WKB formalism and a lower limit for the energy eigenstates of bound states for some potentials

In the present work the conditions appearing in the WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the text books on quantum mechanics, whereas the second one is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the Rayleigh--Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.Comment: Accepted in Modern Physics Letters

Essays on environmentally friendly behaviour and environmental policy

The thesis consists of three chapters of self-contained studies. In Chapter 1, I examine the decision of individuals to secure the provision of an environmental service under a Psychological Games framework. Since environmental services are considered public goods, there is an ongoing depletion of natural resources. While standard economic theory predicts the introduction of a PES is supposed to correct the associated externality by establishing a market which offers a monetary compensation to owners of vital natural resources as a recognition of their effort in providing the environmental services, this chapter argues such intervention might backfire: a motivation crowding-out arises if individuals believe others reciprocate friendly behaviour solely to receive the monetary compensation, ultimately decreasing total environmental protection. Even if environmental protection does take place, the motivation of individuals to secure the provision of the environmental service becomes commodified by the PES. Finally, awarding the PES only to a subset of individuals will also have negative effects on environmental protection, since those not receiving the PES will not want to do for free something that others are being paid to do. Environmental policy implications of this economic instrument are further discussed. In Chapter 2, a sample of the World Values Survey dataset is analyzed to show there are substantial behavioural differences between immigrants and native-born regarding pro-environmental action. In particular, while neither native-born nor immigrants are more willing to sacrifice money to save the environment, immigrants actually engage more on activities like choosing products that are better for the environment, recycling, and reducing water consumption. The engagement in proenvironmental behaviour of immigrants is region-specific and depends on their source region. Moreover, such relatively higher actual engagement in environmentally friendly behaviours can be explained by their high socio-economic status and their high education level, i.e. “selective immigration”. When the behaviour of immigrants by their length of residence in the host country is analyzed, no differences in proenvironmental attitudes or pro-environmental behaviour are found, a result which suggests they do not develop a “sense of belongingness” to the host country. Finally, in line with the standard finding in the literature of acculturation in environmental behaviour, this chapter finds that immigrants conform through time to some of the proenvironmental actions of native-born. In Chapter 3, I investigate the indirect effects on norm activation produced by monetary environmental policy instruments which introduce a situational cue that fosters a change of identity among individuals with potential negative consequences on their pro-environmental behaviour. For that purpose, a two-period identity selection model based on self-verification theory is developed. In each period there are two types of selves an individual can adopt: selfish and pro-environmental. The process of identity selection is driven by the desire of individuals to be consistent across the two periods in order to avoid social disapproval due to self-change. Results show that the monetary environmental policy introduces an asymmetry in the identity selection process that produces a failure of norm activation: while selfish agents preserve their selfish identity after the policy is implemented, pro-environmental agents might change their identity despite they experience social disapproval due to a reduction in the cognitive benefits of keeping such identity produced by the monetary component of the policy. Implications for environmental policy design are discussed

Bouncing Cosmologies in Palatini f(R)f(R) Gravity

We consider the early time cosmology of f(R) theories in Palatini formalism and study the conditions that guarantee the existence of homogeneous and isotropic models that avoid the Big Bang singularity. We show that for such models the Big Bang singularity can be replaced by a cosmic bounce without violating any energy condition. In fact, the bounce is possible even for pressureless dust. We give a characterization of such models and discuss their dynamics in the region near the bounce. We also find that power-law lagrangians with a finite number of terms may lead to non-singular universes, which contrasts with the infinite-series Palatini f(R) lagrangian that one needs to fully capture the effective dynamics of Loop Quantum Cosmology. We argue that these models could also avoid the formation of singularities during stellar gravitational collapse.Comment: 8 pages, 4 figures; added references and a short comment in sec.I