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Preface These notes represent an experiment in the use of information technology in teaching an advanced undergraduate physics course, Quantum Physics at UCSD. The experiment has several goals. • To make all the class material including a complete set of lecture notes available to students on the World-Wide Web. • To make use of some simple multimedia technology to enhance the class notes as a learning tool compared to a conventional textbook. • To present a complex subject to students in several different ways so that each student can use the learning techniques best suited to that individual. • To get some experience with the use of multimedia technologies in teaching advanced courses. • To produce course material that might be appropriate for distance learning or self-paced courses in the future. The current set of notes covers a 3 quarter course at UCSD, from the beginning of Quantum Me- chanics to the quantization of the electromagnetic field and the Dirac equation. The notes for the last quarter should be considered to be a first draft. At this time, the experiment is in progress. One quarter is not sufficient to optimize the course material. While a complete set of html based notes has been produced, only limited additional audio and visual material is now available. It is my personal teaching experience that upper division physics students learn in different ways. Many physics students get very little more than an introduction to the material out of the lecture and prefer to learn from the textbook and homework. Some students claim they cannot learn from the textbook and rely on lectures to get their basic understanding. Some prefer a rather verbose exposition of the material in the text, while others prefer a concise discussion largely based on equations. Modern media have conditioned the students of today in a way that is often detrimental to learning complex subjects from either a lecture or a textbook. I chose to use html and the worldwide web as the primary delivery tool for enhanced class notes. All of the standard software tools and information formats are usable from html. Every computer can access this format using Internet browsers. An important aspect of the design of the notes is to maintain a concise basic treatment of the physics, with derivations and examples available behind hyperlinks. It is my goal, not fully met at this time, to have very detailed derivations, with less steps skipped than in standard textbooks. Eventually, this format will allow more examples than are practical in a textbook. Another important aspect is audio discussion of important equations and drawings. The browser is able to concentrate on an equation while hearing about the details instead of having to go back an forth between text and equation. The use of this needs to be expanded and would benefit from better software tools. Because of the heavy use of complex equations in this course, the html is generated from LaTeX input. This has not proved to be a limitation so far since native html can be included. LaTeX 16 has the ability to produce high quality equations and input is fast compared to other options. The LaTeX2html translator functions well enough for the conversion. Projecting the notes can be very useful in lecture for introductions, for review, and for quick looks at derivations. The primary teaching though probably still works best at the blackboard. One thing that our classrooms really don’t facilitate is switching from one mode to the other. In a future class, with the notes fully prepared, I will plan to decrease the formal lecture time and add lab or discussion session time, with students working moving at their own pace using computers. Projects could be worked on in groups or individually. Instructors would be available to answer questions and give suggestions. Similar sessions would be possible at a distance. The formal lecture could be taped and available in bite size pieces inside the lecture notes. Advanced classes with small numbers of students could be taught based on notes, with less instructor support than is usual. Classes could be offered more often than is currently feasible. Jim Branson 17 1 Course Summary 1.1 Problems with Classical Physics Around the beginning of the 20th century, classical physics, based on Newtonian Mechanics and Maxwell’s equations of Electricity and Magnetism described nature as we knew it. Statistical Me- chanics was also a well developed discipline describing systems with a large number of degrees of freedom. Around that time, Einstein introduced Special Relativity which was compatible with Maxwell’s equations but changed our understanding of space-time and modified Mechanics. Many things remained unexplained. While the electron as a constituent of atoms had been found, atomic structure was rich and quite mysterious. There were problems with classical physics, (See section 2) including Black Body Radiation, the Photoelectric effect, basic Atomic Theory, Compton Scattering, and eventually with the diffraction of all kinds of particles. Plank hypothesized that EM energy was always emitted in quanta E = hν = ̄hω to solve the Black Body problem. Much later, deBroglie derived the wavelength (See section 3.4) for particles. λ = h p Ultimately, the problems led to the development of Quantum Mechanics in which all particles are understood to have both wave and a particle behavior. 1.2 Thought Experiments on Diffraction Diffraction (See section 3) of photons, electrons, and neutrons has been observed (see the pictures) and used to study crystal structure. To understand the experimental input in a simplified way, we consider some thought experiments on the diffraction (See section 3.5) of photons, electrons, and bullets through two slits. For example, photons, which make up all electromagnetic waves, show a diffraction pattern exactly as predicted by the theory of EM waves, but we always detect an integer number of photons with the Plank’s relation, E = hν, between wave frequency and particle energy satisfied. Electrons, neutrons, and everything else behave in exactly the same way, exhibiting wave-like diffrac- tion yet detection of an integer number of particles and satisfying λ = h p . This deBroglie wavelength formula relates the wave property λ to the particle property p. 1.3 Probability Amplitudes In Quantum Mechanics, we understand this wave-particle duality using (complex) probability amplitudes (See section 4) which satisfy a wave equation. ψ(~x, t) = e i(~k·~x−ωt) = e i(~p·~x−Et)/h ̄ 18 The probability to find a particle at a position ~x at some time t is the absolute square of the probability amplitude ψ(~x, t). P(~x, t) = |ψ(~x, t)| 2 To compute the probability to find an electron at our thought experiment detector, we add the probability amplitude to get to the detector through slit 1 to the amplitude to get to the detector through slit 2 and take the absolute square. Pdetector = |ψ1 + ψ2| 2 Quantum Mechanics completely changes our view of the world. Instead of a deterministic world, we now have only probabilities. We cannot even measure both the position and momentum of a particle (accurately) at the same time. Quantum Mechanics will require us to use the