Introduction: Energy Balance in Stationary States
The virial theorem relates the expectation values of kinetic and potential energy in stationary states. It provides powerful constraints without solving the full Schrödinger equation.
The Quantum Virial Theorem
For a stationary state with potential \(V(r) \propto r^n\):
\[2\langle T\rangle = n\langle V\rangle\]where \(T\) is kinetic energy and \(V\) is potential energy.
Key Results
- Harmonic oscillator (\(V \propto r^2\)): \(2\langle T\rangle = 2\langle V\rangle\), so \(\langle T\rangle = \langle V\rangle = E/2\)
- Coulomb potential (\(V \propto -1/r\)): \(2\langle T\rangle = -\langle V\rangle\), so \(\langle T\rangle = -E\) and \(\langle V\rangle = 2E\)
Worked Example: Hydrogen Atom
For hydrogen ground state with \(E = -13.6\) eV:
- \(\langle T\rangle = -E = +13.6\) eV (kinetic energy positive)
- \(\langle V\rangle = 2E = -27.2\) eV (potential energy negative)
The electron has more kinetic energy than |total energy|!
The Quantum Connection
The virial theorem shows why atoms don't collapse: as the electron gets closer to the nucleus, its kinetic energy increases faster than the potential energy decreases. There's a balance point—the Bohr radius—where the total energy is minimized.