Natural representation

The representation that arises first in the study of the hydrogen atom is the natural representation of SO(3) on Recall that we defined the... 

Representations are the primary object of our mathematical analysis. In particular, the natural representation of 50(3) on introduced in this... 

Exercise 4.14 (Used in Proposition 5.1) Show that the degree of a polynomial in three variables is invariant under rotation. In other words, consider the natural representation p of SO(3) on polynomials in three variables and show that the degree of a polynomial is invariant under this representation for any polynomial q and any g e SO(3), show that the degree of q is equal to the degree of p g)q. 

Exercise 4.39 (Used in Section 5.3) Consider the characters )t3 cind /4 the natural representations of SU (2) and P. Find the coefficients of... 

Any physically natural, spherically symmetric set of states corresponds to an invariant subspace and a subrepresentation. For this reason the concepts in this section are fundamental to our analysis of the hydrogen atom. The various shells of the hydrogen atom correspond to subrepresentations of the natural representation of SO 13) on (R ). In particular, the subspaces and Z play a role in the analysis. 

The set of all linear transformations (not necessarily homomorptiisms of representations) from a representation V to a representation W forms a vector space too. This vector space is denoted Hom(T, IT). (Here Hom refers to the fact that a linear transformation can be considered a homomorphism of vector spaces.) There is a natural representation of G on this vector space. 

In other words, the natural representation of G on Hoiiv (V. W) is trivial. Still, ( V, W) does carry important information. In Section 6.4 we will find the vector space dimension of HomoCV, W) to be useful. 

For some representations, the largest and smallest subspaces are the only invariant ones. Consider, for example, the natural representation of the group G = 50(3) on the three-dimensional vector space C . Suppose W is an invariant subspace with at least one nonzero element. We will show that W = C-. In other words, we will show that only itself (all) and the trivial subspace 0 (nothing) are invariant subspaces of this representation. It will suffice to show that the vector (1, 0, 0) lies in W, since W would then have to contain both... 

We can summarize our work above by writing that the natural representation of SO (3) on C is irreducible. In contrast, we have seen in Section 5.1 that the representation of the circle group defined by Formula 5.1 is not irreducible. 

Consider a complex scalar product space V that models the states of a quantum system. Suppose G is the symmetry group and (G, V, p) is the natural representation. By the argument in Section 5.1, the only physically natural subspaces are invariant subspaces. Suppose there are invariant subspaces Gi, U2, W c V such that W = U U2. Now consider a state w of the quantum system such that w e W, but w Uy and w U2. Then there is a nonzero mi e Gi and a nonzero M2 e U2 such that w = ui + U2. This means that the state w is a superposition of states ui and U2. It follows that w is not an elementary state of the system — by the principle of superposition, anything we want to know about w we can deduce by studying mi and M2. 

Proposition 7.2 Suppose f is a nonnegative integer. Then the natural representation of SO (3) on is irreducible. 

Proposition 7.6 Suppose that V is a nontrivial irreducible invariant subspace of the natural representation of S 0(f) on Lf(S f Then there is a nonnegative integer such that V = y. ... 

From Proposition 5.1 we know that y is an invariant subspace. Since the natural representation of S<9(3) on L (W ) is unitary. Proposition 5.4 implies that is a homomorphism of representations. Since V and y are irreducible, it follows from Schur s Lemma and the nontriviality of n [V] that fit gives an isomorphism of representations from V to y. ... 

Ta is a homomorphism of representations for the natural representations of 50(3) on domain and range. 

There is a natural representation of the Lie algebra so 3 using partial differential operators on We can define the three basic angular momentum operators as linear transformations on as follows ... 

Because of the spherical symmetry of physical space, any realistic physical operator (such as the Schrodinger operator) must commute with the angular momentum operators. In other words, for any g e SO(3) and any f in the domain of the Schrodinger operator H we must have H o p(g ] = pig) o H, where p denotes the natural representation of 80(3 on L2(] 3 Exercise 8.15 we invite the reader to check that H does indeed commute with rotation. The commutation of H and the angular momentum operators is the infinitesimal version of the commutation with rotation i.e., we can obtain the former by differentiating the latter. More explicitly, we differentiate the equation... 

We can put these representations (one for each eigenvalue of the Schrodinger operator) together to form a representation of su(2) on the vector space of bound states of the hydrogen atom. We will see in Section 8.6 that there is a physically natural representation of the larger Lie algebra 5o(4) = 5m(2) 5m(2) on the set of bound states of the hydrogen atom. 

Proposition 9.2 For any nonnegative integer n, the natural representation of SOIA) on is irreducible. The dimension of this representation is (n + l). ... 

Our next task is to identify the projective representation of 50(3) on the state space. This representation is determined by the representations on the factors, but the projection must be handled carefully. The spin-1/2 projective representation of SO (3) on (C ) descends from the linear representation on C". The natural representation of SO (3) on (Section 4.4) descends... 

THE SCIENCE OF ECOLOGY emerged at the turn of the last century and brought with it the experimental approaches that were already central to the study of physiology (1-3). Manipulations of whole aquatic ecosystems— excluding aquaculture, which dates back 2500 years (4)—developed more slowly, mainly because of difficulties associated with increased biotic complexity and physical scale in larger systems. One technique initially used to overcome the problems of complexity, scale, and replicability was creation of controlled microcosms that embodied a more or less natural representation of the whole system (5, 6). 

In Fig. 2, we represent the interpolating function in a natural representation, (< (0))1/3 versus 6, which becomes linear near the jamming point. The interpolating function can be approximated by an expression of the type... 

It should be emphasized that, provided all the symmetry-allowed spin functions are included in the wavefunction, the actual ordering of the orbitals is immaterial and may be chosen to highlight different aspects of the same wavefunction. Thus in this present example we may equally well consider the orbital pairs (<7i,singlet-singlet or triplet-triplet coupled. This is in a sense the natural representation to adopt for short since it concentrates attention on the formation of C-H bonds. Some results for this system may be found in Ref. 46. 

To determine the shape of a rigidly rotating spiral wave it is convenient to specify the curvature K as a function of the arc length s. A purely geometrical consideration shows that the function K — K[s), the so-called natural representation of a curve, satisfies the following integro-differential equation [39] ... 

Chemical Graph-Based Repr entations In terms of general chemical applications, chanical graphs [71] are a natural representation to use for assessing molecular similarity. They are typically defined mathematically as an ordered triple of sets, Q = (V,E,L), where Q is the graph, V is the set of its t vertices ( atoms ),... 

We first notice that in the natural representation of the CCR in the case n = 1, mentioned in the first section, we have... 

---