Lecture 12 Particle on a sphere

1. Lecture 12Particle on a sphere (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.

2. The particle on a sphere • Main points: the particle on a sphere leads to a two-dimensional Schrödinger equation and we must use the separation of variables. One of the resulting one-dimensional equations is the particle on a ring. For the other, we seek mathematicians’ help: we introduce associated Legendre polynomials. The product of these and the particle on a ring eigenfunctions are spherical harmonics.

3. The particle on a sphere

4. The particle on a sphere • The potential is zero – only kinetic energy: • In the Cartesian (xyz) coordinates, the ‘del squared’ is • What is it in spherical coordinates?

5. The particle on a sphere • The answer is: • The derivation is analogous to that for cylindrical coordinates. You are invited to derive!

6. The particle on a sphere • The value of r is held fixed. The derivatives with respect to r vanish. • The Schrödinger equation is Two variables θ and φ.

7. The particle on a sphere • Two variables – let us try the separation of variables technique. • SubstitutingFor the separation to take place, we must be able to cleanly separate the equation into two parts, each depending on just one variable.

8. The particle on a sphere • The differentiation with respect to θ for example acts on Θ alone. Therefore • Dividing the both sides by ΘΦ

9. The particle on a sphere • Multiplying both by sin2θ • Subtracting the RHS from both sides Function of θ Function of φ Function of θ Constant Function of φ Function of θ

10. The particle on a sphere • Two independent one-dimensional equations! • We have already solved the first equation. These two parts of the equation must be constant.

11. The particle on a sphere Separation of variables 2D rotation, ml is introduced Custom-made solutions Associated Legendre polynomials l and ml are quantum numbers Custom-made Spherical harmonics

12. The particle on a sphere • To summarize: the Schrödinger equation is • The eigenfunctions are spherical harmonics specified by two quantum numbers l (= 0, 1, 2, …) and ml (= –l, … l), having the form Normalization Associated Legendre eimlφ Spherical harmonics

13. The particle on a sphere • Some low-rank spherical hamonicsare given on the right. • Spherical harmonics are orthogonal functions. They as fundamental to spherical coordinates as sin and cos to Cartesian coordinates.

14. Spherical harmonics • Spherical harmonics are the standing waves of a sphere surface (e.g., soap bubble, earthquake). • Imagine a floating bubble. It vibrates – the amplitudes of the vibration is a linear combination of spherical harmonics. GNU Image from Wikipedia

15. The particle on a sphere • The total energy is determined by the quantum number l only: • Out of this, the energy arising from the φrotation is • The latter cannot exceed the former.

16. The particle on a sphere • Parameter l is called the orbital angular momentum quantum number. • Parameter ml is the magnetic quantum number. • Energy is independent of ml. Therefore, a rotational state with l is (2l +1)-fold degenerate because there are (2l +1) permitted integers ml can take.

17. The particle on a sphere • Let us verify that the associated Legendre polynomial is indeed the solution for l = 1 and ml = 1.

18. Spherical harmonics This is a breathing mode of a bubble This is a a-candy-in-mouth mode of a bubble This is an accordion mode of a bubble

19. Summary • The spherical harmonics are the most fundamental functions in a spherical coordinates. • We have encountered a differential equation whose solution involves associated Legendre polynomials. • The eigenfunctions of the particle on a sphere are spherical harmonics and characterized by two quantum numbers l and ml. • The energy is determined by l only and is proportional to l(l + 1).