mathematics of operators, Fourier Transforms, vector spaces, and much more. 1.4 Wave Packets and Uncertainty The probability amplitude for a free particle with momentum ~p and energy E = p (pc) 2 + (mc2) 2 is the complex wave function ψfree particle(~x, t) = e i(~p·~x−Et)/h ̄ . Note that |ψ| 2 = 1 everywhere so this does not represent a localized particle. In fact we recognize the wave property that, to have exactly one frequency, a wave must be spread out over space. We can build up localized wave packets that represent single particles(See section 5.1) by adding up these free particle wave functions (with some coefficients). ψ(x, t) = 1 √ 2π ̄h Z +∞ −∞ φ(p)e i(px−Et)/h ̄ dp (We have moved to one dimension for simplicity.) Similarly we can compute the coefficient for each momentum φ(p) = 1 √ 2π ̄h Z∞ −∞ ψ(x)e −ipx/h ̄ dx. These coefficients, φ(p), are actually the state function of the particle in momentum space. We can describe the state of a particle either in position space with ψ(x) or in momentum space with φ(p). We can use φ(p) to compute the probability distribution function for momentum. P(p) = |φ(p)| 2 We will show that wave packets like these behave correctly in the classical limit, vindicating the choice we made for ψfree particle(~x, t). The Heisenberg Uncertainty Principle (See section 5.3) is a property of waves that we can deduce from our study of localized wave packets. ∆p∆x ≥ ̄h 2 19 It shows that due to the wave nature of particles, we cannot localize a particle into a small volume without increasing its energy. For example, we can estimate the ground state energy (and the size of) a Hydrogen atom very well from the uncertainty principle. The next step in building up Quantum Mechanics is to determine how a wave function develops with time – particularly useful if a potential is applied. The differential equation which wave functions must satisfy is called the Schr ̈odinger Equation. 1.5 Operators The Schr ̈odinger equation comes directly out of our understanding of wave packets. To get from wave packets to a differential equation, we use the new concept of (linear) operators (See section 6). We determine the momentum and energy operators by requiring that, when an operator for some variable v acts on our simple wavefunction, we get v times the same wave function. p (op) x = ̄h i ∂ ∂x p (op) x e i(~p·~x−Et)/h ̄ = ̄h i ∂ ∂xe i(~p·~x−Et)/h ̄ = pxe i(~p·~x−Et)/h ̄ E (op) = i ̄h ∂ ∂t E (op) e i(~p·~x−Et)/h ̄ = i ̄h ∂ ∂t e i(~p·~x−Et)/h ̄ = Eei(~p·~x−Et)/h ̄ 1.6 Expectation Values We can use operators to help us compute the expectation value (See section 6.3) of a physical variable. If a particle is in the state ψ(x), the normal way to compute the expectation value of f(x) is hf(x)i = Z∞ −∞ P(x)f(x)dx = Z∞ −∞ ψ ∗ (x)ψ(x)f(x)dx. If the variable we wish to compute the expectation value of (like p) is not a simple function of x, let its operator act on ψ(x) hpi = Z∞ −∞ ψ ∗ (x)p (op)ψ(x)dx. We have a shorthand notation for the expectation value of a variable v in the state ψ which is quite useful. hψ|v|ψi ≡ Z∞ −∞ ψ ∗ (x)v (op)ψ(x)dx. We extend the notation from just expectation values to hψ|v|φi ≡ Z∞ −∞ ψ ∗ (x)v (op)φ(x)dx 20 and hψ|φi ≡ Z∞ −∞ ψ ∗ (x)φ(x)dx We use this shorthand Dirac Bra-Ket notation a great deal. 1.7 Commutators Operators (or variables in quantum mechanics) do not necessarily commute. We can compute the commutator (See section 6.5) of two variables, for example [p, x] ≡ px − xp = ̄h i . Later we will learn to derive the uncertainty relation for two variables from their commutator. We will also use commutators to solve several important problems. 1.8 The Schr ̈odinger Equation Wave functions must satisfy the Schr ̈odinger Equation (See section 7) which is actually a wave equation. − ̄h 2 2m ∇ 2ψ(~x, t) + V (~x)ψ(~x, t) = i ̄h ∂ψ(~x, t) ∂t We will use it to solve many problems in this course. In terms of operators, this can be written as Hψ(~x, t) = Eψ(~x, t) where (dropping the (op) label) H = p 2 2m + V (~x) is the Hamiltonian operator. So the Schr ̈odinger Equation is, in some sense, simply the statement (in operators) that the kinetic energy plus the potential energy equals the total energy. 1.9 Eigenfunctions, Eigenvalues and Vector Spaces For any given physical problem, the Schr ̈odinger equation solutions which separate (See section 7.4) (between time and space), ψ(x, t) = u(x)T (t), are an extremely important set. If we assume the equation separates, we get the two equations (in one dimension for simplicity) i ̄h ∂T (t) ∂t = E T (t) Hu(x) = E u(x) The second equation is called the time independent Schr ̈odinger equation. For bound states, there are only solutions to that equation for some quantized set of energies Hui(x) = Eiui(x). For states which are not bound, a continuous range of energies is allowed. 21 The time independent Schr ̈odinger equation is an example of an eigenvalue equation (See section 8.1). Hψi(~x) = Eiψi(~x) If we operate on ψi with H, we get back the same function ψi times some constant. In this case ψi would be called and Eigenfunction, and Ei would be called an Eigenvalue. There are usually an infinite number of solutions, indicated by the index i here. Operators for physical variables must have real eigenvalues. They are called Hermitian operators (See section 8.3). We can show that the eigenfunctions of Hermitian operators are orthogonal (and can be normalized). hψi |ψj i = δij (In the case of eigenfunctions with the same eigenvalue, called degenerate eigenfunctions, we can must choose linear combinations which are orthogonal to each other.) We will assume that the eigenfunctions also form a complete set so that any wavefunction can be expanded in them, φ(~x) = X i αiψi(~x) where the αi are coefficients which can be easily computed (due to orthonormality) by αi = hψi |φi. So now we have another way to represent a state (in addition to position space and momentum space). We can represent a state by giving the coefficients in sum above. (Note that ψp(x) = e i(px−Et)/h ̄ is just an eigenfunction of the momentum operator and φx(p) = e −i(px−Et)/h ̄ is just an eigenfunction of the position operator (in p-space) so they also represent and expansion of the state in terms of eigenfunctions.) Since the ψi form an orthonormal, complete set, they can be thought of as the unit vectors of a vector space (See section 8.4). The arbitrary wavefunction φ would then be a vector in that space and could be represented by its coefficients. φ =   α1 α2 α3 ...   The bra-ket hφ|ψii can be thought of as a dot product between the arbitrary vector φ and one of the unit vectors. We can use the expansion in terms of energy eigenstates to compute many things. In particular, since the time development of the energy eigenstates is very simple, ψ(~x, t) = ψ(~x)e −iEit/h ̄ we can use these eigenstates to follow the time development of an arbitrary state φ φ(t) =   α1e −iE1t/h ̄ α2e −iE2t/h ̄ α3e −iE3t/h ̄ ...   