Review of Quantum Physics

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)