In 1914, Niels Bohr obtained the spectral frequencies of the hydrogen atom after making a number of simplifying assumptions. These assumptions, the cornerstones of the Bohr model, were not fully correct but did yield the correct energy answers. Bohr's results for the frequencies and underlying energy values were confirmed by the full quantum-mechanical analysis which uses the Schrödinger equation, as was shown in 1925–1926. The solution to the Schrödinger equation for hydrogen is analytical. From this, the hydrogen energy levels and thus the frequencies of the hydrogen spectral lines can be calculated. The solution of the Schrödinger equation goes much further than the Bohr model, because it also yields the shape of the electron's wave function (orbital) for the various possible quantum-mechanical states, thus explaining the anisotropic character of atomic bonds.
The Schrödinger equation also applies to more complicated atoms and molecules, albeit they rapidly become impossibly difficult beyond hydrogen or other two-body type problems, such as helium cation He+. In most such cases, the solution is not analytical and either computer calculations are necessary or simplifying assumptions must be made.
Solution of Schrödinger Equation: Overview of Results
The solution of the Schrödinger equation (wave equations) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it is radially symmetric in space and only depends on the distance to the nucleus). Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely, generally from this isotropy of the underlying potential. The eigenstates of the Hamiltonian (that is, the energy eigenstates) can be chosen as simultaneous eigenstates of the angular momentum operator. This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, ℓ and m (both are integers). The angular momentum quantum number ℓ = 0, 1, 2, ... determines the magnitude of the angular momentum. The magnetic quantum number m = −, ..., +ℓ determines the projection of the angular momentum on the (arbitrarily chosen) z-axis.
In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wave functions must be found. It is only here that the details of the 1/r Coulomb potential enter (leading to Laguerre polynomials in r). This leads to a third quantum number, the principal quantum number n = 1, 2, 3, .... The principal quantum number in hydrogen is related to the atom's total energy. Note the maximum value of the angular momentum quantum number is limited by the principal quantum number: it can run only up to n − 1, i.e. ℓ = 0, 1, ..., n − 1.
Due to angular momentum conservation, states of the same ℓ but different m have the same energy (this holds for all problems with rotational symmetry). In addition, for the hydrogen atom, states of the same n but different ℓ are also degenerate (i.e. they have the same energy). However, this is a specific property of hydrogen and is no longer true for more complicated atoms that have a (effective) potential differing from the form 1/r, due to the presence of the inner electrons shielding the nucleus potential.
Calculated Energy Levels
The energy levels of hydrogen are given by solving the Schrödinger equation for the one-electron atom:
where α is the fine-structure constant and j is a number which is the total angular momentum eigenvalue; that is, ℓ ± 1/2 depending on the direction of the electron spin. The quantity in square brackets arises from relativistic (spin-orbit) coupling interactions.
The value of 13.6 eV is called the Rydberg constant and can be found from the Bohr model and is given by: