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Quantum Mechanics Quantum Mechanics Thomas DeGrand July 18, 2021 Quantum Mechanics 2 Contents 1 Introduction 5 2 Quantum mechanics in the language of Hilbert space 11 3 Time dependence in quantum mechanics 47 4 Propagators and path integrals 67 5 Density matrices 83 6 Wave mechanics 97 7 Angular momentum 147 8 Identical particles 185 9 Time independent perturbation theory 195 10 Variational methods 221 11 Time dependent perturbation theory 231 12 Electromagnetic interactions in semiclassical approximation 249 3 Quantum Mechanics 4 13 Scattering 281 14 Classical waves – quantum mechanical particles 335 15 Atoms and molecules 365 Chapter 1 Introduction 5 Quantum Mechanics 6 The subject of most of this book is the quantum mechanics of systems with a small number of degrees of freedom. The book is a mix of descriptions of quantum mechanics itself, of the general properties of systems described by quantum mechanics, and of techniques for describing their behavior. The examples illustrating quantum mechanical properties are often presented as specific physical systems. They will be drawn from many areas of physics, just to illustrate the fact that many apparently different physical systems naturally behave quite similarly. Ehrenfest famously said that “Physics is simple but subtle,” but he did not mean that it was easy. You cannot learn quantum mechanics out of one book, nor out of these notes. You will have to compare the explanations from many sources and integrate them together into your own whole. You should not be surprised to find that any two books are likely to present completely contradictory explanations for the same topic, and that both explanations are at least partially true. The quantum mechanical description of nature is fundamentally different from a classical description, in that it involves probabilistic statements. The usual causal story of classical mechanics is that in specifying a set of initial conditions, one completely specifies the evolu- tion of the system for all time. That is not possible in quantum mechanics, simply because it is not possible to completely specify all the initial conditions. For example, the uncertainty principle ∆p∆x ~ forbids us from simultaneously knowing the coordinates and momentum ≥ of a particle at any time. However, the evolution of the probability itself is causal, and is encoded in the time-dependent Schr¨odinger equation ∂ψ(t) i~ = Hψˆ (t). (1.1) ∂t Once we specify the wave function (probability amplitude) ψ(t) at some time t0, we know it for all later times. This is of course difficult for us, macroscopic beings that we are, to deal with. Once we have gotten over our classical discomfort with the quantum world, we notice several striking features which recur again and again: The wide applicability of the same ideas, or the same physical systems, to many areas • of physics. The presence of symmetries and their consequences for dynamics. • Quantum Mechanics 7 I like to take advantage of the first of these features and think about applications of quantum mechanics in terms of a few paradigmatic systems, which are approximations to nearly all physical systems you might encounter. It is worthwhile to understand these systems completely and in many different ways. The two most important such systems are The simple harmonic oscillator. Most weakly coupled systems behave like a set of • coupled harmonic oscillators. This is easiest to visualize classically, for a system of particles interacting with a potential. If the system has some equilibrium structure, and if the energy of the system is small, then the particles will be found close to their equilibrium locations. This means that the potential energy is close to its minimum. We can do a Taylor expansion about the minimum, and then 1 2 V (x) V (x )+ V ′′(x )(x x ) + . (1.2) ≃ 0 2 0 − 0 which is the potential of an oscillator. The same situation will happen for a quantum mechanical system. An obvious example would be the motion of nuclear degrees of freedom in a molecule, the origin of the “vibrational spectrum” of molecular excitations, typically seen in the infrared. But there are many other less-obvious examples. Classical systems described by a set of equations of motion d2y i = ω2y (1.3) dt2 − i i can be thought of as collections of harmonic oscillators. Classical wave systems have such equations of motion. Their quantum analogues are also oscillators, and so their quantum descriptions will involve oscillator-like energies and degrees of freedom. Do you recall that the quantum mechanical energy spectrum for an oscillator is a set of equally spaced levels, E = nǫ (up to an overall constant) where ǫ is an energy scale, ǫ = ~ωi in Eq. 1.3, for example, and n is an integer 0, 1, 2,....? Do you also recall the story of the photon, that the quantum electromagnetic field is labeled by an integer, the number of