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Unformatted text preview: 1 Introduction to Quantum and Statistical Mechanics Note 12 The Hydrogen Atom in Quantum Mechanics Assigned Reading: Hagelstein, Senturia and Orlando, Chaps. 28 and 29 12.1 Overview The quantum mechanical view of the hydrogen atom is fundamental to the development of modern physics in many aspects: • The Bohr-Rutherford’s tragedy in electrons circulating the nucleus can be avoided. In classical mechanics, the electron has to be in a circular motion to provide the centrifugal force against the Coulomb attraction, but it will radiate energy according to Maxwell’s equation. What is needed in the atomic model is a “ stationary ” state which provides the centrifugal force. • It is a theoretical triumph of quantum mechanics in explaining the experimental discrete spectral radiation lines of hydrogen under excitation (and thus all of the eigenstates are named after the original spectral lines) as shown in Fig. 12.1 that fits in the Rydberg formula in Eqs. (12.1) and (12.2). You can see the resemblance to the energy levels in the particle in a box, as the Coulomb potential is very sharp and high. 1 7 2 2 2 1 10 0974 . 1 1 1 1 − × = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = m R n n R λ (12.1) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 2 2 2 1 1 1 6 . 13 n n eV E photon (12.2) • The quantum mechanical formulation of the energy levels in the hydrogen atom (which can be calibrated by the atomic emission spectra in Fig. 12.1) provides the quantitative foundation of physical chemistry as the explanation of the periodic table. • By the Uncertainty Principle which predicts the stationary states with nonzero Δ x and Δ p , we have a consistent explanation for the simplest atomic structure with the Maxwell equation. Logically, the quantum mechanical treatment of the hydrogen atom should immediately follows that of the simple harmonic oscillator and the multi-dimension problems, without the need of the approximation methods and the density of states. It is treated afterwards since the mathematical treatment is a bit more involved, and you will be mostly shown the analytical solution, instead of making derivations. 2 In the following, we will first provide the quantum mechanical treatment of the angular momentum, which is necessary for a problem with radial symmetry. We will study the eigenstates and eigenvalues of the angular momentum, which contains spherical harmonics, with some limited resemblance to the Legendre harmonics that we used in the simple harmonic oscillators. With the angular momentum, we will formulate the hydrogen atom in the center-of- mass coordinates and perform separation of variables by the radial and spherical parts. We will finally put everything together to arrive at the quantitative description of atomic orbitals that you have known in high school chemistry, 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, etc....
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- Angular Momentum, Hydrogen atom, Niels Henrik David Bohr