac{1}{4\pi\epsilon_0}\frac{e^2}{r}\]
Stationary equation
\[-\frac{\hbar^2}{2m_e}\nabla^2\phi+U(r)\phi=E\phi\]
Derivation 2: Identify quantum numbers
Separation in spherical coordinates produces three quantum numbers.
Allowed orbital quantum numbers
\[\ell=0,1,2,\ldots,n-1\]
Allowed magnetic quantum numbers
\[m_\ell=-\ell,-\ell+1,\ldots,+\ell\]
Derivation 3: Use the energy and angular-momentum rules
Hydrogen energy depends only on \(n\) in the basic Coulomb model.
Energy levels
\[E_n=-\frac{13.6\,\mathrm{eV}}{n^2}\]
Angular momentum magnitude
\[L=\sqrt{\ell(\ell+1)}\hbar\]
Angular momentum projection
\[L_z=m_\ell\hbar\]
Rules
Hydrogen energy
\[E_n=-\frac{13.6\,\mathrm{eV}}{n^2}\]
Orbital angular momentum
\[L=\sqrt{\ell(\ell+1)}\hbar\]
Angular momentum projection
\[L_z=m_\ell\hbar\]
Allowed magnetic quantum numbers
\[m_\ell=-\ell,-\ell+1,\ldots,+\ell\]
Examples
Question
Find the hydrogen energy for
\[n=3\]
Answer
\[E_3=-\frac{13.6}{9}=-1.51\,\mathrm{eV}\]
Checks
- Bound-state energies are negative.
- Larger \(n\) means less tightly bound and larger typical radius.
- For a given \(n\), allowed \(\ell\) values run from 0 to \(n-1\).
- For a given \(\ell\), there are \(2\ell+1\) allowed \(m_\ell\) values.