Thus far with Schrödinger's equation, we have considered only situations where the spatial probability distribution was steady in time. In our rationalization of the time-independent Schrödinger equation, we imagined we had, for example, a steady electron beam, where the electrons had a definite energy; this beam was diffracting off some object, such as a crystal or through a pair of slits. The result was some steady diffraction pattern (at least, the probability distribution did not vary in time). We then went on to use this equation to examine some other specific problems, including potential wells, where this requirement of definite energy led to the unusual behavior that only specific, “quantized” energies were allowed.

In particular, we analyzed the problem of the harmonic oscillator and found stationary states of that oscillator. On the face of it, stationary states of an oscillator (other than the trivial one of the oscillator having zero energy) make little sense given our classical experience with oscillators – a classical oscillator with energy oscillates.

Clearly, we must expect quantum mechanics to model situations that are not stationary. The world about us changes, and if quantum mechanics is to be a complete theory, it must handle such changes. To understand such changes, at least for the kinds of systems where Schrödinger's equation might be expected to be valid, we need a time-dependent extension of Schrödinger's equation.

One of the most important practical applications of quantum mechanics is the understanding and engineering of crystalline materials. Of course, the full understanding of crystalline materials is a major part of solid-state physics and merits a much longer discussion than we give here. We will, however, try to introduce some of the most basic quantum mechanical principles and simplifications in crystalline materials. This will also allow us to perform many quantum mechanical calculations of important processes in semiconductors.

Crystals

A crystal is a material whose measurable properties are periodic in space. We can think about it using the idea of a unit cell. If we think of the unit cell as a “block” or “brick,” then a definition of a crystal structure is one that can fill all space by the regular stacking of these unit cells. If we imagine that we marked a black spot on the same position of the surface of each block, these spots or points would form a crystal lattice. We can, if we wish, define a set of vectors, R L , that we call lattice vectors. The set of lattice vectors consists of all of the vectors that link points on this lattice; that is,

Here, a1, a2, and a3 are the three linearly independent vectors that take us from a given point in one unit cell to the equivalent point in the adjacent unit cell.

In this appendix, we summarize the core background mathematics that we presume in the rest of the book. A major purpose here is to clarify the mathematical notations and terminology. Readers coming from different backgrounds may be more used to one notation or another and other books that the reader may consult may use different notation and terms, so we clarify the ones we use here and their relations to others. This appendix may also serve as a refresher or to patch over some holes temporarily in the reader's knowledge until the reader has more time to study the relevant mathematics in more detail. This short discussion here is certainly not a complete one and no attempt is made to give rigorous and complete mathematics.

Quantum mechanics is sometimes presented as being a very mathematical subject. It is true that many aspects of quantum mechanics can only be well defined using a mathematical vocabulary. In fact, we can assure the reader that the mathematics of quantum mechanics is not harder than that found in classical physics or any analytic branch of engineering and the required background is essentially the same as in those fields. Because quantum mechanics is very fundamentally based on linear mathematics, quantum mechanics, in practice, is arguably simpler mathematically than many other areas of science and engineering.

We have seen how to solve some simple quantum mechanical problems exactly and, in principle, we know how to solve for any quantum mechanical problem that is the solution of Schrödinger's equation. Some extensions of Schrödinger's equation are important for many problems, especially those including the consequences of electron spin. Other equations also arise in quantum mechanics, beyond the simple Schrödinger equation description, such as appropriate equations to describe photons and relativistically correct approaches. We postpone discussion of any such more advanced equations.

For all such equations, however, there are relatively few problems that are simple enough to be solved exactly. This is not a problem peculiar to quantum mechanics; there are relatively few classical mechanics problems that can be solved exactly either. Problems that involve multiple bodies or that involve the interaction between systems are often quite difficult to solve.

One could regard such difficulties as being purely mathematical, say that we have done our job of setting up the necessary principles to understand quantum mechanics, and move on, consigning the remaining tasks to applied mathematicians or possibly to some brute-force computer technique. Indeed, the standard set of techniques that can be applied, for example, to the solution of differential equations can be (and are) applied to the solution of quantum mechanical differential equation problems. The problem with such an approach is that if we blindly apply the mathematical techniques, we may lose much insight as to how such more complicated systems work.

