Review of Quantum Physics

1.2. EXAMPLE: INFINITE SQUARE WELL 5The wavefunction is an abstract object, but we can calculate the probability density function fromthe wave function usingP(x) = |Ψ(x)| 2 . (1.13)Here P(x)dx tells us the probability that we will find the particle between x and x+dx. Probabilitiesrange from 0 (almost impossible) to 1 (almost certain). If we are to have a normalized eigenstate φ nthen we must ensure that ∫P(x)dx = 1 (1.14)where V denotes an integral over all space. If we apply this to our eigenstates then∫ a0∫ aV|φ n (x)| 2 dx = 1 (1.15)0N 2 sin 2 nπx ∫ a(a dx = 1N2 1−cos 2nπx )dx2 a(1.16)0= N 2a 2 = 1 (1.17)⇒ N =√2a(1.18)⇒ φ n (x) =√2asinnπxa . (1.19)Another important property of these eigenstates that we will use later is their orthogonality.∫ a0φ ∗ n(x)φ m (x)dx = δ nm (1.20)where δ nm is 0 unless n = m. An illustration of some of the energy eigenvalues and eigenstates isshown in Fig. 1.3.As we have stated above quantum mechanics has a probabilistic interpretation. Consequently it isoften more useful to calculate the expected value of an observable. In this example we might find theexpectation of the position ˆx or the momentum ˆp of the particle in the well. For simplicity here weassume that the particle is in the ground state φ 0 (x)〈ˆp〉 =∫ a0= 2 ¯ha i√ ) √2 πx(¯h d 2 πxsin sin dx (1.21)a a i dx a a} {{ }} {{ }} {{ }ˆp φ 0∫ a0φ ∗ 0sin πx πa aφ ∗ 0πxcos dx = 0 (by symmetry). (1.22)aSimilarly for the average position of the particle:∫ a√ √2 πx 2 πx〈ˆx〉 = sin x0 a a }{{}sina a} {{ } ˆx } {{ }dx (1.23)φ 0= 2 a= 2 a∫ a0∫ a0xsin 2 πx dx (1.24)a(x1−cos 2πx )dx = a 2 a 2 . (1.25)