simply by computing the coefficients αi at t = 0. We can define the Hermitian conjugate (See section 8.2) O† of the operator O by hψ|O|ψi = hψ|Oψi = hO †ψ|ψi. Hermitian operators H have the property that H† = H. 22 1.10 A Particle in a Box As a concrete illustration of these ideas, we study the particle in a box (See section 8.5) (in one dimension). This is just a particle (of mass m) which is free to move inside the walls of a box 0 < x < a, but which cannot penetrate the walls. We represent that by a potential which is zero inside the box and infinite outside. We solve the Schr ̈odinger equation inside the box and realize that the probability for the particle to be outside the box, and hence the wavefunction there, must be zero. Since there is no potential inside, the Schr ̈odinger equation is Hun(x) = − ̄h 2 2m d 2un(x) dx2 = Enun(x) where we have anticipated that there will be many solutions indexed by n. We know four (only 2 linearly independent) functions which have a second derivative which is a constant times the same function: u(x) = e ikx , u(x) = e −ikx , u(x) = sin(kx), and u(x) = cos(kx). The wave function must be continuous though, so we require the boundary conditions u(0) = u(a) = 0. The sine function is always zero at x = 0 and none of the others are. To make the sine function zero at x = a we need ka = nπ or k = nπ a . So the energy eigenfunctions are given by un(x) = C sin nπx a where we allow the overall constant C because it satisfies the differential equation. Plugging sin nπx a back into the Schr ̈odinger equation, we find that En = n 2π 2 ̄h 2 2ma2 . Only quantized energies are allowed when we solve this bound state problem. We have one remaining task. The eigenstates should be normalized to represent one particle. hun|uni = Za 0 C ∗ sin nπx a C sin nπx a dx = |C| 2 a 2 So the wave function will be normalized if we choose C = q 2 a . un(x) = r 2 a sin nπx a We can always multiply by any complex number of magnitude 1, but, it doesn’t change the physics. This example shows many of the features we will see for other bound state problems. The one difference is that, because of an infinite change in the potential at the walls of the box, we did not need to keep the first derivative of the wavefunction continuous. In all other problems, we will have to pay more attention to this. 1.11 Piecewise Constant Potentials in One Dimension We now study the physics of several simple potentials in one dimension. First a series of piecewise constant potentials (See section 9.1.1). for which the Schr ̈odinger equation is − ̄h 2 2m d 2u(x) dx2 + V u(x) = Eu(x) 23 or d 2u(x) dx2 + 2m ̄h 2 (E − V )u(x) = 0 and the general solution, for E > V , can be written as either u(x) = Aeikx + Be−ikx or u(x) = A sin(kx) + B cos(kx) , with k = q2m(E−V ) h ̄ 2 . We will also need solutions for the classically forbidden regions where the total energy is less than the potential energy, E < V . u(x) = Aeκx + Be−κx with κ = q2m(V −E) h ̄ 2 . (Both k and κ are positive real numbers.) The 1D scattering problems are often analogous to problems where light is reflected or transmitted when it at the surface of glass. First, we calculate the probability the a particle of energy E is reflected by a potential step (See section 9.1.2) of height V0: PR = √ E− √ √ E−V0 E+ √ E−V0 2 . We also use this example to understand the probability current j = h ̄ 2im [u ∗ du dx − du∗ dx u]. Second we investigate the square potential well (See section 9.1.3) square potential well (V (x) = −V0 for −a < x < a and V (x) = 0 elsewhere), for the case where the particle is not bound E > 0. Assuming a beam of particles incident from the left, we need to match solutions in the three regions at the boundaries at x = ±a. After some difficult arithmetic, the probabilities to be transmitted or reflected are computed. It is found that the probability to be transmitted goes to 1 for some particular energies. E = −V0 + n 2π 2 ̄h 2 8ma2 This type of behavior is exhibited by electrons scattering from atoms. At some energies the scattering probability goes to zero. Third we study the square potential barrier (See section 9.1.5) (V (x) = +V0 for −a < x < a and V (x) = 0 elsewhere), for the case in which E < V0. Classically the probability to be transmitted would be zero since the particle is energetically excluded from being inside the barrier. The Quantum calculation gives the probability to be transmitted through the barrier to be |T | 2 = (2kκ) 2 (k 2 + κ 2) 2 sinh2 (2κa) + (2kκ) 2 → ( 4kκ k 2 + κ 2 ) 2 e −4κa where k = q 2mE h ̄ 2 and κ = q2m(V0−E) h ̄ 2 . Study of this expression shows that the probability to be transmitted decreases as the barrier get higher or wider. Nevertheless, barrier penetration is an important quantum phenomenon. We also study the square well for the bound state (See section 9.1.4) case in which E < 0. Here we need to solve a transcendental equation to determine the bound state energies. The number of bound states increases with the depth and the width of the well but there is always at least one bound state. 24 1.12 The Harmonic Oscillator in One Dimension Next we solve for the energy eigenstates of the harmonic oscillator (See section 9.2) potential V (x) = 1 2 kx2 = 1 2mω2x 2 , where we have eliminated the spring constant k by using the classical oscillator frequency ω = q k m . The energy eigenvalues are En = n + 1 2 ̄hω. The energy eigenstates turn out to be a polynomial (in x) of degree n times e −mωx2/h ̄ . So the ground state, properly normalized, is just u0(x) = mω π ̄h 1 4 e −mωx 2 /h ̄ . We will later return the harmonic oscillator to solve the problem by operator methods. 1.13 Delta Function Potentials in One Dimension The delta function potential (See section 9.3) is a very useful one to make simple models of molecules and solids. First we solve the problem with one attractive delta function V (x) = −aV0δ(x). Since the bound state has negative energy, the solutions that are normalizable are Ceκx for x < 0 and Ce−κx for x > 0. Making u(x) continuous and its first derivative have a discontinuity computed from the Schr ̈odinger equation at x = 0, gives us exactly one bound state with E = − ma2V 2 0 2 ̄h 2 . Next we use two delta functions to model a molecule (See section 9.4), V (x) = −aV0δ(x + d) − aV0δ(x − d). Solving this problem by matching wave functions at the boundaries at ±d, we find again transcendental equations for two bound state energies. The ground state energy is more negative than that for one delta function, indicating that the molecule would be bound. A look at the wavefunction shows that the 2 delta function state can lower the kinetic energy compared to the state for one delta function, by reducing the curvature of the wavefunction. The excited state has more curvature than the atomic state so we would not expect molecular binding in that state. Our final 1D potential, is a model of a solid (See section 9.5). V (x) = −aV0 X∞ n=−∞ δ(x − na) This has a infinite, periodic array of delta functions, so this might be applicable to a crystal. The solution to this is a bit tricky but it comes down to cos(φ) = cos(ka) + 2maV0 ̄h 2 k sin(ka). Since the right hand side of the equation can be bigger than 1.0 (or less than -1), there are regions of E = h ̄ 2k 2 2m which do not have solutions. There are also bands of energies with solutions. These energy bands are seen in crystals (like Si). 