Universidade Federal de Goiás Instituto de Física e Química An Introduction to the Mathematical Aspects of Quantum Mechanics: Course Notes Petrus Henrique Ribeiro dos Anjos Paulo Eduardo Gonçalves de Assis Catalão,Go 2015 Contents 1 Quantum States and Observables 5 1.1 QuantumStates.......................... 6 1.1.1 Uncertainty Principle . 11 1.2 Inner Product and Hilbert Spaces . 12 1.3 Observables ............................ 15 1.4 Probability and Functions of Observables . 18 1.5 Self-adjointoperators. 20 1.6 RiezRepresentation. 22 1.7 SpectralTheorem......................... 24 1.8 Exercises.............................. 28 2 The Spectrum 29 2.1 SpectrumandResolvent . 30 2.2 Findingthespectrum. 35 2.2.1 ThespectrumoftheHamiltonian . 36 2.3 Exercises.............................. 41 3 Quantum Dynamics 43 3.1 Time evolution and Schrödinger Equation . 43 3.2 Applications to Two-Level Systems . 49 3.3 Schrödinger’swaveequation . 52 3.4 Time dependence of Expected Values . 57 3.4.1 Newton’s 2nd Law and Quantum Mechanics . 58 4 CONTENTS 3.5 QuantumPictures ........................ 60 3.6 Exercises.............................. 62 4 Approximation Methods 63 4.1 Non-perturbativemethods . 63 4.1.1 VariationalMethods . 63 4.1.2 ExtensiontoExcitedStates . 66 4.1.3 A 2 2 example ..................... 67 × 4.1.4 Method os Successive Powers . 68 4.1.5 WKB - Semiclassical approximation . 68 4.2 Time-independent Perturbation Theory . 71 4.2.1 Time-independent perturbation: Non-degenerate . 72 4.2.2 Time-independent perturbation: Degenerate . 75 4.3 TheAnharmonicOscillator . 77 4.4 Time-dependent Perturbation . 79 4.4.1 Time-dependent Perturbation Theory . 80 4.5 Furhter Applications and Fermi’s Golden Rule . 83 4.6 Dirac’sinteractionpicture . 84 4.7 Exercises.............................. 85 Referências 87 Chapter 1 Quantum States and Observables In classical physics the mathematical description of a phenomenon is somewhat clear. From the early days of modern science, the movement of a macroscopic body could be completely characterized by the specification of its position at a given instant of time. This process was easily achieved with the use of simple tools such as rulers and clocks. Predictions could be com- puted based on formulations relying on simple mathematical objects called functions. The development of differential and integral calculus, in fact, owes a lot to the works of scientists like Isaac Newton. With the advance of our understanding of the microscopic world, physi- cist were forced to abandon the classical formalism to match experiments. Intead, a new theory came to light and both the basic objects to describe it and the mathematical tools to formulate it had to be revised. Part of this mathematical toolbox was already well stablished at the beginning of the last century but some of it had to be developed alongside the results of experiments. Here we present an introduction to the mathematical aspects of quantum mechanics. 5 6 CHAPTER 1. QUANTUM STATES AND OBSERVABLES 1.1 Quantum States Definition 1.A. The state of a single particle 1-D quantum system is a complex valued continuous function ψ(x, t) such that i. The probability (x I) of finding that the position of the particle Pψ ∈ belong to the interval I R at time t is given by ⊂ (x I)= ψ(x, t) 2dx Pψ ∈ | | ZI ii. The probability (p I) of finding that the momentum of the particle Pψ ∈ belong to te interval I R at time t is given by ⊂ 1 p (p I)= ψ˜( , t) 2dp, Pψ ∈ ~ | ~ | ZI where f˜ is the Fourier transform of ψ, given by 1 f˜(k)= f(x)eikxdx √2π R Z A general property of the Fourier transform called Parceval’s identity [1], show us that Note that p dp ψ(x, t) 2dx = ψ˜( , t) 2 . (1.1.1) R | | R ~ | ~ Z Z And to our probability interpretation hold, we that ψ(x, t) 2dx =1, (1.1.2) R | | Z which means that the particle lies somewhere in the real line. As an exercise, we suggest that you show that the defined on 1.A Pψ actually obeys other probabilities rules. More precisely, you should show that with our definition, for any countable sequence of disjoint intervals I1,I2, ..., we have that (x I )= (x I ). (1.1.3) Pψ ∈ n Pψ ∈ ∪ n n n [ X 1.1. QUANTUM STATES 7 So the position x (or the momentum p) of the particle can assume any value in the real line, and to each interval we assign a probability (x I) Pψ ∈ that the value of x lies