Hydrogenic energy levels

Key points: The energy of the bound electron is determined by n, the principal quantum number; in addition, l specifies the magnitude of the orbital angular momentum and ml specifies the orientation of that angular momentum. Each of the wavefunctions obtained by solving the Schrödinger equation for a hydrogenic atom is uniquely labelled by a set of three integers called quantum numbers. These quantum numbers are designated n, l, and ml : n is called the principal quantum number, l is the orbital angular momentum quantum number (formerly the ‘azimuthal quantum number’), and ml is called the magnetic quantum number. Each quantum number specifies a physical property of the electron: n specifies the energy, l labels the magnitude of the orbital angular momentum, and ml labels the orientation of that angular momentum. The value of n also indicates the size of the orbital, with high n, high-energy orbitals more diffuse than low n compact, tightly bound, low-energy orbitals. The value of l also indicates the angular shape of the orbital, with the number of lobes increasing as l increases. The value of ml also indicates the orientation of these lobes.

The allowed energies are specified by the principal quantum number, n. For a hydro genic atom of atomic number Z, they are given by

(The fundamental constants in this expression are given inside the back cover.) The calcu lated numerical value of R is 1.097 x 10⁷ m1, in excellent agreement with the empirical value determined spectroscopically. For future reference, the value of hcR corresponds to 13.6 eV. The zero of energy (at n=∞) corresponds to the electron and nucleus being widely separated and stationary. Positive values of the energy correspond to unbound states of the electron in which it may travel with any velocity and hence possess any energy. The energies given by eqn 1.3 are all negative, signifying that the energy of the electron in a bound state is lower than a widely separated stationary electron and nucleus. Finally, because the energy is proportional to 1/n², the energy levels converge as the energy increases (becomes less negative, Fig. 1.6). The value of l specifies the magnitude of the orbital angular momentum through {l(l 1)}^1/2h , with l 0, 1, 2, . . . . We can think of l as indicating the rate at which the electron circulates around the nucleus. As we shall see shortly, the third quantum number ml specifies the orientation of this momentum, for instance whether the circulation is clockwise or anticlockwise.

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