PHIL 1102 Lecture Notes - Lecture 2: Modus Ponens, False Premise, Deductive Reasoning

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23 Jul 2015

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Deductive arguments are those in which it is claimed that the conclusion must follow the premises. That is, it is impossible for the premises to be all true and the conclusion to be false. Inductive arguments are those in which it is claimed that the premises merely make the conclusion probable. That is, if the premises of an inductive argument is true, then it is unlikely that the conclusion is false. We will be spending most of the course on deductive arguments, but an easy, standard example of a deductive argument form is modus ponens: For other examples, think of a mathematical proof. Numbers greater than 1 whose only factors are 1 and the number itself and 7 has only 1 and 7 as factors, then 7 is a prime number. The definition we have given of a deductive argument presupposes the concept of validity.

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(1) We want to consider an atomic system containing two electrons.For now, we will consider that the electrons do not interact witheach other through the repulsive Coulomb interaction and we willneglect all interactions explicitly involving spin, such as thespin-orbit interaction. We can then write the one-electron-atomeigenfunctions as a product of a spatial function and a spinfunction in the form,
?nlmlms =?nlmlsms,
where the sigma function is the spin eigenfunction, the exactrepresentation of which you will not need. For an electron labeledby 1, applying our formalism in class, this can be expressed in theform,
?a(1)=?a(1)smsa (1), where a represents both the spatial and spinquantum numbers and a represents only the
spatial quantum numbers. Two-electron product eigenfunctions can bewritten in the form,
?aÃ = ?a (1)?Ã (2)= ?a (1)?b (2)smsa (1)smsb (2), and, finally, inthe more compact notation that we adapted in class,
?aÃ=?absms ms . ab
Of course, this is not an acceptable total eigenfunction for ourtwo-electron system since it is not antisymmetric with respect toan interchange of the electron labels.

(a) Write down both symmetric and antisymmetric forms for thespatial, two-electron eigenfunctions.
(b) Do the same for the spin, two-electron eigenfunctions,remembering that ms can assume the two values + 1/2 and â1/2.
(c) Now write down all of the antisymmetric total, two-electroneigenfunctions formed by the products. Label the eigenfunctionsthat are spatially symmetric with the letter s and those that arespatially antisymmetric with the letter t. These two-electron,total eigenfunctions are degenerate under our initial assumptions.

The emergence of line spectra from a gas when a current passesthrough it was an observed phenomenon waiting for an explanation inthe early 20th century. The atomic line spectra coming fromelements such as hydrogen had been analyzed since the late 19thcentury. By studying the wavelength of the emerging radiation ofhydrogen, experimenters found (often by trial and error) that thewavelengths in those spectra were described by the formula

\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2}\right),
where R is known as the Rydberg constant. It has a value of R =1.097 \times 10^{7}\; \rm m^{-1}. The variables n_1 and n_2 areinteger numbers (n_1 = 1,2,3,4,\dots). As experiments continued,scientists began to see more and more characteristic lines emergingfrom the hydrogen spectrum. Each of them corresponded to awavelength predicted by this formula with some integer values forn_1 and n_2.

In 1913, Niels Bohr provided an explanation for the observationsmade in experiments by proposing that each electron in an atom hadonly certain allowable energy levels. Furthermore, he postulatedthat when an electron changed its energy level, it must emit orabsorb a single photon with the energy equal to the energydifference between the two levels.

With this interpretation, the variables n_1 and n_2 acquired newsignificance. Bohr's model predicted that n_2 was the initialenergy state of the atom (n_initial) and that n_1 was the finalstate of the atom (n_final). A photon with the predicted wavelengthwas created when an atom made a transition from its initial energystate to a lower final energy state. The equation now read

\frac{1}{\lambda} = R \left( \frac{1}{n_{\rm final}^2} -\frac{1}{n_{\rm initial}^2} \right).
Bohr's model also predicted that a photon of the appropriatewavelength (given by the same formula) could promote an atom from alower energy level to a higher one.

In this problem, we explore the photon characteristics necessaryfor transitions between energy levels in hydrogen and singlyionized {\rm He}^+, whose energy levels are accurately predicted bythe Bohr model.
What is the change in energy DeltaE of the hydrogen atom as theelectron makes the transition from the n=3 energy level to the n=1energy level?
Express your answer numerically in electron volts.

b. When an atom makes a transition from a higher energy level to alower one, a photon is released. What is the wavelength of thephoton that is emitted from the atom during the transition from n=3to n=1?
Express your answer numerically in angstroms.

c.

When the hydrogen atom makes the transition from the n=2 to the n=1energy level, it emits a photon. This photon can be absorbed by toa singly ionized helium atom ({\rm He}^+) in its second energylevel (n_{\rm initial} = 2), causing it to move to a higher energylevel. To what final energy level number n_final can the photonpromote the helium ion?

Note that {\rm He}^+ has two protons in its nucleus and a singleelectron.
Express your answer as an integer.

Please answer fully, Thank you
a.

PHIL 1102 Lecture Notes - Lecture 2: Modus Ponens, False Premise, Deductive Reasoning

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