25 1.14 Harmonic Oscillator Solution with Operators We can solve the harmonic oscillator problem using operator methods (See section 10). We write the Hamiltonian in terms of the operator A ≡ r mω 2 ̄h x + i p √ 2m ̄hω . H = p 2 2m + 1 2 mω2x 2 = ̄hω(A †A + 1 2 ) We compute the commutators [A, A† ] = i 2 ̄h (−[x, p] + [p, x]) = 1 [H, A] = ̄hω[A †A, A] = ̄hω[A † , A]A = − ̄hωA [H, A† ] = ̄hω[A †A, A† ] = ̄hωA† [A, A† ] = ̄hωA† If we apply the the commutator [H, A] to the eigenfunction un, we get [H, A]un = − ̄hωAun which rearranges to the eigenvalue equation H(Aun) = (En − ̄hω)(Aun). This says that (Aun) is an eigenfunction of H with eigenvalue (En − ̄hω) so it lowers the energy by ̄hω. Since the energy must be positive for this Hamiltonian, the lowering must stop somewhere, at the ground state, where we will have Au0 = 0. This allows us to compute the ground state energy like this Hu0 = ̄hω(A †A + 1 2 )u0 = 1 2 ̄hωu0 showing that the ground state energy is 1 2 ̄hω. Similarly, A† raises the energy by ̄hω. We can travel up and down the energy ladder using A† and A, always in steps of ̄hω. The energy eigenvalues are therefore En = n + 1 2 ̄hω. A little more computation shows that Aun = √ nun−1 and that A †un = √ n + 1un+1. These formulas are useful for all kinds of computations within the important harmonic oscillator system. Both p and x can be written in terms of A and A† . x = r ̄h 2mω (A + A † ) p = −i r m ̄hω 2 (A − A † ) 26 1.15 More Fun with Operators We find the time development operator (See section 11.5) by solving the equation i ̄h ∂ψ ∂t = Hψ. ψ(t) = e −iHt/h ̄ψ(t = 0) This implies that e −iHt/h ̄ is the time development operator. In some cases we can calculate the actual operator from the power series for the exponential. e −iHt/h ̄ = X∞ n=0 (−iHt/ ̄h) n n! We have been working in what is called the Schr ̈odinger picture in which the wavefunctions (or states) develop with time. There is the alternate Heisenberg picture (See section 11.6) in which the operators develop with time while the states do not change. For example, if we wish to compute the expectation value of the operator B as a function of time in the usual Schr ̈odinger picture, we get hψ(t)|B|ψ(t)i = he −iHt/h ̄ψ(0)|B|e −iHt/h ̄ψ(0)i = hψ(0)|e iHt/h ̄Be−iHt/h ̄ |ψ(0)i. In the Heisenberg picture the operator B(t) = e iHt/h ̄Be−iHt/h ̄ . We use operator methods to compute the uncertainty relationship between non-commuting variables (See section 11.3) (∆A)(∆B) ≥ i 2 h[A, B]i which gives the result we deduced from wave packets for p and x. Again we use operator methods to calculate the time derivative of an expectation value (See section 11.4). d dthψ|A|ψi = i ̄h hψ|[H, A]|ψi + ψ ∂A ∂t ψ ψ (Most operators we use don’t have explicit time dependence so the second term is usually zero.) This again shows the importance of the Hamiltonian operator for time development. We can use this to show that in Quantum mechanics the expectation values for p and x behave as we would expect from Newtonian mechanics (Ehrenfest Theorem). dhxi dt = i ̄h h[H, x]i = i ̄h h[ p 2 2m , x]i = D p m E dhpi dt = i ̄h h[H, p]i = i ̄h [V (x), ̄h i d dx] = − dV (x) dx Any operator A that commutes with the Hamiltonian has a time independent expectation value. The energy eigenfunctions can also be (simultaneous) eigenfunctions of the commuting operator A. It is usually a symmetry of the H that leads to a commuting operator and hence an additional constant of the motion. 27 1.16 Two Particles in 3 Dimensions So far we have been working with states of just one particle in one dimension. The extension to two different particles and to three dimensions (See section 12) is straightforward. The coordinates and momenta of different particles and of the additional dimensions commute with each other as we might expect from classical physics. The only things that don’t commute are a coordinate with its momentum, for example, [p(2)z, z(2)] = ̄h i while [p(1)x, x(2)] = [p(2)z, y(2)] = 0. We may write states for two particles which are uncorrelated, like u0(~x(1))u3(~x(2)), or we may write states in which the particles are correlated. The Hamiltonian for two particles in 3 dimensions simply becomes H = − ̄h 2 2m(1) ∂ 2 ∂x2 (1) + ∂ 2 ∂y2 (1) + ∂ 2 ∂z2 (1)! + − ̄h 2 2m(2) ∂ 2 ∂x2 (2) + ∂ 2 ∂y2 (2) + ∂ 2 ∂z2 (2)! + V (~x(1), ~x(2)) H = − ̄h 2 2m(1) ∇ 2 (1) + − ̄h 2 2m(2) ∇ 2 (1) + V (~x(1), ~x(2)) If two particles interact with each other, with no external potential, H = − ̄h 2 2m(1) ∇2 (1) + − ̄h 2 2m(2) ∇2 (1) + V (~x(1) − ~x(2)) the Hamiltonian has a translational symmetry, and remains invariant under the translation ~x → ~x +~a. We can show that this translational symmetry implies conservation of total momentum. Similarly, we will show that rotational symmetry implies conservation of angular momentum, and that time symmetry implies conservation of energy. For two particles interacting through a potential that depends only on difference on the coordinates, H = ~p 2 1 2m + ~p 2 2 2m + V (~r1 − ~r2) we can make the usual transformation to the center of mass (See section 12.3) made in classical mechanics ~r ≡ ~r1 − ~r2 R~ ≡ m1~r1 + m2~r2 m1 + m2 and reduce the problem to the CM moving like a free particle M = m1 + m2 H = − ̄h 2 2M ∇~ 2 R plus one potential problem in 3 dimensions with the usual reduced mass. 1 μ = 1 m1 + 1 m2 H = − ̄h 2 2μ ∇~ 2 r + V (~r) So we are now left with a 3D problem to solve (3 variables instead of 6). 28 1.17 Identical Particles Identical particles present us with another symmetry in nature. Electrons, for example, are indis- tinguishable from each other so we must have a symmetry of the Hamiltonian under interchange (See section 12.4) of any pair of electrons. Lets call the operator that interchanges electron-1 and electron-2 P12. [H, P12] = 0 So we can make our energy eigenstates also eigenstates of P12. Its easy to see (by operating on an eigenstate twice with P12), that the possible eigenvalues are ±1. It is a law of physics that spin 1 2 particles called fermions (like electrons) always are antisymmetric under interchange, while particles with integer spin called bosons (like photons) always are symmetric under interchange. Antisymmetry under interchange leads to the Pauli exclusion principle that no two electrons (for example) can be in the same state. 1.18 Some 3D Problems Separable in Cartesian Coordinates We begin our study of Quantum Mechanics in 3 dimensions with a few simple cases of problems that can be separated in Cartesian coordinates (See section 13). This is possible when the Hamiltonian can be written H = Hx + Hy + Hz. One nice example of separation of variable in Cartesian coordinates is the 3D harmonic oscillator V (r) = 1 2 mω2 r 2 which has energies which depend on three quantum numbers. Enxnynz = nx + ny + nz + 3 2 ̄hω It really behaves like 3 independent one dimensional harmonic oscillators. Another problem that separates is the particle in a 3D box. Again, energies depend on three quantum numbers Enxnynz = π 2 ̄h 2 2mL2 n 2 x + n 2 y + n 2 z for a cubic box of side L. We investigate the effect of the Pauli exclusion principle by filling our 3D box with identical fermions which must all be in different states. We can use this to model White Dwarfs or Neutron Stars. In classical physics, it takes three coordinates to give the location of a particle in 3D. In quantum mechanics, we are finding that it takes three quantum numbers to label and energy eigenstate (not including spin).
