text: An alternative approach for applying the Hellmann–Feynman theorem is to promote a fixed or discrete parameter which appears in a Hamiltonian to be a continuous variable solely for the mathematical purpose of taking a derivative. Possible parameters are physical constants or discrete quantum numbers. As an example, the radial Schrödinger equation for a hydrogen-like atom is
  which depends upon the discrete azimuthal quantum number . Promoting  to be a continuous parameter allows for the derivative of the Hamiltonian to be taken:
  The Hellmann–Feynman theorem then allows for the determination of the expectation value of  for hydrogen-like atoms:
  In order to compute the energy derivative, the way  depends on  has to be known. These quantum numbers are usually independent, but here the solutions must be varied so as to keep the number of nodes in the wavefunction fixed. The number of nodes is , so .