photons in a particular allowed state? This integer is the n of the harmonic oscillator. We can take the analogy between “counting” and oscillator states still further. Imagine that we had a system which seemed to have nothing to do with an oscillator, like a hydrogen atom. It has an energy spectrum E = ǫi, where i labels the quantum number of the state. Now imagine that we have a whole collection of hydrogen atoms, and imagine also that these atoms do not interact. The energy of the collection is E = niǫi, where again ni, the number of particles in the collection with P Quantum Mechanics 8 energy ǫi, is an integer. If we forget about where the numbers ǫi came from, we will be making an oscillator-like description of the collection. The two state system. In an undergraduate textbook, the paradigm for this system is • the spin-1/2 particle in an external magnetic field. The electron is either spin-up or spin-down, and energy eigenstates are states where the spin aligns along the magnetic field. But this description is often used in situations not involving spin, typically as a way of approximating some complicated dynamics. For example, it often happens that a probe can excite a system from one state to a single different state, or that the probe can excite the state to many different ones, but that there is only one strong transition, and all other ones are small. Then it makes sense to replace the more complicated system by one which contains only two states, and assume that the system is confined only to those states. These states could be different atomic levels (perhaps interacting with a radiation field), different kinds of elementary particles which can exchange roles (neutrino oscillations are a recent example) or could even be systems which are naively very different (cold gases, which can condense as either molecules or as individual atoms, for another recent example). And of course the “qubit” of quantum computing jargon is a two state system. Of course, there are other paradigmatic systems. Often, we study collections of particles which are weakly interacting. A good approximate starting point for these systems is to ignore the interactions, giving a set of free particles as zeroth order states. This is often how we describe scattering. Hydrogen, despite its appearance in every undergraduate quantum mechanics course, is not so paradigmatic. Hydrogen is not even a very typical atom and its “1/n2” Rydberg spectrum is unique, a consequence of a particular special symmetry. It is useful to know about it, though, because it is simple enough that we can solve for its properties completely. The second feature – physics arising from symmetry – appears again and again in these notes. If a physical system has an underlying symmetry, seemingly different processes can be related. An example is the relative intensity of spectral lines of families of states with the same total angular momentum and different z-components: the pattern of spectral lines for an atom decaying from an excited state, placed in an external magnetic field. Often, without knowing anything about a dynamical system other than the symmetries it obeys, one can determine what processes are forbidden. (People speak of “selection rules” as a shorthand for these consequences; “forbidden” means that the rate is zero.) Such predictions are more Quantum Mechanics 9 reliable than ones of the absolute rates for processes actually occur. Knowing how to use symmetry to make predictions is at the bottom of most of the technical parts of this course. There is a deep connection between symmetry and conservation laws. We usually describe this in quantum mechanics by saying that the Hamiltonian possess a symmetry and the presence of a symmetry of the Hamiltonian directly leads to a conservation law. And most of the time we say “Here is a Hamiltonian, what are the consequences?” This is a perfectly good question, except that it may be putting the cart before the horse. Where do Hamiltonians come from, anyway? I think most people would say that it is the symmetries which are at the bottom; a Hamiltonian is a construct which builds the symmetries into the dynamics from the beginning. And not just the Hamiltonian, but one should use dynamical variables that naturally encode the symmetry, or transform in some simple way under the symmetry. Figuring out an appropriate set of variables for dealing with perhaps the most important symmetry transformation – rotational invariance – will be a big part of the course. Then, the job of the physicist is to recognize some symmetry, write down a mathematical system which encodes it, and then look for the consequence of the symmetry in new places. I’ve wandered away from the course, to try to disturb you. Let me say one more disturbing thing, before we get to work: Typically, physics at different energy scales decouples.
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