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Preface These notes represent an experiment in the use of information technology in teaching an advanced undergraduate physics course, Quantum Physics at UCSD. The experiment has several goals. • To make all the class material including a complete set of lecture notes available to students on the World-Wide Web. • To make use of some simple multimedia technology to enhance the class notes as a learning tool compared to a conventional textbook. • To present a complex subject to students in several different ways so that each student can use the learning techniques best suited to that individual. • To get some experience with the use of multimedia technologies in teaching advanced courses. • To produce course material that might be appropriate for distance learning or self-paced courses in the future. The current set of notes covers a 3 quarter course at UCSD, from the beginning of Quantum Me- chanics to the quantization of the electromagnetic field and the Dirac equation. The notes for the last quarter should be considered to be a first draft. At this time, the experiment is in progress. One quarter is not sufficient to optimize the course material. While a complete set of html based notes has been produced, only limited additional audio and visual material is now available. It is my personal teaching experience that upper division physics students learn in different ways. Many physics students get very little more than an introduction to the material out of the lecture and prefer to learn from the textbook and homework. Some students claim they cannot learn from the textbook and rely on lectures to get their basic understanding. Some prefer a rather verbose exposition of the material in the text, while others prefer a concise discussion largely based on equations. Modern media have conditioned the students of today in a way that is often detrimental to learning complex subjects from either a lecture or a textbook. I chose to use html and the worldwide web as the primary delivery tool for enhanced class notes. All of the standard software tools and information formats are usable from html. Every computer can access this format using Internet browsers. An important aspect of the design of the notes is to maintain a concise basic treatment of the physics, with derivations and examples available behind hyperlinks. It is my goal, not fully met at this time, to have very detailed derivations, with less steps skipped than in standard textbooks. Eventually, this format will allow more examples than are practical in a textbook. Another important aspect is audio discussion of important equations and drawings. The browser is able to concentrate on an equation while hearing about the details instead of having to go back an forth between text and equation. The use of this needs to be expanded and would benefit from better software tools. Because of the heavy use of complex equations in this course, the html is generated from LaTeX input. This has not proved to be a limitation so far since native html can be included. LaTeX 16 has the ability to produce high quality equations and input is fast compared to other options. The LaTeX2html translator functions well enough for the conversion. Projecting the notes can be very useful in lecture for introductions, for review, and for quick looks at derivations. The primary teaching though probably still works best at the blackboard. One thing that our classrooms really don’t facilitate is switching from one mode to the other. In a future class, with the notes fully prepared, I will plan to decrease the formal lecture time and add lab or discussion session time, with students working moving at their own pace using computers. Projects could be worked on in groups or individually. Instructors would be available to answer questions and give suggestions. Similar sessions would be possible at a distance. The formal lecture could be taped and available in bite size pieces inside the lecture notes. Advanced classes with small numbers of students could be taught based on notes, with less instructor support than is usual. Classes could be offered more often than is currently feasible. Jim Branson 17 1 Course Summary 1.1 Problems with Classical Physics Around the beginning of the 20th century, classical physics, based on Newtonian Mechanics and Maxwell’s equations of Electricity and Magnetism described nature as we knew it. Statistical Me- chanics was also a well developed discipline describing systems with a large number of degrees of freedom. Around that time, Einstein introduced Special Relativity which was compatible with Maxwell’s equations but changed our understanding of space-time and modified Mechanics. Many things remained unexplained. While the electron as a constituent of atoms had been found, atomic structure was rich and quite mysterious. There were problems with classical physics, (See section 2) including Black Body Radiation, the Photoelectric effect, basic Atomic Theory, Compton Scattering, and eventually with the diffraction of all kinds of particles. Plank hypothesized that EM energy was always emitted in quanta E = hν = ̄hω to solve the Black Body problem. Much later, deBroglie derived the wavelength (See section 3.4) for particles. λ = h p Ultimately, the problems led to the development of Quantum Mechanics in which all particles are understood to have both wave and a particle behavior. 1.2 Thought Experiments on Diffraction Diffraction (See section 3) of photons, electrons, and neutrons has been observed (see the pictures) and used to study crystal structure. To understand the experimental input in a simplified way, we consider some thought experiments on the diffraction (See section 3.5) of photons, electrons, and bullets through two slits. For example, photons, which make up all electromagnetic waves, show a diffraction pattern exactly as predicted by the theory of EM waves, but we always detect an integer number of photons with the Plank’s relation, E = hν, between wave frequency and particle energy satisfied. Electrons, neutrons, and everything else behave in exactly the same way, exhibiting wave-like diffrac- tion yet detection of an integer number of particles and satisfying λ = h p . This deBroglie wavelength formula relates the wave property λ to the particle property p. 1.3 Probability Amplitudes In Quantum Mechanics, we understand this wave-particle duality using (complex) probability amplitudes (See section 4) which satisfy a wave equation. ψ(~x, t) = e i(~k·~x−ωt) = e i(~p·~x−Et)/h ̄ 18 The probability to find a particle at a position ~x at some time t is the absolute square of the probability amplitude ψ(~x, t). P(~x, t) = |ψ(~x, t)| 2 To compute the probability to find an electron at our thought experiment detector, we add the probability amplitude to get to the detector through slit 1 to the amplitude to get to the detector through slit 2 and take the absolute square. Pdetector = |ψ1 + ψ2| 2 Quantum Mechanics completely changes our view of the world. Instead of a deterministic world, we now have only probabilities. We cannot even measure both the position and momentum of a particle (accurately) at the same time. Quantum Mechanics will require us to use the mathematics of operators, Fourier