Note-4. Schrodinger equation in three dimensions We will consider problems where the partial differential equations are separable. 4-1. Free particles in a box--separable in Cartesin coordinates If the particle is confined in a box L3, clearly the wavefunction is given by 2 n x n y n z un1n2n3 ( x, y, z ) ( ) 3 / 2 sin 1 sin 2 sin 3 (1) L L L L and the energies are given by 2 2 2 E 2 ( n1 n2 n3 ) 2 2 (2) 2mL Thus there are three quantum numbers, n1,n2, n3 to denote a give state and since the energy depends is given by (2), there are degeneracy in that different eigenstates can have the same energy. 4-2. Separable in Spherical coordinates, ( r, , ) If the potential seen by the particle depends only on the distance r, then the Schrodinger equation is separable in Spherical coordinates. Some key equations: 2 2 H V (r) 2m [H , L] 0 [ H , L2 ] 0 [ Lx , L y ] iLz Choose eigenstates of H, L2 and Lz gives quantum numbers n, , m . ( This is just a convention.) The eigenstates are given by Em ( r, , ) RE ( r )Ym ( , ) where L2Ym 2 ( 1)Y m and LzYm mYm The radial wavefunction RE (r ) satisfies d2 2 d 2 l(l 1) 2 2E 2 dr R nl (r ) 2 V(r ) R nl (r ) 2 R nl (r ) 0 r dr 2r 2 Note that at large r, the centrifugal potential goes like 1/r2. Thus it is important to distinguish potential V(r) which drops faster than 1/r2 or not. In general, if the potential decreases exponentially with r at large r, we called that it is a short-range potential. For Coulomb interaction it goes like 1/r, it is a long-range potential. By writing un ( r ) rRn ( r ) The equation for un (r ) is d 2 u nl (r ) 2 l(l 1) 2 2 E V( r ) u nl (r ) 0 dr 2 2r 2 Here n is a quantum number for the radial equation. In general, the energy E depends on n and . 4-3. Spherical harmonics A good place to find the summary of spherical harmonics is http://mathworld.wolfram.com/SphericalHarmonic.html Spherical harmonics are used when there are spherical symmetry. For diatomic molecules, for example, there is no spherical symmetry, then one also uses, for example (for m>0 only), 1 i Y,cos [Ym Ym ] Y,sin [Ym Ym ] * * 2 2 For 1 , they are called px and py, respectively. In this case, Y10 is proportional to pz. The parity of Ym is (1) . For small m for a given , the function peaks more along the z-direction (quantization axis). For the maximum m, it peaks perpendicular to it. Terminology in spectroscopy: 0, 1, 2, 3, 4, …. are called s, p, d f, and g, …. 4-4. Free-particle solutions For V(r)=0, d 2 2 d l(l 1) 2 2 R (r ) k 2 R (r ) 0 dr r dr r Using kr, d 2 R 2 dR l(l 1) RR 0 d 2 d 2 There are two independent solutions, j ( ) and n ( ) , the spherical Bessel functions and spherical Neumann functions, respectively. To first order, at large , they are like sine functions and cosine functions, respectively. At small , j ( ) is finite but n ( ) diverges. One can also uses a separate set of two independent solutions, called spherical Hankel functions h ( ) j ( ) i n ( ) (1, 2 ) These two functions represents a spherical wave going outward (outgoing wave) or a wave coming toward the center (ingoing wave) at large distances. Take a look at these functions if they are unfamiliar to you on the web or in your textbooks. 4-5. The 3D infinite potential well If the particle is confined to a sphere of radius a, clearly the radial wavefunction which if finite at r=0 is given by j (kr) . The condition that it vanishes at r=a requires that j ( ka) 0 Thus the allowed energies are related to the zero's of the spherical Bessel functions. 4-6. The expansion theorem for a plane wave Recall that Ym ( , ) are eigenstates of L2 and Lz in the Hilbert space of the two spherical angles. Thus any function in , can be expanded in terms of the complete set of functions of Ym ( , ) . A plane wave giving by e ik r e ikr cos can be expanded as e ikr cos 2l 1 i l jl kr P1 cos l0 This equation will be useful for discussion scattering where the incident wave is a plane wave and the scattered wave is a spherical wave. =========================================== Homework #4 4-1. Check the definition of L+ and L- and their operations on Ym ( , ) . Calculate Ylm1 L x Ylm2 and Ylm1 L y Ylm 2 . 4-2. Calculate Ylm1 L2x Ylm 2 and Ylm1 L2y Ylm 2 . 4-3. The Hamiltonian for an axially symmetric rotator is given by L2x L2y L2z H 2I1 2l 2 What are the eigenvalues of H? Sketch the spectrum, assuming that I1 I 2 . 4-4. The three-dimensional flux is given by j * ( r ) ( r ) * ( r ) ( r ) 2i ˆ Calculate the radial flux integrated over all angles, that is, dir j for wave functions ikr Ylm , . The unit vector in the radial direction is iˆr . e of the form (r ) C r 4-5. Learn how to count. (You can use the results from any books to start this problem.) (a) For a cubic box of dimension L on each side, what is the energy of the highest occupied level in the ground state for a systems of 20 noninteracting electrons? (b) Answer the same question if the cube is replaced by a spherical potential well where the potential is zero inside r=L and infinite outside. 4-6. Use computer to graph (or sketch by hand) in polar plots | Ym ( , ) |2 for ( , m ) (1,0), (1,1), (2,0), (2,1) and (2,2).