Multiplication by a constant

However, we shall usually omit circumflexes over operators that are simply multiplication by a constant.] Repeated application of the definition of operator multiplication shows that the associative law holds for all operators ... 

The last part of Problem 7.1 demonstrates that a given function does not necessarily correspond to the solution of only one differential equation. In later sections we shall address the question of how to determine the number of different functions (where each function differs from another by more than simply multiplication by a constant) that are solutions of a given differential equation. 

Often the data analysis requires multiplication by some number B such as Avo-gadro s number, which is typically known to much higher accuracy than any of the measured data. In that case, the rule for multiplication above simplifies immediately to AC/C = A A/A—the fractional error after multiplication by a constant is unchanged. 

Proof. The easiest proof is that the function X2 g2 is bijective, and so is multiplication by a constant... 

An operator for multiplication by a constant commutes with any other operator. 

The operator 1 (multiplication by 1) is the unit operator. The operator 6 (multiplication by 0) is the null operator. We usually omit the circumflex over operators that are simply multiplication by a constant. We can transfer operators from one side of an operator equation to the other (Problem 3.9) thus (3.5) is equivalent to Dx — xD —1 = 0, where circumflexes over the null and unit operators were omitted. 

Multiplication by a constant with no error. A common laboratory activity is to prepare a solntion by weighing a mass of solid and then dissolving it in water. The concentration is usnally reported in terms of the number of moles of solid rather than the mass. If the mass of NaCl (= 58.44) is 40.00(0.01) g, the number of moles is 40.00/58.44 = (40.00)(0.0171) = 0.6845 moles. We can assnme that there is no random error associated with the molecular weight of NaCl, so the error for this value is found nsing Eq. (2.21). 

Let n = 2x M) > 0. On the manifold M, a function /, holomorphic in the conformal structure on M given by the Riemannian metric T is existent and unique up to multiplication by a constant. This function has simple zeros at the points of the set does not have other zeros, and has poles of order n at infinity. Let M be a Riemann surface of the function y/J and x Af —> M a projection. Then /J is a single-valued function on Af. Define /y/7 formula... 

When multiplying a thermochemical equation by a constant, both the stoichiometric coefficients and AH are multiplied by the constant. Thus, for the above, formal reaction equation (a), multiplication by a constant k will often... 

Because of their important role in quantum mechanical problems, it is useful to imder-stand operators and their workings. An operator is a mathematical device that relates two functions. An operator equation is any equation that includes an operator. Thus, g x) = 3f(x) can be regarded as an operator equation since the functions g(x) and f x) Me related by the operation of multiplication by 3. The equation can be expressed as g(x) = 0(x), where 0 = 3. Though operators can also be as simple as multiplication by a constant such as 3, they can be multiplication by a variable, such as x, or differentiation with respect to one or more variables. Anything that can relate one function to another function can be labeled an operator. 

The Hamiltonian operator is Hermitian, the operation of multiplication by a constant is Hermitian, and that made it possible to exchange the order in which the operator was applied. With this simplification, the first-order equation yields an expression for the first-... 

---