Transforms, vector spaces, and much more. 1.4 Wave Packets and Uncertainty The probability amplitude for a free particle with momentum ~p and energy E = p (pc) 2 + (mc2) 2 is the complex wave function ψfree particle(~x, t) = e i(~p·~x−Et)/h ̄ . Note that |ψ| 2 = 1 everywhere so this does not represent a localized particle. In fact we recognize the wave property that, to have exactly one frequency, a wave must be spread out over space. We can build up localized wave packets that represent single particles(See section 5.1) by adding up these free particle wave functions (with some coefficients). ψ(x, t) = 1 √ 2π ̄h Z +∞ −∞ φ(p)e i(px−Et)/h ̄ dp (We have moved to one dimension for simplicity.) Similarly we can compute the coefficient for each momentum φ(p) = 1 √ 2π ̄h Z∞ −∞ ψ(x)e −ipx/h ̄ dx. These coefficients, φ(p), are actually the state function of the particle in momentum space. We can describe the state of a particle either in position space with ψ(x) or in momentum space with φ(p). We can use φ(p) to compute the probability distribution function for momentum. P(p) = |φ(p)| 2 We will show that wave packets like these behave correctly in the classical limit, vindicating the choice we made for ψfree particle(~x, t). The Heisenberg Uncertainty Principle (See section 5.3) is a property of waves that we can deduce from our study of localized wave packets. ∆p∆x ≥ ̄h 2 19 It shows that due to the wave nature of particles, we cannot localize a particle into a small volume without increasing its energy. For example, we can estimate the ground state energy (and the size of) a Hydrogen atom very well from the uncertainty principle. The next step in building up Quantum Mechanics is to determine how a wave function develops with time – particularly useful if a potential is applied. The differential equation which wave functions must satisfy is called the Schr ̈odinger Equation. 1.5 Operators The Schr ̈odinger equation comes directly out of our understanding of wave packets. To get from wave packets to a differential equation, we use the new concept of (linear) operators (See section 6). We determine the momentum and energy operators by requiring that, when an operator for some variable v acts on our simple wavefunction, we get v times the same wave function. p (op) x = ̄h i ∂ ∂x p (op) x e i(~p·~x−Et)/h ̄ = ̄h i ∂ ∂xe i(~p·~x−Et)/h ̄ = pxe i(~p·~x−Et)/h ̄ E (op) = i ̄h ∂ ∂t E (op) e i(~p·~x−Et)/h ̄ = i ̄h ∂ ∂t e i(~p·~x−Et)/h ̄ = Eei(~p·~x−Et)/h ̄ 1.6 Expectation Values We can use operators to help us compute the expectation value (See section 6.3) of a physical variable. If a particle is in the state ψ(x), the normal way to compute the expectation value of f(x) is hf(x)i = Z∞ −∞ P(x)f(x)dx = Z∞ −∞ ψ ∗ (x)ψ(x)f(x)dx. If the variable we wish to compute the expectation value of (like p) is not a simple function of x, let its operator act on ψ(x) hpi = Z∞ −∞ ψ ∗ (x)p (op)ψ(x)dx. We have a shorthand notation for the expectation value of a variable v in the state ψ which is quite useful. hψ|v|ψi ≡ Z∞ −∞ ψ ∗ (x)v (op)ψ(x)dx. We extend the notation from just expectation values to hψ|v|φi ≡ Z∞ −∞ ψ ∗ (x)v (op)φ(x)dx 20 and hψ|φi ≡ Z∞ −∞ ψ ∗ (x)φ(x)dx We use this shorthand Dirac Bra-Ket notation a great deal. 1.7 Commutators Operators (or variables in quantum mechanics) do not necessarily commute. We can compute the commutator (See section 6.5) of two variables, for example [p, x] ≡ px − xp = ̄h i . Later we will learn to derive the uncertainty relation for two variables from their commutator. We will also use commutators to solve several important problems. 1.8 The Schr ̈odinger Equation Wave functions must satisfy the Schr ̈odinger Equation (See section 7) which is actually a wave equation. − ̄h 2 2m ∇ 2ψ(~x, t) + V (~x)ψ(~x, t) = i ̄h ∂ψ(~x, t) ∂t We will use it to solve many problems in this course. In terms of operators, this can be written as Hψ(~x, t) = Eψ(~x, t) where (dropping the (op) label) H = p 2 2m + V (~x) is the Hamiltonian operator. So the Schr ̈odinger Equation is, in some sense, simply the statement (in operators) that the kinetic energy plus the potential energy equals the total energy. 1.9 Eigenfunctions, Eigenvalues and Vector Spaces For any given physical problem, the Schr ̈odinger equation solutions which separate (See section 7.4) (between time and space), ψ(x, t) = u(x)T (t), are an extremely important set. If we assume the equation separates, we get the two equations (in one dimension for simplicity) i ̄h ∂T (t) ∂t = E T (t) Hu(x) = E u(x) The second equation is called the time independent Schr ̈odinger equation. For bound states, there are only solutions to that equation for some quantized set of energies Hui(x) = Eiui(x). For states which are not bound, a continuous range of energies is allowed. 21 The time independent Schr ̈odinger equation is an example of an eigenvalue equation (See section 8.1). Hψi(~x) = Eiψi(~x) If we operate on ψi with H, we get back the same function ψi times some constant. In this case ψi would be called and Eigenfunction, and Ei would be called an Eigenvalue. There are usually an infinite number of solutions, indicated by the index i here. Operators for physical variables must have real eigenvalues. They are called Hermitian operators (See section 8.3). We can show that the eigenfunctions of Hermitian operators are orthogonal (and can be normalized). hψi |ψj i = δij (In the case of eigenfunctions with the same eigenvalue, called degenerate eigenfunctions, we can must choose linear combinations which are orthogonal to each other.) We will assume that the eigenfunctions also form a complete set so that any wavefunction can be expanded in them, φ(~x) = X i αiψi(~x) where the αi are coefficients which can be easily computed (due to orthonormality) by αi = hψi |φi. So now we have another way to represent a state (in addition to position space and momentum space). We can represent a state by giving the coefficients in sum above. (Note that ψp(x) = e i(px−Et)/h ̄ is just an eigenfunction of the momentum operator and φx(p) = e −i(px−Et)/h ̄ is just an eigenfunction of the position operator (in p-space) so they also represent and expansion of the state in terms of eigenfunctions.) Since the ψi form an orthonormal, complete set, they can be thought of as the unit vectors of a vector space (See section 8.4). The arbitrary wavefunction φ would then be a vector in that space and could be represented by its coefficients. φ =   α1 α2 α3 ...   The bra-ket hφ|ψii can be thought of as a dot product between the arbitrary vector φ and one of the unit vectors. We can use the expansion in terms of energy eigenstates to compute many things. In particular, since the time development of the energy eigenstates is very simple, ψ(~x, t) = ψ(~x)e −iEit/h ̄ we can use these eigenstates to follow the time development of an arbitrary state φ φ(t) =   α1e −iE1t/h ̄ α2e −iE2t/h ̄ α3e −iE3t/h ̄ ...   