Hydrogen, atom, quantum state stationary states

Solution of (12) gives the complete non-relativistic quantum-mechanical description of the hydrogen atom in its stationary states. The wave function is interpreted in terms of... 

Ket notation is sometimes used for functions in quantum mechanics. In this notation, the function / is denoted by the symbol j/) /—1/>. Ket notation is convenient for denoting eigenfunctions by listing their eigenvalues. Thus nlm) denotes the hydrogen-atom stationary-state wave function with quantum numbers , /, and m. 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

Soon after Bohr developed his initial configuration Arnold Sommerfeld in Munich realized the need to characterize the stationary states of the electron in the hydrogen atom by. means of a second quantum number—the so-called angular-momentum quantum number, Bohr immediately applied this discovery to many-electron atoms and in 1922 produced a set of more detailed electronic configurations. In turn, Sommerfeld went on to discover the third or inner, quantum number, thus enabling the British physicist Edmund Stoner to come up with an even more refined set of electronic configurations in 1924. 

The electron which responds to both quantum and classical potential fields exhibits this dual nature in its behaviour. Like a photon, an electron spreads over the entire region of space-time permitted by the boundary conditions, in this case stipulated by the classical potential. At the same time it also responds to the quantum field and reaches a steady, so-called stationary, state when the quantum and classical forces acting on the electron, are in balance. The best known example occurs in the hydrogen atom, which is traditionally described to be in the product state tpH = ipP ipe, hence with broken holistic symmetry. In many-electron atoms the atomic wave function is further fragmented into individual quantum states for pairs of electrons with paired spins. 

The incompatibility of Rutherford s planetary model, based soundly on experimental data, with the principles of classical physics was the most fundamental of the conceptual challenges facing physicists in the early 1900s. The Bohr model was a temporary fix, sufficient for the interpretation of hydrogen (H) atomic spectra as arising from transitions between stationary states of the atom. The stability of atoms and molecules finally could be explained only after quantum mechanics had been developed. 

Figure 7.9 Quantum staircase. In this analogy for the energy levels of the hydrogen atom, an electron can absorb a photon and jump up to a higher step (stationary state) or emit a photon and jump down to a lower one. But the electron cannot lie between two steps.

Until now in our quantum mechanical discussions we have described the stationary or time-independent states of a system. Furthermore, our language was such as to imply that we had sufficient information about the system to know that it was in a particular state described by a particular set of quantum numbers. For example, in the case of the harmonic oscillator we spoke as though we knew that the oscillator was in the pth state with wave function i// , and energy = (v A- j)hv or, in the case of the hydrogen atom, that it was in a state described by the set of numbers n, /, m. This approach is very useful in a first discussion of quantum mechanical properties of various kinds of systems. However, we do not have reason to presuppose that a system is in a particular quantum state. 

The quantum-mechanical treatment of the hydrogen atom has been thoroughly worked out. A number of stationary (non-time variable) states are possible. Each state may be... 

Bohr postulated that there can be only certain discrete orbits for the electron around a nucleus—called stationary states—and that to go from one state to another, an atom must absorb or emit a packet of just the right amount of energy—a quantum. He then proceeded to predict the position of the lines in the hydrogen spectrum based on Balmer s formula, Planck s energy packets, the mass and charge on an electron, and his quantized orbits. 

A rough understanding of the spectrum and structure of atoms was first achieved by Niels Bohr in Copenhagen. He realized that a classical treatment of the hydrogen atom, similar to planet motion, would not give rise to a discrete spectrum, not even a stable atom. Therefore, he introduced stationary states for the first time. This was a hint of what was to come in the form of quantum mechanics about 10 years later. His principal assumptions are worth citing, since the main differences between classical theory and what is reqnired by a novel theory are clearly stated ... 

Contents Experimental Basis of Quantum Theory. -Vector Spaces and Linear Transformations. - Matrix Theory. -- Postulates of Quantum Mechanics and Initial Considerations. - One-Dimensional Model Problems. - Angular Momentum. - The Hydrogen Atom, Rigid, Rotor, and the H2 Molecule. - The Molecular Hamiltonian. - Approximation Methods for Stationary States. - General Considerations for Many-Electron Systems. - Calculational Techniques for Many-Electron Systems Using Single Configurations. - Beyond Hartree-Fock Theory. 

The Orbital Model As an Approximation. The quantum mechanical revolution also implies a more technical modification to our view of microscopic phenomena. Whereas the old quantum theory, as perfected by Pauli, required the assignment of as many as four quantum numbers to each electron in a many-electron atom, the arrival of quantum mechanics showed that even this more abstract notion is strictly inconsistent in any atom other than hydrogen. This result can be expressed by saying that individual electrons in a many-electron atom are not of themselves in stationary states whereas the atom as a whole does possess stationary states. This change in perspective is far from trivial and shows definitively that the orbital model is an approximation in many-electron systems. It also requires that the scientific term orbital is strictly non-referring with the exception of when it applies to the hydrogen atom or other one-electron systems. ... 

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