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Basic Types of Business Entities . . . . . . . . . . . . . . . The sole proprietorship . . . . . . . . . . . . . . . . . The partnership . . . . . . . . . . . . . . . . . . . . . Limited partnerships . . . . . . . . . . . . . . . The corporation . . . . . . . . . . . . . . . . . . . . . S corporations . . . . . . . . . . . . . . . . . . . The limited liability company . . . . . . . . . . . Federal Tax Identi cation Numbers . . . . . . . . . . . . . . Tax Penalties for Late Filers and Non lers . . . . . . . . . . Choosing a QuickBooks Start Date for Your Business Entity A Checklist of Issues . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 43 44 45 45 46 46 46 48 50 51 52

(6.32) Then the total rate is simply (6.33) As you can see from (6.33), when la} is the ground state, Wab will never be positive so that the summation will vanish. In other words, in the ground state the contribution of the vacuum fluctuations cancels the radiation reaction contribution exactly and the atom does not decay. This shows that the ground state of the atom is stable only because of the vacuum fluctuations of the field. The self-reaction term alone would lead to the collapse of the ground state as classical electrodynamics predicts and the atomic commutation relations [ri,pj] = ilioij would not hold. 4 So far, our treatment of spontaneous emission has been perturbative. We will now go beyond the perturbative approach and we will show that an initially excited atom decays exponentially to the ground state in free space. Before we do so, however, it is convenient to simplify the problem further.

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to classical theory, an atom cannot have a minimum-energy state even when only the Coulomb interaction is included in the Hamiltonian. It is the quantum commutation relations of the atomic operators that prevent the electron from colliding with the proton by giving it enough kinetic energy to overcome the Coulomb attraction every time the electron is localized within a certain neighborhood of the proton [409]. When the coupling with the radiation field is included to account for atomic emission, the atomic commutation relations are violated unless the field commutation relations are included [131, 556].

The tradi tional pharmaceutical business model is based on the sale of a single, very high margin blockbuster drug that can generate enough revenue in a few years to pay for a decade or more of research and development on perhaps dozens of drug candidates, while also creating a handsome pro t for the pharmaceutical company and its shareholders The tools model involves.

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The canonical variables we have used to describe this system are not unique. They are related to other possible sets of canonical variables by unitary transformations; the quantum analog of classical contact transformations. We will perform such a transformation on these canonical variables to obtain a more convenient set of variables. When written in terms of this new set, the Hamiltonian will acquire a form more amenable to physical interpretation. We will then be able to use physical insight to make some approximations that will greatly simplify the problem. Let us consider the unitary transformation T that takes pinto p - (q/c)A (0), that is, TpTt = p- ~A(O). (6.34)

This is a translation in momentum space of -(q/c)A (0). We will now show that the generator of infinitesimal translations in one dynamical variable is the canonical conjugate variable [243]. By applying the infinitesimal translations successively, we will obtain an expression for a general finite translation that will yield T as a particular case. Let Tv. (a) be a unitary transformation that produces a translation in the space of a given dynamical variable u:

(6.35)

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We can write the unitary transformation Tv. (0') corresponding to an infinitesimal translation 0 as (6.36) Tv.(O') = 1 + iO'X, where X is a Hermitian operator which is the generator of infinitesimal translations. From (6.37) Tv.(O')uTJ(c5) = u + c5 we obtain the following commutation relation between X and u

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