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16. , 17. 1 n!. 33). What happens to this sequence if different initial conditions (other than F(0) = 0, F(1) = 1) are introduced? 18. Derive other Fibonacci-type series using other symmetric generating matrices in 2 −1 ]) and other initial conditions. G L(2; Z) (for example, [ −1 1 19. The energy levels |nlm of the nonrelativistic hydrogen atom exhibit an n 2 -fold degeneracy under the Lie group S O(4). 6 eV). If the Coulomb symmetry is broken by placing one or more electrons in the Coulomb potential, the overall symmetry reduces to that of the rotation group: there is a symmetry reduction S O(4) ↓ S O(3).

The homeomorphisms φα ◦ φβ−1 : φβ (Uα ∩ Uβ ) → φα (Uα ∩ Uβ ) of open sets in R n to open sets in R n are 1:1, invertible, and differentiable. Remarks The charts φα allow construction of coordinate systems on the open sets Uα . It is often not possible to find a single coordinate system on the entire manifold, as the example of the sphere in Fig. 1 shows. Since the “transition functions” φα ◦ φβ−1 map R n → R n , all the definitions of elementary multivariable calculus are applicable to them. For example, the adjective “differentiable” can be replaced by other adjectives (C k , smooth, analytic, .

Matrices in this group have the structure [ γα βδ ], where α, β, γ , δ are complex numbers and αδ − βγ = 1. Define the matrix X by X = H (x, y, z, ct) = ct + z x + iy x − iy ct − z = ct I2 + σ · x where x is the three vector x = (x, y, z) and σ = (σ1 , σ2 , σ3 ) = (σx , σ y , σz ) are the Pauli spin matrices. a. Show that X is hermitian: X † ≡ (X t )∗ = X . b. Show that the most general 2 × 2 hermitian matrix can be written in the form used to construct X . c. If g ∈ S L(2; C), show that g † Xg = X = H (x , y , z , ct ).