Review of Quantum Physics

6 CHAPTER 1. REVIEW OF QUANTUM PHYSICSE n Ψ(x) P(x) = |Ψ(x)| 2E 4 = 16 ¯h2 π 2|φ 4 | 22ma 2 φ 1φ 4|φ 1 | 2E 3 = 9 ¯h2 π 22ma 2φ 3|φ 3 | 2E 2 = 4 ¯h2 π 22ma 2E 1 = ¯h2 π 22ma 2φ 2|φ 2 | 2Figure 1.3: The infinite square well energy eigenvalues, E n with their quadratic spacing, and thecorresponding eigenstate wave functions and probability distributions for the first 4 eigenvalues.This result is reassuring: the well is symmetric, so we might expect that it sits in the middle onaverage.The uncertainty principle is a well known feature of quantum mechanics. Two conjugate variables (inthis case position, x and momentum p) cannot simultaneously be know precisely. The uncertaintyprinciple states that∆x∆p ≥ ¯h/2 (1.26)where (∆x) 2 = 〈ˆx 2 〉−〈ˆx〉 2 and (∆p) 2 = 〈ˆp 2 〉−〈ˆp〉 2 . As an example of this we will calculate 〈ˆx 2 〉 and〈ˆp 2 〉 and demonstrate its application.∫ a√ √2〈ˆx 2 πx 2〉 = sin x 2 πx0 a a }{{}sin dx (1.27)a a} {{ } ˆx } 2 {{ }φ 0= 2 a∫ a0∫ aφ ∗ 0x 2 sin 2 πxa= 1 a 0= 1 ( a3a( 1= a 2 3 − 1 )2π 2(x 2 1−cos 2πxa3 − a32π 2 )dx (1.28))dx (1.29)(1.30)(1.31)