AZIMUTHAL (ANGULAR MOMENTUM)QUANTUM NUMBER I 11 where the operator is(dd).the eigenfunction is f).and the eigenvalue is(n). Generally.it is implied that wave functions.hence orbitals.are eigenfunctions. 1.23 EIGENVALUES The values of A calculated from the wave equation,Eq.1.17.If the eigenfunction is an orbital,then the eigenvalue is related to the orbital energy. 1.24 THE SCHRODINGER EQUATION FOR THE HYDROGEN ATOM An (eigenvalue)equation,the solutions of which in spherical coordinates are (r,6,p)=R(r)O(6)(p) (1.24) The eigenfunctions o.also called orbitals,are functions of the three variables shown where r is the distance of a point from the origin,and 0 and are the two angles required to locate the point( ee Fig.1.15).For ome e purposes,the spatial or radial and the r part c the Schrodi pendently.Associate par with each eigenfunction (orbit is an eig ue (orb energy).An exact solution of the Schrodinger equation is possible only for the hydrogen atom,or any one-electron system.In many-electron systems wave func- tions are generally approximated as products of modified one-electron functions (orbitals).Each solution of the Schrodinger equation may be distinguished by a set of three quantum numbers,n,1,and m,that arise from the boundary conditions. 1.25 PRINCIPAL QUANTUM NUMBER An integer 1.2.3.....that governs the size of the orbital (wave function)and deter- mic theor and the e la the of the orbita and the farther it extends from the nucleus AZIMUTHAL (ANGULAR MOMENTUM) The quantum number with values of /=0.1.2.....(n-1)that determines the shape of the orbital.The value of I implies particular angular momenta of the electron resulting from the shape of the orbital.Orbitals with the azimuthal quantum numbers 1=0,1.2,and 3 are called s.p.d.and forbitals,respectively.These orbital desig nations are taken from atomic where the words"sharp","principal" "diffuse" and“fnda tal"de cribe l es in atomic spectr ca This ber does not enter into the expression for the energy of an orbital.However,whenwhere the operator is (d 2 /dx 2 ), the eigenfunction is f(x), and the eigenvalue is (4π2 /λ2 ). Generally, it is implied that wave functions, hence orbitals, are eigenfunctions. 1.23 EIGENVALUES The values of λ calculated from the wave equation, Eq. 1.17. If the eigenfunction is an orbital, then the eigenvalue is related to the orbital energy. 1.24 THE SCHRÖDINGER EQUATION FOR THE HYDROGEN ATOM An (eigenvalue) equation, the solutions of which in spherical coordinates are φ(r, θ, ϕ)
R(r) Θ(θ) Φ(ϕ) (1.24) The eigenfunctions φ, also called orbitals, are functions of the three variables shown, where r is the distance of a point from the origin, and θ and ϕ are the two angles required to locate the point (see Fig. 1.15). For some purposes, the spatial or radial part and the angular part of the Schrödinger equation are separated and treated inde￾pendently. Associated with each eigenfunction (orbital) is an eigenvalue (orbital energy). An exact solution of the Schrödinger equation is possible only for the hydrogen atom, or any one-electron system. In many-electron systems wave func￾tions are generally approximated as products of modified one-electron functions (orbitals). Each solution of the Schrödinger equation may be distinguished by a set of three quantum numbers, n, l, and m, that arise from the boundary conditions. 1.25 PRINCIPAL QUANTUM NUMBER n An integer 1, 2, 3, . . . , that governs the size of the orbital (wave function) and deter￾mines the energy of the orbital. The value of n corresponds to the number of the shell in the Bohr atomic theory and the larger the n, the higher the energy of the orbital and the farther it extends from the nucleus. 1.26 AZIMUTHAL (ANGULAR MOMENTUM) QUANTUM NUMBER l The quantum number with values of l
0, 1, 2, . . . , (n 1) that determines the shape of the orbital. The value of l implies particular angular momenta of the electron resulting from the shape of the orbital. Orbitals with the azimuthal quantum numbers l
0, 1, 2, and 3 are called s, p, d, and f orbitals, respectively. These orbital desig￾nations are taken from atomic spectroscopy where the words “sharp”, “principal”, “diffuse”, and “fundamental” describe lines in atomic spectra. This quantum num￾ber does not enter into the expression for the energy of an orbital. However, when AZIMUTHAL (ANGULAR MOMENTUM) QUANTUM NUMBER l 11 c01.qxd 5/17/2005 5:12 PM Page 11