Think forward to be most effective today. Lessons and experience from the past should inform the decisions of today but those decisions must be made anticipating future conditions. The demand for the services and broad functionality of the environmental manager are only increasing. Thinking and planning for future contexts while considering the lessons, knowledge, and experiences of the past, all to produce the best practices for today, is essential for the effective environmental manager. Identifying, assessing, and orienting on future trends is critical to ensuring your program is operating with the most up-to-date and relevant technology and design; the environmental manager should embrace them and prepare for them.

Proofs by induction and definitions by recursion are fundamental tools when working with the natural numbers. However, there are many other places where variants of these ideas apply. In fact, more delicate and exotic proofs by induction and definitions by recursion are two central tools in mathematical logic. We will eventually develop transfinite versions of these ideas in Chapter 9 to give us ways to continue into exotic, infinite realms, and these techniques are essential in both set theory and model theory. In this chapter, we develop the more modest tools of induction and recursion along structures that are generated by one-step processes, like the natural numbers. Occasionally, these types of induction and recursion are called structural.

In this age, in this country, public sentiment is everything. With it, nothing can fail; against it, nothing can succeed. Whoever molds public sentiment goes deeper than he who enacts statutes, or pronounces judicial decisions.

Now that we have successfully worked through several important aspects of propositional logic, it is time to move on to a much more substantial and important logic: first-order logic. We gave a basic overview of the fundamental ideas in the introduction. Fundamentally, many areas of mathematics deal with mathematical structures consisting of special constants, relations, and functions, together with certain axioms that these structures obey. We want our logic to be able to handle different types of situations, so we allow ourselves to vary the number and types of these objects. For example, in group theory, we have a special identity element and a binary function corresponding to the group operation. If we want, we can also add in a unary function corresponding to the inverse operation. For ring theory, we have two constants for 0 and 1, along with two binary functions for addition and multiplication (and possibly a unary function for additive inverses). For partial orderings, we just have one binary relation. Any such choice gives rise to a language.

Matching standards to performance. The effective environmental manager must be aware of the mandates and professional standards that they and their programs are accountable to, as well as be able to demonstrate the performance and achievements of the programs they manage. Knowing your lateral operational limits and performing effectively and efficiently is not only critical to the career trajectory of the environmental manager but, more importantly, requisite for them to protect human health and the environment.

A goal without a plan is just a wish. Antoine de Saint-Exupéry, c. 1940

Strategic planning and diffusion of innovations. The effective environmental manager must be able to conduct strategic planning and be able to diffuse innovations into the communities and regulated entities that they work with. Strategic planning is a skill not often taught to, yet often expected of, managers. And getting people to try new ways of doing this is not easy and requires effective planning to accomplish. You can develop the most effective solution possible for a problem but if you cannot diffuse it, i.e., get stakeholder buy-in, and do not create a plan to support it – logistically, operationally, or otherwise – then it will not work.

Situational analysis – i.e., understanding the situation. The successful environmental manager must be able to effectively analyze complex and often conflicting information, directions, and stakeholders when managing the environment. This skill we call situational analysis. This is a skill that requires the environmental manager to analyze and translate the management factors in terms of a situation summary – which issues, legal trends, specific laws, jurisdictions, policy participants, policy processes, barriers to communication are applicable – and make management recommendations to address the situation. This is an “everyday” or field skill that managers should use; however, as you will note in subsequent chapters, it is also closely related to strategic planning. It is a skill equally important to public, private and not-for-profit environmental managers in terms of regulating pollution, complying with regulations and understanding those you wish to influence.

We solve the quantum mechanical harmonic oscillator problem using an operator approach. We define the lowering and raising operators. We use the quantum mechanical harmonic oscillator to review the fundamental ideas of quantum mechanics.We study some examples of time dependence in the harmonic oscillator including the coherent state. We apply the quantum mechanical harmonic oscillator to the study of the vibrations of the nuclei of molecules.