Review of Quantum Physics

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8 CHAPTER 1. REVIEW OF QUANTUM PHYSICSfrom the coefficients |c i | 2 = 1/2 for i = 1,3. The crucial part as we shall see later is that once thesystem has been measured in a particular state, it collapses into that eigenstate, so that any furthermeasurements recover the same result.Lastlywe could leavethis stateto evolve, which it will do accordingto the time dependent Schrödingerequation.ĤΨ = i¯h ∂ Ψ. (1.44)∂tAs we will see each of the energy eigenstates has a time varying factor of exp { −iE nt} ¯h , so we can writeΨ(x,t) = ∑ nc n φ n (x)e −iEnt/¯h . (1.45)In this example we have thatΨ(x,t) = √ 1√e −iE 0 t 2 πx¯h sin 2 a a + √ 1√e −9iE 0 t 2 3πx¯h sin 2 a a(1.46)P(x,t) = |Ψ(x,t)| 2 (1.47)= 1 (sin 2 πx 3πx+sin2a a a +2cos 8E )0t πx 3πxsin sin (1.48)¯h a awhere E 0 = ¯h2 n 2 π 22ma 2 . We plot the probability of finding the particle at a particular point x in thewell as a function of time P(x,t) = |Ψ(x,t)| 2 in Fig. 1.4. Note that the first two terms are justthe probability distributions for the n = 1 and n = 3 energy eigenstates, and the last term is the“interference” between these two states. This extended example illustrates most the main points ofthe postulates of quantum mechanics. We will discuss these, together with some better notation inthe next section.1.3 Dirac NotationThe notation of the last example was rather cumbersome. We had to repeatedly write down integralsigns in a long winded way. Dirac or bra-ket notation is a general, if slightly more abstract way ofwriting down the relevant equations. Essentially we denote the wave function by |Ψ〉, which is knownas a ket. Its complex conjugate is denoted by a bra 〈Ψ|. The integrals we encountered in the lastsection can be denoted: ∫φ ∗ (x)ψ(x)dx = 〈φ|ψ〉. (1.49)A compact notation for the energy eigenstates issimilarly if we take the conjugate of this we haveso the operator Ĥ can act to the left!VĤ|φ n 〉 = E n |φ n 〉, (1.50)〈φ n |Ĥ = E n〈φ n | (1.51)Often it is even more compact to label the eigenstate ket with simply the eigenvalue. For example ifwe consider the momentum operator:ˆp|p〉 = p|p〉 (1.52)
1.3. DIRAC NOTATION 9tP(x,t)=| ψ | 2xFigure 1.4: The time dependant probability distribution given in Eq. (1.48), for our particular initialstate.This equation (especially when read aloud!) may seem confusing, but the compactness afforded bythe notation is very useful for writing complicated expressions.We will see the real utility of this notation later when we discuss matrix mechanics, as it allows aunified of it with wave mechanics, and perturbation theory where the notation would otherwise bevery dense.From here onwards we will try use Dirac notation so that we become accustomed with it, but will notbe too formal.Analogy with vectorsThere is a strong analogy between Dirac notation, and vector notation. The table below illustratesthe analogies between the two different notations. If we use the matrix mechanics representation ofquantum physics, then this analogy becomes even stronger. However, this analogy is no coincidenceas they are both examples of linear vector spaces.
Page 5 and 6: 1.2. EXAMPLE: INFINITE SQUARE WELL
Page 7: 1.2. EXAMPLE: INFINITE SQUARE WELL
Page 11 and 12: 1.4. THE POSTULATES 11Classical Var
Page 13 and 14: 1.4. THE POSTULATES 13so here A n a
Page 15 and 16: 1.4. THE POSTULATES 15Labelling qua
Page 17 and 18: 1.5. PROBLEMS 17Thus if Â commutes
Page 19 and 20: 1.5. PROBLEMS 199B. A particle is r
Page 21: 1.6. ANSWERS TO CHAPTER 1 PROBLEMS

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