When we think of processing information, we are typically used to a classical world in which we represent information in terms of the classical state of an object. For example, we could represent a number as the length of some particular rod in meters or the value of some electrical potential in volts; these would be analog representations. More typically in information processing, we represent numbers and other information digitally, in binary form as a sequence of “bits” that are either “1” or “0”. We can use all sorts of physical representations for the 1 and 0, such as an object being “up” or “down,” a device being “on” (e.g., passing current) or “off” (e.g., not passing current), or a voltage being “high” or “low.”

In the quantum mechanical world, we have additional options in representing information; in particular we can use quantum mechanical superpositions, such as a superposition of “up” and “down.” Though we could easily also have the equivalent of a superposition in a classical world for one physical system (a system that was half up and half down could simply be represented by it being horizontal), in quantum mechanics, we can have kinds of superpositions of multiple systems (so-called entangled states) that have no classical analog and the act of measurement on a quantum mechanical system in a superposition can have quite a different result from that in a classical system (i.e., the process of “collapse” into an eigenstate).

We now embark on one of Aristophanes' most famous and delightful plays. With its female lead (though played by a man, of course), it brings back women into the text, alive and very decidedly kicking.

The grammatical input here is challenging, consisting of the aorist optative active and middle (GE pp. 210–211, #212), δίδωμι (GE pp. 212–214, #214), ἀμελής and γλυκύς (GE pp. 214–215, #215) and the relative pronoun (GE pp. 216–219, #216–219). It is worth mastering this material very thoroughly. δίδωμι, of which GE says comfortingly that ‘there is little here that is difficult to recognise’, is the valuable gateway to a family of verbs which end in -μι.

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1 ἣ Your first relative pronoun: feminine because it goes with Lysistrata, nominative because it is the subject of the verb in its clause.

3 καταλύσασαι What part of the verb is this? Women are the subject of the sentence.

5 ἴδοιμι From ὁράω. The second aorist is εἶδον, the aorist stem is ἰδ- and this is the aorist optative.

6–7 καὶ Κλεονίκη … κἀμοί = καί ὰμοί; what does και mean in both these phrases?

8 οὔσαις What part of the verb? It's a participle, and we are still talking about women.

οἵ What gender, what case? Why? See GE p. 216, #216. Note that in the nominative case m.s., f.s., and pl. the relative pronoun has an accent, whereas the article has none.

Quo usque tandem abutere, Catilina, patientia nostra? quam diu etiam furor iste tuus nos eludet? quem ad finem sese effrenata iactabit audacia? nihilne te nocturnum praesidium Palati, nihil urbis uigiliae, nihil timor populi, nihil concursus bonorum omnium, nihil hic munitissimus habendi senatus locus, nihil horum ora uultusque mouerunt? patere tua consilia non sentis, constrictam iam horum omnium scientia teneri coniurationem tuam non uides? quid proxima, quid superiore nocte egeris, ubi fueris, quos conuocaueris, quid consili ceperis quem nostrum ignorare arbitraris? o tempora, o mores! senatus haec intellegit, consul uidet; hic tamen uiuit. uiuit? immo uero etiam in senatum uenit, fit publici consili particeps, notat et designat oculis ad caedem unum quemque nostrum, nos autem fortes uiri satis facere rei publicae uidemur, si istius furorem ac tela uitemus. ad mortem te, Catilina, duci iussu consulis iam pridem oportebat, in te conferri pestem istam quam tu in nos omnes iam diu machinaris.

An uero uir amplissimus, P. Scipio, pontifex maximus, Ti. Gracchum mediocriter labefactantem statum rei publicae priuatus interfecit, Catilinam orbem terrae caede atque incendiis uastare cupientem nos consules perferemus? nam illa nimis antiqua praetereo, quod C. Serui-lius Ahala Sp. Maelium nouis rebus studentem manu sua occidit. fuit, fuit ista quondam in hac re publica uirtus ut uiri fortes acrioribus suppliciis ciuem perniciosum quam acerbissimum hostem coercerent. habemus senatus consultum in te, Catilina, uehemens et graue; non deest rei publicae consilium neque auctoritas huius ordinis; nos, nos, dico aperte, consules desumus.

