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The hydrogen atom has special significance in quantum mechanics as a simple physical system for which an exact solution to the Schrödinger equation exists, from which the experimentally observed frequencies and intensities of the hydrogen spectral lines can be calculated.
In 1913, Niels Bohr had deduced the spectral frequencies of the hydrogen atom making several assumptions (see The Bohr Model). The results of Bohr for the frequencies and underlying energy values are confirmed by the full quantum-mechanical analysis which uses the Schrödinger equation, as was shown in 1925/26. The full analysis goes further, because it also yields the shape of the electron's wave function ("orbital") for the different possible quantum-mechanical states. This allows to determine the intensity of spectral lines (which correspond to transitions between these states), among other things. In addition, the full analysis is applicable also to more complicated atoms with more than one electron, as well as molecules etc. However, in all of these cases approximations have to be made and computer calculations are usually necessary.
The solution of the Schrödinger equation for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it 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 states are not only eigenstates of the HamiltonianIn physics, Hamiltonian has two distinct but closely related meanings. In classical mechanics, the Hamiltonian is a function describing the state of a mechanical system in terms of position and momentum variables. See Hamiltonian mechanics. In quantum mec, but also eigenstates of the angular momentum operator . This corresponds to the fact that angular momentum is conserved in the motion of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum quantum numberA quantum number is a number used to parametrise certain properties of particles or other systems in quantum mechanics. Combinations of quantum numbers can be used to identify eigenstates of the system. Each quantum number represents a specific degree ofs, l and m (integer numbers). The "angular momentum" quantum number l = 0, 1, 2, ... determines the magnitude of the angular momentum. The "magnetic" quantum number m = −l, .. , +l determines the projection of the angular momentum on the (arbitrarily chosen) z-axis.
In addition, the radial dependence of the wave functions has to be found. It is only here that the details of the 1/r Coulomb potential enter (leading to Laguerre polynomialsIn mathematics, the Laguerre polynomials named after Edmond Laguerre (1834 1886), are a polynomial sequence defined by : These polynomials are orthogonal to each other with respect to the inner product given by : Generalization The orthogonality property in r). This leads to a third quantum number, the "main" quantum number n = 1, 2, 3, ... Note that the angular momentum quantum number 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 symmetryIn physics and mathematics, rotational symmetry is the invariance of an object or a system of equations under the rotations for example the SO(3) transformations of the three-dimensional space. Laws of physics are rotationally invariant if they do not dis). In addition, for the hydrogen atom, the states of the same n are also degenerate (i.e. they have the same energy); but this is a specialty and it is no longer true for more complicated atoms which have an (effective) potential differing from the form 1/r (due to the presence of the inner electrons shielding the nucleus potential).
Taking into account the spinSpin has several meanings, including those primarily discussed as spinning For physical rotation or an analogous phenomenon in sub-atomic physics, see spin (physics) For the periodical, see Spin Magazine Computer: For unproductive repetition in a computer of the electron adds a last quantum number, the projection of the electrons spin along the z axis, which can take on two values. Therefore, any eigenstate of the electron in the hydrogen atom is described fully by four quantum numbers. According to the usual rules of quantum mechanics, the actual state of the electron may be any superpositionThe term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning. of these states. This explains also why the choice of z-axis for the quantizationGenerally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. In signal processing, quantization is the process of approximating a continuous signal by a set of discrete symbols or integer values. of angular momentum is immaterial: An orbital of given l and m' obtained for another preferred axis z' can always be represented as a suitable superposition of the various states of different m (but same l) that have been obtained for z.