simply by computing the coefficients αi at t = 0. We can define the Hermitian conjugate (See section 8.2) O† of the operator O by hψ|O|ψi = hψ|Oψi = hO †ψ|ψi. Hermitian operators H have the property that H† = H. 22 1.10 A Particle in a Box As a concrete illustration of these ideas, we study the particle in a box (See section 8.5) (in one dimension). This is just a particle (of mass m) which is free to move inside the walls of a box 0 < x < a, but which cannot penetrate the walls. We represent that by a potential which is zero inside the box and infinite outside. We solve the Schr ̈odinger equation inside the box and realize that the probability for the particle to be outside the box, and hence the wavefunction there, must be zero. Since there is no potential inside, the Schr ̈odinger equation is Hun(x) = − ̄h 2 2m d 2un(x) dx2 = Enun(x) where we have anticipated that there will be many solutions indexed by n. We know four (only 2 linearly independent) functions which have a second derivative which is a constant times the same function: u(x) = e ikx , u(x) = e −ikx , u(x) = sin(kx), and u(x) = cos(kx). The wave function must be continuous though, so we require the boundary conditions u(0) = u(a) = 0. The sine function is always zero at x = 0 and none of the others are. To make the sine function zero at x = a we need ka = nπ or k = nπ a . So the energy eigenfunctions are given by un(x) = C sin nπx a where we allow the overall constant C because it satisfies the differential equation. Plugging sin nπx a back into the Schr ̈odinger equation, we find that En = n 2π 2 ̄h 2 2ma2 . Only quantized energies are allowed when we solve this bound state problem. We have one remaining task. The eigenstates should be normalized to represent one particle. hun|uni = Za 0 C ∗ sin nπx a C sin nπx a dx = |C| 2 a 2 So the wave function will be normalized if we choose C = q 2 a . un(x) = r 2 a sin nπx a We can always multiply by any complex number of magnitude 1, but, it doesn’t change the physics. This example shows many of the features we will see for other bound state problems. The one difference is that, because of an infinite change in the potential at the walls of the box, we did not need to keep the first derivative of the wavefunction continuous. In all other problems, we will have to pay more attention to this. 1.11 Piecewise Constant Potentials in One Dimension We now study the physics of several simple potentials in one dimension. First a series of piecewise constant potentials (See section 9.1.1). for which the Schr ̈odinger equation is − ̄h 2 2m d 2u(x) dx2 + V u(x) = Eu(x) 23 or d 2u(x) dx2 + 2m ̄h 2 (E − V )u(x) = 0 and the general solution, for E > V , can be written as either u(x) = Aeikx + Be−ikx or u(x) = A sin(kx) + B cos(kx) , with k = q2m(E−V ) h ̄ 2 . We will also need solutions for the classically forbidden regions where the total energy is less than the potential energy, E < V . u(x) = Aeκx + Be−κx with κ = q2m(V −E) h ̄ 2 . (Both k and κ are positive real numbers.) The 1D scattering problems are often analogous to problems where light is reflected or transmitted when it at the surface of glass. First, we calculate the probability the a particle of energy E is reflected by a potential step (See section 9.1.2) of height V0: PR = √ E− √ √ E−V0 E+ √ E−V0 2 . We also use this example to understand the probability current j = h ̄ 2im [u ∗ du dx − du∗ dx u]. Second we investigate the square potential well (See section 9.1.3) square potential well (V (x) = −V0 for −a < x < a and V (x) = 0 elsewhere), for the case where the particle is not bound E > 0. Assuming a beam of particles incident from the left, we need to match solutions in the three regions at the boundaries at x = ±a. After some difficult arithmetic, the probabilities to be transmitted or reflected are computed. It is found that the probability to be transmitted goes to 1 for some particular energies. E = −V0 + n 2π 2 ̄h 2 8ma2 This type of behavior is exhibited by electrons scattering from atoms. At some energies the scattering probability goes to zero. Third we study the square potential barrier (See section 9.1.5) (V (x) = +V0 for −a < x < a and V (x) = 0 elsewhere), for the case in which E < V0. Classically the probability to be transmitted would be zero since the particle is energetically excluded from being inside the barrier. The Quantum calculation gives the probability to be transmitted through the barrier to be |T | 2 = (2kκ) 2 (k 2 + κ 2) 2 sinh2 (2κa) + (2kκ) 2 → ( 4kκ k 2 + κ 2 ) 2 e −4κa where k = q 2mE h ̄ 2 and κ = q2m(V0−E) h ̄ 2 . Study of this expression shows that the probability to be transmitted decreases as the barrier get higher or wider. Nevertheless, barrier penetration is an important quantum phenomenon. We also study the square well for the bound state (See section 9.1.4) case in which E < 0. Here we need to solve a transcendental equation to determine the bound state energies. The number of bound states increases with the depth and the width of the well but there is always at least one bound state. 24 1.12 The Harmonic Oscillator in One Dimension Next we solve for the energy eigenstates of the harmonic oscillator (See section 9.2) potential V (x) = 1 2 kx2 = 1 2mω2x 2 , where we have eliminated the spring constant k by using the classical oscillator frequency ω = q k m . The energy eigenvalues are En = n + 1 2 ̄hω. The energy eigenstates turn out to be a polynomial (in x) of degree n times e −mωx2/h ̄ . So the ground state, properly normalized, is just u0(x) = mω π ̄h 1 4 e −mωx 2 /h ̄ . We will later return the harmonic oscillator to solve the problem by operator methods. 1.13 Delta Function Potentials in One Dimension The delta function potential (See section 9.3) is a very useful one to make simple models of molecules and solids. First we solve the problem with one attractive delta function V (x) = −aV0δ(x). Since the bound state has negative energy, the solutions that are normalizable are Ceκx for x < 0 and Ce−κx for x > 0. Making u(x) continuous and its first derivative have a discontinuity computed from the Schr ̈odinger equation at x = 0, gives us exactly one bound state with E = − ma2V 2 0 2 ̄h 2 . Next we use two delta functions to model a molecule (See section 9.4), V (x) = −aV0δ(x + d) − aV0δ(x − d). Solving this problem by matching wave functions at the boundaries at ±d, we find again transcendental equations for two bound state energies. The ground state energy is more negative than that for one delta function, indicating that the molecule would be bound. A look at the wavefunction shows that the 2 delta function state can lower the kinetic energy compared to the state for one delta function, by reducing the curvature of the wavefunction. The excited state has more curvature than the atomic state so we would not expect molecular binding in that state. Our final 1D potential, is a model of a solid (See section 9.5). V (x) = −aV0 X∞ n=−∞ δ(x − na) This has a infinite, periodic array of delta functions, so this might be applicable to a crystal. The solution to this is a bit tricky but it comes down to cos(φ) = cos(ka) + 2maV0 ̄h 2 k sin(ka). Since the right hand side of the equation can be bigger than 1.0 (or less than -1), there are regions of E = h ̄ 2k 2 2m which do not have solutions. There are also bands of energies with solutions. These energy bands are seen in crystals (like Si). 