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Path Integrals in Quantum Mechanics Path Integrals in Quantum Mechanics Emma Wikberg Project work, 4p Department of Physics Stockholm University 23rd March 2006 Abstract The method of Path Integrals (PI’s) was developed by Richard Feynman in the 1940’s. It offers an alternate way to look at quantum mechanics (QM), which is equivalent to the Schrödinger formulation. As will be seen in this project work, many "elementary" problems are much more difficult to solve using path integrals than ordinary quantum mechanics. The benefits of path integrals tend to appear more clearly while using quantum field theory (QFT) and perturbation theory. However, one big advantage of Feynman’s formulation is a more intuitive way to interpret the basic equations than in ordinary quantum mechanics. Here we give a basic introduction to the path integral formulation, start- ing from the well known quantum mechanics as formulated by Schrödinger. We show that the two formulations are equivalent and discuss the quantum mechanical interpretations of the theory, as well as the classical limit. We also perform some explicit calculations by solving the free particle and the harmonic oscillator problems using path integrals. The energy eigenvalues of the harmonic oscillator is found by exploiting the connection between path integrals, statistical mechanics and imaginary time. Contents 1 Introduction and Outline 2 1.1 Introduction . 2 1.2 Outline . 2 2 Path Integrals from ordinary Quantum Mechanics 4 2.1 The Schrödinger equation and time evolution . 4 2.2 The propagator . 6 3 Equivalence to the Schrödinger Equation 8 3.1 From the Schrödinger equation to PI’s . 8 3.2 From PI’s to the Schrödinger equation . 12 4 Free Particle 15 5 The Classical Limit 17 6 Harmonic Oscillator 19 7 Imaginary time and statistical mechanics 25 7.1 General considerations . 25 7.2 Application to the harmonic oscillator . 26 8 Appendix 28 8.1 Free particle the easy way . 28 8.2 Details on the computation of the harmonic oscillator . 29 8.2.1 Why det(P) = det(N), j det(O)j = 1 . 29 8.2.2 Calculation of det(N) ................... 29 8.2.3 Calculation of elements of N¡1 . 31 1 Chapter 1 Introduction and Outline 1.1 Introduction One of the most often cited experiments of quantum physics is the double slit experiment. Quantum mechanical particles, e.g. electrons, give rise to an interference pattern, just like waves, when they are allowed to pass through a pair of slits. The interference phenomenon occurs even when they are shot through the slits one particle at a time. This strange behavior is due to the wave nature of the particles, and one says that the electrons pass through both slits at the same time, hence interfering with themselves. The fact that quantum mechanical objects, in contrast to classical, do not follow a certain trajectory, but rather a superposition of such, is counterintuitive in itself. However, in the path integral formulation of quantum mechanics, this property is naturally embedded in the equations. Because of this, Feynman’s theory treats quantum mechanical problems in a perhaps more intuitive way than Schrödingers wave equation. Moreover, the PI formalism treats the classical trajectory of a particle in the same way as any other path, leading to an easy way to find and understand the classical limit of the theory. These advantageous properties will perhaps make the method appear more useful than the calculations do at this introductory stage. 1.2 Outline The outline of this project work is as follows. In chapter two we introduce path integrals starting from the well known Schrödinger formalism. We ex- plore the general expression for the time evolution of a state, leading to the definition of the propagator. We interpret the propagator and discuss the meaning of "integrating over all paths". At the end of the chapter we arrive 2 at the path integral expression of the propagator for a general Hamiltonian. A formal derivation of the equivalence of the Schrödinger equation and the path integral formulation is contained in chapter three. As a first example of a PI we calculate the propagator for a free particle in chapter four. In the appendix this problem is solved in an easier way, using ordinary quantum mechanics. Chapter five covers the classical limit at a qualitative level. More complicated are the calculations done in chapter six, treating the harmonic oscillator. However, almost every step is performed in detail, there- fore even this part of the work should be possible to follow for everyone fa- miliar with elementary quantum mechanics and linear algebra. Some details in the derivation have been moved to the appendix to make the text, at least a bit, easier to read. In chapter seven, the interesting connection between path integrals in imaginary time and statistical mechanics is discussed. The similarity between the partition function and the propagator makes it easy to calculate the energy eigenvalues for the harmonic oscillator. This is also done in this chapter. 