Tandem aliquando, Quirites, L. Catilinam furentem audacia, scelus anhelantem, pestem patriae nefarie molientem, uobis atque huic urbi ferro flammaque minitantem ex urbe uel eiecimus uel emisimus uel ipsum egredientem uerbis prosecuti sumus. abiit, excessit, euasit, erupit. nulla iam pernicies a monstro illo atque prodigio moenibus ipsis intra moenia comparabitur. atque hunc quidem unum huius belli domestici ducem sine controuersia uicimus. non enim iam inter latera nostra sica illa uersabitur; non in campo, non in foro, non in curia, non denique intra domesticos parietes pertimescemus. loco ille motus est, cum est ex urbe depulsus. palam iam cum hoste nullo impediente 10bellum iustum geremus. sine dubio perdidimus hominem magnificeque uicimus, cum illum ex occultis insidiis in apertum latrocinium coniecimus. quod uero non cruentum mucronem ut uoluit extulit, quod uiuis nobis egressus est, quod ei ferrum e manibus extorsimus, quod incolumes ciues, quod stantem urbem reliquit, quanto tandem illum maerore esse afflictum et profligatum putatis? iacet ille nunc prostratus, Quirites, et se perculsum atque abiectum esse sentit et retorquet oculos profecto saepe ad hanc urbem quam e suis faucibus ereptam esse luget. quae quidem mihi laetari uidetur, quod tantam pestem euomuerit forasque proiecerit.

CATILINARIAN

Following a visit by assassins to his house on the pretext of the morning salutatio, C., as consul, summoned a meeting of the senate to the temple of Jupiter Stator that same day (8 or possibly 7 November; cf. appendix 2). The speech for this occasion was written up and published (cf. the Introduction section 4) and is known as the First Catilinarian. It has as its subject not a bill proposed for enactment but rather Catiline himself and his future.

C. was evidently launched on a narrative of the frustrated assassination attempt (cf. 2.12.5–7 hesterno die, Quirites, cum domi meae paene interfectus essem, senatum in aedem Iouis Statoris conuocaui, rem omnem ad patres conscriptos detuli) when Catiline entered the chamber; our speech responds to Catiline's arrival; cf. Stroh (2000) 70. This scenario will account for the focus on Catiline as principal addressee of the speech, the other senators' shunning of Catiline, and the fact that the speech contains only two brief references to the attempted assassination (§§9.11–13 and 32.3).

C. described the effect of the speech this way: L. Catilinam … ex urbe uel (1) eiecimus uel (2) emisimus uel (3) ipsum egredientem uerbis prosecuti sumus (Cat. 2.1.1–4; see ad loc.).

This is a vital section, in which you will meet new tenses of the verb; so far the narrative has been confined to the present tense alone.

In 5A–B the imperfect tense is introduced; and in 5C–D the future.

If you are not happy with the term imperfect, it may be helpful to look at the explanation in GE p. 93, #103.

Note: In Section 6 you will met another past tense called the aorist. The meaning of ‘imperfect’ in this context is ‘incomplete’, ‘unfinished’, while ‘aorist’ means ‘undefined’. For example:

Greek also has a perfect tense: this expresses a completed action in present time, ‘I have just come in’, ‘I have shut the door’. It is much less common than the aorist, and it is not introduced until Section 13E.

Before beginning 5A you may find it helpful to write out the present tense (active and middle) of παύω, and another table for εἰμί, leaving room to the right of the table to enter imperfect endings as they occur.

This is a long section with a lively story which introduces several important items of grammar, the dative case (GE pp. 176–180, #189–190), the aorist infinitive (GE pp. 187–189, #195–197), the aorist imperative (GE pp. 189–191, #198–200), γιγνώσκω (GE p. 201, #209), and the principal parts of common and very irregular verbs (GE p. 202, #211). None of this is difficult, but it pays to learn the new grammar carefully after each section in which it is introduced.

Before you start you may like to look at the table (GE pp. 176–177, #103) where the forms of the dative are set out. Notice the predominance of ι:

The various uses of the dative are well described in GE pp. 179–180, #190.

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10 τί βουλόμενος As before, ‘wanting what, do you …?’ ἀπολεῖς με … ‘you will destroy me’, ‘you will be the death of me’. The future of ἀπόλλυμι.

11 βοῆ χρῶμαι χράομαι (‘I use’) takes the dative (GE p.180, #190 (e)). The meaning is the same as if βοῶ had been written.

