50 algebra defining commutation relations

These relations are called the defining commutation relations for the Lie algebra and the coefficients ciJk are called the structure constants. A different choice of basis leads to a different but equivalent set of structure constants. We also say that the are generators of the Lie algebra. The rather difficult problem of classifying different Lie algebras will not be considered here (see, e.g., Wybourne, 1974 Gilmore, 1974), since those we shall use arise quite naturally from physical considerations. 

Once a Lie algebra has been defined in the abstract sense via the defining commutation relations Eq. (3), it is of practical interest to find physical realizations of the generators in terms of position and momentum operators, which also satisfy these defining commutation relations. We shall call such a set of concrete operators a realization of the Lie algebra. In practice, as we shall see, we often work backwards by starting with a set of concrete operators... 

As an example consider the important three-dimensional Lie algebra often denoted as so(3) or su(2), whose defining commutation relations can be cast in the form... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

Given a Lie algebra with defining commutation relations Eq. (3), we can consider the generators , as operators acting on some n-dimensional vector space W. If j> i = 1,..., n is a basis set for W then... 

Thus, they are closed under commutation and form a three-dimensional Lie algebra. By taking the appropriate linear combinations of these operators we can obtain operators satisfying the so(2, 1) defining commutation relations,... 

Except for the factor —2H, Eqs. (162a)-(162c) are the defining commutation relations for the Lie algebra so(4) given in Section IV. ... 

A tensor operator under the algebra G 3 G, T, is defined as that operator satisfying the commutation relations... 

To define a Lie algebra homomorphism, it suffices to define it on basis elements of 01 and check that the commutation relations are satisfied. Because the homomorphism is linear, it is defined uifiquely by its value at basis elements. Because the bracket is linear, if the brackets of basis elements satisfy the equality in Definition 8.7, then any linear combination of basis elements will satisfy equality in Definition 8.7. 

The operator a(i) in the Heisenberg algebra, of course, corresponds to the operator Pa[i constructed in Chapter 8. But our commutator relation (8.14) differs from the standard one, we need to modify operators. In fact, it is more natural to change also the sign of the bilinear form. Hence we define... 

This is known as a commutation relation and in algebra, of course, the result would be zero. Classical physics would also predict that the result is zero, but in quantum mechanics it isn t. We can think about this in the following manner. Classical physics is a macroscopic approximation to quantum physics in the limit of large dimensions, quantum physics goes over to classical physics. The commutator, defined as xp - px, is small, but... 

By purely algebraic methods it can be shown that the whole quantum theory of angular momentum follows from these equations. In particular, if 2 is a Hermitian operator which is defined to obey the commutation relations... 

What is the cause of this uncertainty We noted earlier that x and P are related and have a mutual effect. Another way to quantify the effect of position and momentum is by using a commutator. In real arithmetic and algebra with real numbers, we are used to interchanging the order of factors as in 2x3 = 3x2 = 6, that is the commutator [3,2] = 0, but when we use calculus operators that interchange of order may not work. Let us define a quantity called the commutator as a bracket that represents the amount by which interchanging the order of two successive operators makes a difference (on some... 

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