in I. We can show how to compute the mathematical expectation of x. As a warm up, assume that x is restricted to a bounded interval [a, b]. We can divide [a, b] into smaller subintervals Ik, and consider the following object: xkIk, Xk where xk is an arbitrary point of Ik. We desire that this sum converge to a limit as the maximum length goes to zero, and furthermore the convergence is independent of our choices of intervals Ik and point xk. If all this holds, we call the limit x¯ the mathematical expectation of x. If x is not restricted to a bounded interval, we can fix an arbitrary bounded interval [a, b], calculate (as above) a limit for this bounded interval, and finally take the limit where [a, b] goes to the real line. Now, in the latter case, if this limit (a) exist, (b) is independent of our choice of interval [a, b] and (c) is independent of how we grow the interval; we call this limit x¯ the mathematical expectation of x. There are a lot of “if” ’s here, and in order to our physical theory works we need to start to prove some things. Fortunately the following holds: Lemma 1.1.1. Let the system be in a state ψ and f(x) be a continuous function such that f(x) ψ(x, t) 2dx < + , R | || | ∞ Z then f¯ the mathematical expectation of f(x) is given by f¯ = f(x) ψ(x, t) 2dx R | | Z Proof. Let I R be a bounded interval and I , ...I be a partition of I into ⊂ 1 N smaller non-overlaping intervals with maximum length δ. For let for each k pick an arbitrary x I , then k ∈ k 8 CHAPTER 1. QUANTUM STATES AND OBSERVABLES 2 2 f(x) ψ(x, t) dx f(xk) ψ(x Ik) 6 f(x) f(xk) ψ(x, t) dx. I | | − P ∈ Ik | − || | Z k k Z X X The continuity of f(x) implies that we can make f(x) f(x ) 6 ǫ | − k | 2 by taking δ sufficiently small. Therefore f(x) f(xk) ψ(x, t) dx 6 Ik | − || | 2 ǫ ψ(x, t) dx, then R Ik | | R 2 2 f(x) ψ(x, t) dx f(xk) ψ(x Ik) 6 ǫ ψ(x, t) dx. I | | − P ∈ I | | Z k Z X To conclude the demonstration we note that +l f(x) ψ(x) 2dx = lim f(x) ψ(x) 2dx R | | l + l | | Z → ∞ Z− A direct consequence of the lemma above is the Corolary 1.1.1. Let the system be in a state ψ, the following holds x¯ = x ψ(x, t) 2dx R | | Z Analogue statements holds concerning the momentum, as follows. Their proof are left as exercises. Lemma 1.1.2. Let the system be in a state ψ and g(p) denote a continuous function such that p dp g(p) ψ˜( , t) 2 < + , R | || ~ | ~ ∞ Z the following holds p dp g(p)ψ˜( , t) 2 =g ¯ R ~ | ~ Z Corolary 1.1.2. Let the system be in a state ψ, the following holds p dp p¯ = pψ˜( , t) 2 R ~ | ~ Z 1.1. QUANTUM STATES 9 Since we use Fourier transform to describe the probabilities concerning the momentum, it will be helpful to recall some properties of this integral transformation. We use Fourier transform here without too much concern on the conditions that f(x) must satisfy in order that the transforated function exists, for a complete discussion of Fourier transform see Ref. [1]. First we recall an useful property of Fourier transform: ∂ f(k)= ikf˜(k) (1.1.4) ∂x − g This property follows directly form integration by parts and the observation that if f˜ exists then f(x) 0 as x 0. We also need an inverse Fourier | |→ | |→ transform, and it is given by 1 ikx f(x)= f˜(k)e− dk, (1.1.5) √2π R Z . To have some insight into equation 1.1.5, it worth consider a very helpful identity: 1 ik(x x′) δ(x x′)= e− − dk, (1.1.6) − 2π R Z where δ(x) denotes the quite ubiquitous entity that physicist call “Dirac Delta Function. Formally, this object can be defined as a “function” such that for any continuous function f and ǫ> 0 +ǫ f(x)δ(x)dx = f(0). (1.1.7) ǫ Z− When dealing with “Dirac Delta Function”, we need to keep in mind that: 1. It is not a function: Even an initial analysis show that δ(x) is not a function, it is actually an entity that the mathematicians call a distri- bution, and we suggest to see ref. [2] for a rigorous treatment. 2. Dirac is not the first to use it: For exemple, an infinitesimal formula for an infinitely tall, unit impulse delta function explicitly appears in a work of Cauchy in 1827 (see ref [3]). And indeed, several other 10 CHAPTER 1. QUANTUM STATES AND OBSERVABLES authors have dealt with objects with similar characteristics (among them Poisson , Kirchoff , Lord Kelvin , again see ref. [3]). And in particular, in the late nineteenth century Heaviside had derived the main properties of δ(x).