3 Chapter 2 Path Integrals from ordinary Quantum Mechanics 2.1 The Schrödinger equation and time evolu- tion In ordinary quantum mechanics, as formulated by Schrödinger, Heisenberg and Dirac in the 1920’s, we study the Schrödinger equation for a state jÃ(t)i: @ H^ jÃ(t)i = ih¹ jÃ(t)i @t ¡iH^ (t¡t0)=¹h ) jÃ(t)i = e jÃ(t0)i (2.1) ^ p^2 where the (time independent) Hamiltonian operator is H = 2m + V (^x). We will in this project work only treat the one dimensional case. Here we have also introduced the notation we will use in this report; x^ and p^ are the position and momentum operators, respectively. These have corresponding eigenstates and eigenvalues; x^jxi = xjxi (2.2) and p^jpi = pjpi: (2.3) The two sets of eigenstates form two complete basis and satisfy orthogo- nality and completeness relations (what is stated for x below is also valid for p): hx0jxi = ±(x0 ¡ x); (2.4) 4 Z dxjxihxj = 1: (2.5) @ For later purposes, let us also state that in the x basis, p^ = ¡ih¹ @x , which together with the normalizing condition (2.4) for the momentum operator implies 1 hxjpi = p eipx=¹h: (2.6) 2¼h¹ Since the set of position (momentum) eigenstates form a complete basis, one can expand any state jÃi in terms of these; Z jÃ(t)i = Ã(x; t)jxidx; (2.7) where Ã(x; t) = hxjÃ(t)i. In terms of the wave function Ã(x; t) the Schrödinger equation reads @ HÃ(x; t) = ih¹ Ã(x; t); (2.8) @t where h¹2 @2 H = ¡ + V (x): (2.9) 2m @x2 In this project work, we will only deal with such time independent Hamil- tonians, for which the potential V is constant in time. It should be familiar from an elementary course in quantum mechanics, that the differential equa- tion (2.8) is in these cases separable with respect to space and time variables. The separation constants En are the energy eigenvalues and the energy eigen- states associated with them, Ãn(x) = hxjÃni satisfy the equation à ! h¹2 @2 ¡ + V (x) à (x) = E à (x); (2.10) 2m @x2 n n n or in ket notation ^ HjÃni = EnjÃni;Ãn(x) = hxjÃni: (2.11) The energy eigenstates form a complete basis and fulfill relations similar to (2.4) and (2.5). The solutions to the time part of (2.8) are exponential factors e¡iEn(t¡t0)=¹h. The general solution thus becomes X ¡iEn(t¡t0)=¹h Ã(x; t) = cne Ãn(x) (2.12) n 5 The complex coefficients cn are determined by the initial conditions, i.e. the form of the wave function at t = t0. The orthogonality requirement on the energy eigenstates implies that Z ¤ cn = Ãn(x)Ã(x; t0)dx: (2.13) Thus, once we have found the energy eigenvalues and eigenfunctions, and know the wave function at some time t0, we can find the wave function at any later time t using equation (2.12) and (2.13). 2.2 The propagator In the previous section we used ordinary Schrödinger quantum mechanics to find a general expression for a wave function Ã(x; t), presuming that we knew the wave function Ã(x; t0). We will now write down a different expression for the time evolution of a wave function, introducing the so called propagator, which is the most important quantity of the path integral formulation. Start from equation (2.1) and multiply by a position eigenket from the left: ¡iH^ (t¡t0)=¹h Ã(x; t) = hxjÃ(t)i = hxje jÃ(t0)i (2.14) We may also insert an identity operator (eq. (2.5)), yielding Z ^ Ã(x; t) = hxje¡iH(t¡t0)=¹h dx jx i hx jÃ(t )i = 0 0 | 0 {z 0 } Ã(x0;t0) Z Z ¡iH^ (t¡t0)=¹h = dx0hxje jx0iÃ(x0; t0) ´ dx0 U(x; t; x0; t0)Ã(x0; t0); (2.15) where Ã(x0; t0) is the (known) wave function at t = t0. The quantity ¡iH^ (t¡t0)=¹h U(x; t; x0; t0) = hxje jx0i (2.16) is the so called propagator. What is the meaning of this quantity? First of ¡iH^ (t¡t0)=¹h all, notice that e jx0i is the state of some object at time t, supposed ¡iH^ (t¡t0)=¹h it was in the state jx0i at time t0. Hence, hxje jx0i = U(x; t; x0; t0) is the probability amplitude that we will find the object in state jxi at time t, if it was in jx0i at t0. To make this even more precise; If we make a measurement of a particle’s position at time t0 and find it to be x0, then the propagator tells us the probability amplitude for finding the particle at the 6 point x if we make a new measurement at a later time t.
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