Information AboutHydrogen Atom |
| CATEGORIES ABOUT HYDROGEN ATOM | |
| fundamental physics concepts | |
| atoms | |
| quantum models | |
| hydrogen physics | |
The hydrogen atom has special significance in Quantum Mechanics as a simple physical system for which the solution to the Schrödinger Equation is Analytical , from which the positions of Energy Levels (thus, the frequencies of the hydrogen Spectral Line s) 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 solution of the Schrödinger equation goes much further, because it also yields the shape of the electron's wave function ("orbital") for the various possible quantum-mechanical states - thus explaining anisotropic character of atomic bonds. The Schrödinger equation also applies to more complicated atoms and Molecule s, however, in most cases the solution is not analytical and either computer calculations are necessary or some simplifying assumptions must be made. SOLUTION OF SCHRöDINGER EQUATION: OVERVIEW OF RESULTS The solution of the Schrödinger equation for the hydrogen atom uses the fact that the of the Hamiltonian , but also eigenstates of the Angular Momentum Operator (so called Spherical Harmonics ). 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 Number s, ''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 Polynomials in ''r''). This leads to a third quantum number, the principal 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 Symmetry ). 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 Spin 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 Superposition of these states. This explains also why the choice of z-axis for the Quantization 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. MATHEMATICAL SUMMARY OF EIGENSTATES OF HYDROGEN ATOM See Also: hydrogen-like atom The normalized position Wavefunction s, given in Spherical Coordinates are:
The Eigenvalue s are:
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:: <math> L Z N, L, M
Ang |
\hbar m n, l, m
ang </math> |
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:: <math> H N, L, M
Ang |
E_n n, l, m
ang </math> |
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Hydrogen |
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