25 1.14 Harmonic Oscillator Solution with Operators We can solve the harmonic oscillator problem using operator methods (See section 10). We write the Hamiltonian in terms of the operator A ≡ r mω 2 ̄h x + i p √ 2m ̄hω . H = p 2 2m + 1 2 mω2x 2 = ̄hω(A †A + 1 2 ) We compute the commutators [A, A† ] = i 2 ̄h (−[x, p] + [p, x]) = 1 [H, A] = ̄hω[A †A, A] = ̄hω[A † , A]A = − ̄hωA [H, A† ] = ̄hω[A †A, A† ] = ̄hωA† [A, A† ] = ̄hωA† If we apply the the commutator [H, A] to the eigenfunction un, we get [H, A]un = − ̄hωAun which rearranges to the eigenvalue equation H(Aun) = (En − ̄hω)(Aun). This says that (Aun) is an eigenfunction of H with eigenvalue (En − ̄hω) so it lowers the energy by ̄hω. Since the energy must be positive for this Hamiltonian, the lowering must stop somewhere, at the ground state, where we will have Au0 = 0. This allows us to compute the ground state energy like this Hu0 = ̄hω(A †A + 1 2 )u0 = 1 2 ̄hωu0 showing that the ground state energy is 1 2 ̄hω. Similarly, A† raises the energy by ̄hω. We can travel up and down the energy ladder using A† and A, always in steps of ̄hω. The energy eigenvalues are therefore En = n + 1 2 ̄hω. A little more computation shows that Aun = √ nun−1 and that A †un = √ n + 1un+1. These formulas are useful for all kinds of computations within the important harmonic oscillator system. Both p and x can be written in terms of A and A† . x = r ̄h 2mω (A + A † ) p = −i r m ̄hω 2 (A − A † ) 26 1.15 More Fun with Operators We find the time development operator (See section 11.5) by solving the equation i ̄h ∂ψ ∂t = Hψ. ψ(t) = e −iHt/h ̄ψ(t = 0) This implies that e −iHt/h ̄ is the time development operator. In some cases we can calculate the actual operator from the power series for the exponential. e −iHt/h ̄ = X∞ n=0 (−iHt/ ̄h) n n! We have been working in what is called the Schr ̈odinger picture in which the wavefunctions (or states) develop with time. There is the alternate Heisenberg picture (See section 11.6) in which the operators develop with time while the states do not change. For example, if we wish to compute the expectation value of the operator B as a function of time in the usual Schr ̈odinger picture, we get hψ(t)|B|ψ(t)i = he −iHt/h ̄ψ(0)|B|e −iHt/h ̄ψ(0)i = hψ(0)|e iHt/h ̄Be−iHt/h ̄ |ψ(0)i. In the Heisenberg picture the operator B(t) = e iHt/h ̄Be−iHt/h ̄ . We use operator methods to compute the uncertainty relationship between non-commuting variables (See section 11.3) (∆A)(∆B) ≥ i 2 h[A, B]i which gives the result we deduced from wave packets for p and x. Again we use operator methods to calculate the time derivative of an expectation value (See section 11.4). d dthψ|A|ψi = i ̄h hψ|[H, A]|ψi + ψ ∂A ∂t ψ ψ (Most operators we use don’t have explicit time dependence so the second term is usually zero.) This again shows the importance of the Hamiltonian operator for time development. We can use this to show that in Quantum mechanics the expectation values for p and x behave as we would expect from Newtonian mechanics (Ehrenfest Theorem). dhxi dt = i ̄h h[H, x]i = i ̄h h[ p 2 2m , x]i = D p m E dhpi dt = i ̄h h[H, p]i = i ̄h [V (x), ̄h i d dx] = − dV (x) dx Any operator A that commutes with the Hamiltonian has a time independent expectation value. The energy eigenfunctions can also be (simultaneous) eigenfunctions of the commuting operator A. It is usually a symmetry of the H that leads to a commuting operator and hence an additional constant of the motion. 27 1.16 Two Particles in 3 Dimensions So far we have been working with states of just one particle in one dimension. The extension to two different particles and to three dimensions (See section 12) is straightforward. The coordinates and momenta of different particles and of the additional dimensions commute with each other as we might expect from classical physics. The only things that don’t commute are a coordinate with its momentum, for example, [p(2)z, z(2)] = ̄h i while [p(1)x, x(2)] = [p(2)z, y(2)] = 0. We may write states for two particles which are uncorrelated, like u0(~x(1))u3(~x(2)), or we may write states in which the particles are correlated. The Hamiltonian for two particles in 3 dimensions simply becomes H = − ̄h 2 2m(1) ∂ 2 ∂x2 (1) + ∂ 2 ∂y2 (1) + ∂ 2 ∂z2 (1)! + − ̄h 2 2m(2) ∂ 2 ∂x2 (2) + ∂ 2 ∂y2 (2) + ∂ 2 ∂z2 (2)! + V (~x(1), ~x(2)) H = − ̄h 2 2m(1) ∇ 2 (1) + − ̄h 2 2m(2) ∇ 2 (1) + V (~x(1), ~x(2)) If two particles interact with each other, with no external potential, H = − ̄h 2 2m(1) ∇2 (1) + − ̄h 2 2m(2) ∇2 (1) + V (~x(1) − ~x(2)) the Hamiltonian has a translational symmetry, and remains invariant under the translation ~x → ~x +~a. We can show that this translational symmetry implies conservation of total momentum. Similarly, we will show that rotational symmetry implies conservation of angular momentum, and that time symmetry implies conservation of energy. For two particles interacting through a potential that depends only on difference on the coordinates, H = ~p 2 1 2m + ~p 2 2 2m + V (~r1 − ~r2) we can make the usual transformation to the center of mass (See section 12.3) made in classical mechanics ~r ≡ ~r1 − ~r2 R~ ≡ m1~r1 + m2~r2 m1 + m2 and reduce the problem to the CM moving like a free particle M = m1 + m2 H = − ̄h 2 2M ∇~ 2 R plus one potential problem in 3 dimensions with the usual reduced mass. 1 μ = 1 m1 + 1 m2 H = − ̄h 2 2μ ∇~ 2 r + V (~r) So we are now left with a 3D problem to solve (3 variables instead of 6). 28 1.17 Identical Particles Identical particles present us with another symmetry in nature. Electrons, for example, are indis- tinguishable from each other so we must have a symmetry of the Hamiltonian under interchange (See section 12.4) of any pair of electrons. Lets call the operator that interchanges electron-1 and electron-2 P12. [H, P12] = 0 So we can make our energy eigenstates also eigenstates of P12. Its easy to see (by operating on an eigenstate twice with P12), that the possible eigenvalues are ±1. It is a law of physics that spin 1 2 particles called fermions (like electrons) always are antisymmetric under interchange, while particles with integer spin called bosons (like photons) always are symmetric under interchange. Antisymmetry under interchange leads to the Pauli exclusion principle that no two electrons (for example) can be in the same state. 1.18 Some 3D Problems Separable in Cartesian Coordinates We begin our study of Quantum Mechanics in 3 dimensions with a few simple cases of problems that can be separated in Cartesian coordinates (See section 13). This is possible when the Hamiltonian can be written H = Hx + Hy + Hz. One nice example of separation of variable in Cartesian coordinates is the 3D harmonic oscillator V (r) = 1 2 mω2 r 2 which has energies which depend on three quantum numbers. Enxnynz = nx + ny + nz + 3 2 ̄hω It really behaves like 3 independent one dimensional harmonic oscillators. Another problem that separates is the particle in a 3D box. Again, energies depend on three quantum numbers Enxnynz = π 2 ̄h 2 2mL2 n 2 x + n 2 y + n 2 z for a cubic box of side L. We investigate the effect of the Pauli exclusion principle by filling our 3D box with identical fermions which must all be in different states. We can use this to model White Dwarfs or Neutron Stars. In classical physics, it takes three coordinates to give the location of a particle in 3D. In quantum mechanics, we are finding that it takes three quantum numbers to label and energy eigenstate (not including spin).
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