17 ἐμπεσεῖται Future of ἐμπίπτω – πίπτωῦμαι ἕπεσον.

18 τυγχάνεις εἰδώς εἰδως -ότος is the participle of οἶ the infinitive is εἰδέναι.

20 οὑτοιί οῦτος αὕτη τοῦτο + ι is even more demonstrative. ‘These spectators here’. (Note that the neuter is τουτί ταυτί.)

How to use this course

You have three books:

1. Reading Greek: Text and Vocabulary (RGT)

2. Reading Greek: Grammar and Exercises (GE)

3. Independent Study Guide to Reading Greek (ISG)

In addition there is a CD Speaking Greek which illustrates pronunciation, and records readings from passages in RGT.

You may also like to have The World of Athens (second edition 2008), which gives you background information about Athenian history and life.

The first two books, RGT and GE, were designed to be used together so that

1. You read the passage in RGT with the aid of the running vocabulary for the section.

2. Then you look at the explanations of the grammar in GE.

3. Then you learn the grammar and the lists of ‘vocabulary to be learnt’ in GE.

4. Then you do the exercises for the section to make sure that you have understood the grammar. You may want to do each section of the exercises as you study the grammar in GE, so that you get immediate practice in a new feature of language. It is not essential to do all the exercises on morphology and syntax, but they will help you to practise the language and to make sure that you understand the grammar. However, you should always do the final ‘Test Exercise’ in each section, as it is an important check on your grasp of the section.

In this section you will meet a number of different features of grammar and syntax that are clearly described in GE pp. 315–318, #291–294. Notice particularly μή + aorist subjunctive = ‘don't …’ (GE #292), and φοβοῦμαι μή + subjunctive = ‘I am afraid that something may happen’ (GE #293).

Also there are verbal nouns ending in -τέος which express obligation (‘must’), GE #294–295.

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3 ἀθύμως ἔχοντι Agrees with ʾΑριστάρχῳ. ἔχω + adverb expresses the state that someone is in, e.g. καλῶς ἔχω ‘I am well’, κακῶς ἔχω ‘I am in a bad way.’

10 τοὺς ἐξηγητάς These were state officials who advised what to do in cases of murder. Apollodoros' next question shows two of the procedures which they might advise on.

13 ἐπεποιήκει What tense? see GE p. 315, #291.

ἠδικήκει What tense?

17 διεξελθόντι δέ μοι … οὺκ ἔφασαν ἐξεῖναι … Literally ‘to me having related … they said that it was not possible to …’

ἐπεπόνθη What verb? What tense? Try πάσχω πείσομαι ἔπαθον πεπόνθα.

Translation for 16A

Apollodoros is going straight towards Ilisos, walking along the road outside the wall, beneath the wall itself. When he is at the gate, there he meets Aristarchos the son of Ariston, who is very depressed. Apollodoros, seeing Aristarchos coming towards him, addresses him.

There is relatively little grammatical input in this section. You learn the present and imperfect tenses passive (GE pp. 226–227, #220), which are the same as the middle, the genitive absolute (GE p. 228, #222), comparative adverbs (GE pp. 229–230, #225) and the optative of φημί (GE p. 231, #227).

The concept of the passive will present no problem to Latinists. GE p. 227, #221 will be useful.

Aristophanes' comedy Akharnians tells us much about Athenian politics and the feelings of the countrymen of Attica cooped up inside the city during the Peloponnesian War. It holds out a tantalising vision of peace.

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1 κυρία ἐκκλησία See RGT p. 91, Section 8A line 25 and the map on RGT p. 92.

2 ἡ Πνὺξ αὑτηί αύτηί is the extra demonstrative form. Cf. ούτοσί.

ἐρῆμος Why no feminine ending? It's a two-termination adjective, i.e. it has no separate feminine ending. Compound adjectives (e.g. ἀ-θάνατος, εὐ-δόκιμος) mostly fall into this category, as well as a number of other adjectives. See GE pp. 230–231, #226.

5 σχοινίον A rope with vermilion dye was swept across the agora to push people towards the Pnyx (the hill on which the ecclesia was held). The Assembly itself was proclaimed by a trumpet call; any citizen arriving with vermilion dye, and therefore touched by the rope, could be fined for arriving late.