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Path Integrals and Quantum Mechanics Path Integrals and Quantum Mechanics Martin Sandstr¨om Department Of Physics Ume_aUniversity Supervisor: Jens Zamanian October 1, 2015 Abstract In this thesis we are investigating a different formalism of non-relativistic quantum me- chanics called the path integral formalism. It is a generalization of the classical least action principle. The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state. Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normal- ization constant to find the explicit expression of the path integral. We look in to the subject of two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit. Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a phys- ical meaning in quantum mechanics. It is also shown that the path integral formulation is equivalent to the Schr¨odingerdescription of quantum mechanics, by deriving the Schr¨odinger equation from the path integral. Further applications of the path integral are discussed. 1 Contents 1 Introduction 3 2 Least Action in Classical Mechanics 5 3 Least Action in Quantum Mechanics 7 3.1 Summing over Paths . 7 3.2 Propagator with a Quadratic Lagrangian . 8 3.3 Separating the Propagator into Multiple Propagations . 10 3.4 Writing the full Propagator with a Time-Slicing Method . 11 3.5 Orthonormality Condition between Propagators . 12 3.6 The Relation of the Path Integral to the Schr¨odingerEquation . 14 3.7 Obtaining the Hamilton-Jacobi Equation from a Wave Function . 15 3.8 The Propagator in terms of Solutions of the Schr¨odingerEquation . 16 4 Applications of the Path Integral 16 4.1 Slits . 16 4.1.1 Single Slit of a Square Type . 16 4.1.2 Single Slit of Gaussian Type . 19 4.1.3 Double Slit of a Gaussian Type . 22 4.2 The Energies for the Quantum Harmonic Oscillator . 23 4.3 The Aharonov-Bohm Effect . 24 5 Conclusion 29 A Derivation of Propagators 30 A.1 The Wave-function in Terms of the Spatial Solutions of the Schr¨odingerEquation . 30 A.2 The Free Particle Propagator . 32 A.3 The Propagator for the Harmonic Oscillator . 33 A.4 The Propagator for an Entangled Path with the Origin Removed . 36 2 1 Introduction In the late 19th century an inconsistency between statistical mechanics and electrodynamics was discovered that showed classical mechanics to be inadequate. The inconsistency between statistical mechanics and electrodynamics was clear to Einstein, Rayleigh and Jeans in 1905 when they realized independently that in thermodynamics all degrees of freedom of a system have an average 1 energy of 2 kT , where k is the Boltzmann constant and T the environmental temperature. In electrodynamics the total number of frequency modes inside a cavity is inversely proportional to the square of the frequency, which followed the Rayleigh-Jeans law1, 2 Bf (T ) / kT f (1.0.1) where Bf is the spectral radiance defined as the radiance through a surface per unit frequency. The higher the frequencies are, the spectral radiance associated with each frequency of the oscillators gets larger because of the dependence of the square of the frequency. It is an attempt to describe the spectral radiance of electromagnetic radiation at all wavelengths from a black body at a given temperature. Since all possible frequencies must be considered, the smaller the wavelength are the spectral radiance associated with each wavelength of the oscillators gets larger because the law is inversely proportional to the the fourth power of the wavelength λ = c=f. A contradiction existed in that when we sum over all the modes in the partition function2 X ¯ Z = e−βE; (1.0.2) E¯ with each average energy E¯ equal to kT , the series does not converge. Max Planck resolved this by assuming that the energy of each harmonic oscillator came in a discrete form En = nhf, or quanta, where n is a non-negative integer, f is the frequency of the oscillator and h a constant of proportionality now known as Planck's constant. Then if we sum over all integers n the series converges to a finite number 1 X 1 Z = e−βnhf = ; (1.0.3) eβhf − 1 n called the Planck distribution. Later on the physicist Albert Einstein proposed an experiment that showed that when light at a certain frequency strikes a metal surface, electrons are released from it. The idea expressed in mathematical form is hf = Ek − Ev; (1.0.4) where hf is the energy of a light quanta, Ek the kinetic energy of the electrons and Ev the energy involving the applied voltage for the electric field. Another experiment, which shows that classical mechanics is insufficient in explaining phenomena, is the famous double-slit experiment. A source of particles are traveling freely to a wall where two separated slits have been carved out. If we designate one of the slits by A and the second by B, the particles have two options when they arrive to the wall: Either they can pass through slit A or slit B. At a distance after the slits we place detectors that counts how many particles that hits them. If we plot the number of detections with the position where the detector is positioned the shape of two bumps will be noticeable, where the height of them is the number of particles detected. Clearly the number of particles, at every position along the screen, vary each time we wish to perform the experiment. So all we can detect is the average number of particles hitting the screen after we have performed the experiment many times from equal initial conditions. If we move the experiment to a very small scale where the slits and the separation between them are very small our intuition tell us that nothing special will happen and the two bumps will again 1See David Bohm, Quantum Theory, p. 6 2See for instance David Chandler, Introduction to Modern Statistical Mechanics. 3 be visible after performing the same experiments. However, this is not what we see because; when we again plot the total number of detections to position a wave-like pattern will appear in the distribution, just as what we expect when two water waves interfere with each other. This effect after scaling of an experiment is something that classical theories can not explain and a modification was in need. In the 1920's a new theory called Quantum Mechanics was developed and this theory agreed with experiment. Many formulations were developed; for example Werner Heisenberg's theory treats matrices using linear algebra and Erwin Schr¨odinger'stheory deals with wave-functions commonly noted as . The wave-function contains the description of the state of a system and the absolute square of is interpreted as the probability to find the system in a particular state. The wave-function satisfies the equation @ i (x; t) = H^ (x; t); (1.0.5) ~@t known as the Schr¨odingerequation, where H^ is the Hamiltonian operator and denotes the wave-function. 4 In this thesis we are going to focus on another equivalent formulation of Quantum Mechanics that is based on the influential book of Quantum Mechanics by P.A.M. Dirac3 (and later developed by Richard P. Feynman4). Dirac assumed that a solution to the Schr¨odinger equation had the form i Z = A exp fiScl=~g = A exp Lcl dt ; (1.0.6) ~ where Scl is the classical action. The classical action is found by using Lagrange's equations and substituting the resulting equation of motion back in the action and solve the time integral. This form of the wave-function can also be found in a paper from 1926 by Erwin Schr¨odinger5 where for the wave-function to have the statistical property it can be written as a product of wave- functions corresponding to states occurring in succession. As the actions must have the additive property under the statistical property, we can find the relation between them as S = K log (1.0.7) where S is the classical action and K is a constant with the dimensions of action. One easily sees that it can be re-written in the form of (1.0.6) but with K undetermined. If one substitutes this solution into the Schr¨odingerequation we obtain a differential equation for Scl and the normal- ization A. If we let ~ ! 0 the resulting differential equation must reduce to the Hamilton-Jacobi equation from classical mechanics. 2 Least Action in Classical Mechanics In this section we are going to look at how we can find the path x(t) that minimizes the action. If we vary from the the path x(t) a little bit then the action does not not change to a first order variation of the path and accordingly to the principle of least action we obtain the equation of motion. The action is a functional in the sense that it takes the whole form of the path x(t) as input and outputs a number. For example, the action Z t2 S[x(t)] = dt L(x(t); x_(t)) (2.0.8) t1 is a functional of the whole path x(t) where L is the Lagrangian of the system. A functional can be viewed as a function of a function i.e.
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