What are you looking for Book " Advanced Quantum Mechanics "? It emphasizes the role of quantum theory for an understanding of materials and electromagnetic radiation. The book presents major advances in fundamentals of quantum physics from to the present.
No familiarity with relativistic quantum mechanics or quantum field theory is presupposed; however, the reader is assumed to be familiar with non-relativistic quantum mechanics, classical electrodynamics, and classical mechanics. The author's clear presentation focuses on key concepts, particularly experimental work in the field. Dick emphasizes the importance of advanced quantum mechanics for materials science and all experimental techniques which employ photon absorption, emission, or scattering.
Important aspects of introductory quantum mechanics are covered in the first seven chapters to make the subject self-contained and accessible for a wide audience. Advanced Quantum Mechanics, Materials and Photons can therefore be used for advanced undergraduate courses and introductory graduate courses which are targeted towards students with diverse academic backgrounds from the Natural Sciences or Engineering.
To enhance this inclusive aspect of making the subject as accessible as possible Appendices A and B also provide introductions to Lagrangian mechanics and the covariant formulation of electrodynamics. The expectation value of a Hermitian operator is real, that of an anti-Hermitian operator is purely imaginary. An important special case of 4.
In the Schwarz inequality 4. Substitution of 4. Substituting into 4. The solution of this equation is a Gaussian wave packet. At the initial time, this has a minimal uncertainty product. Only for the harmonic oscillator do the minimal wave packets coincide with the coherent states and remain minimal in the course of the time development of the system.
In addition to these, there are also uncertainty relations concerning energy and time, whose derivation cannot be given in this simple, formal manner. We would like to present a few variants of the energy—time uncertainty relation. Position and momentum are observ- ables to which Hermitian operators are assigned and which are measured at a particular time t. The time merely plays the role of a parameter. The energy—time uncertainty relation 4. Momentum transfer to a pointer Figure 4. The measuring apparatus has momentum zero before the measurement; the mo- mentum transferred to the apparatus mass M in the course of the measure- ment is 2p.
Quanti- tatively, this relation follows from time dependent perturbation theory. Let us denote the proportionality factor by b. Theorem 3. Then A and B commute. From Theorem 2 and Theorem 3 it follows that observables represented by commuting operators can simultaneously possess precise values, such that measurements of these quantities give unique results.
Examples of commuting operators are x1 , x2 , x3 or p1 , p2 , p3 or x1 , p2 , p3 , but not x1 , p1. A complete set of eigenfunctions of the operator A is called a basis of A.
Complete set of operators or equivalently complete set of observables. The set of Hermitian operators A, B,. These eigenfunc- tions can then be characterized by the corresponding eigenvalues a, b,. Remark: If for a given set of operators the eigenstates are still degenerate, there exists an additional symmetry of these operators and the generator of this symmetry operation also commutes with this set of operators. If O is a function of the operators A, B. Theorem 5. An operator O that commutes with a complete set of operators is a function of these operators.
Since the operator function o A, B,. We return again to the fact that commuting operators have common eigenfunctions. In such an eigenstate they simultaneously have precise, unique values, i. We will analyze this situation in more detail in our discussion of the Stern—Gerlach experiment as an example of a measurement Chap.
A subsequent measurement of B will change the state of the system, because after an ideal measurement whose accuracy allows the unique determination of an eigenvalue of B, the state of the system will change to the corresponding eigenstate of B. For the results of a further measurement of A, only probabilistic statements can be made, which are determined by the expansion of the eigenstate of B thus obtained in terms of eigenfunctions of A.
Only if A and B commute can sharp values be assigned to both of them simultaneously and measurements of the corresponding observables do not disturb interfere with each other. One expresses this also by saying that the two observables are compatible. The Uncertainty Relation Problems 4.
Angular Momentum For application to centrally symmetric potentials, we would now like to in- vestigate properties of the angular momentum, which is also decisive in such problems in classical mechanics. The similarity of the commuta- tors 5.
The angular momentum L is the generator of rotations. We would like to illustrate this geometrically. Expanding 5. Setting this equal to 5. With this, 5. The generators of symmetry operations are summarized in Table 8. However, since L2 is a scalar, then according to 5.
We would now like to determine the eigenvalues of the common eigenfunctions of L2 and Lz. With 5. From 5. Further information can be obtained from the normalization using 5. Because of the intimate connection with rotations it is advantageous to transform to spher- ical polar coordinates Fig.
Illustration concerning the Fig. Ylm is even for even l and odd for odd l. The components Lx and Ly are not diagonal in the states Ylm. The functions Yll 2 i. Linear combinations of the states Ylm are also important. For example, one refers to Fig. Angular Momentum Problems 5. See also Appendix C.
The Central Potential I In this chapter, we will consider motion in central potentials. We then determine the bound states for the most important case of an attractive Coulomb potential. Finally, we transform the two-body problem into a one-body problem with a potential, so that our treatment of motion in a Coulomb potential also covers the nonrelativistic hydrogen atom.
Because of the rotational symmetry of the potential, one transforms to spherical coordinates. Since by 5. In ii , the boundary conditions of p. Since H as a scalar is invariant under rotations, i.
As in the present situation in which the conservation of angular momentum L follows from the rota- tional symmetry of the Hamiltonian, a continuous symmetry always leads to a corresponding conservation law. In any case, we now would like to continue the investigation of 6. We now must determine boundary and normalization conditions for u r.
The Central Potential I Fig. We know from Chap. However, an odd bound state exists only if the potential reaches a minimum strength.
Therefore, V r must exceed a minimum strength, in order that in three dimensions a bound state exist. The centrifugal term is increasingly repulsive with increasing l. The Central Potential I where because of the normalization condition only the exponentially decreas- ing solution is relevant for u r , i.
Equation 6. In order that 6. The energy eigenval- ues of the bound states of the Coulomb potential result from 6. The Central Potential I Table 6. Table 6. For this purpose, we multiply 6. In order to see this, and for com- pleteness, we insert a short overview of the most important for our purposes properties of the Laguerre polynomials.
Let us summarize: With 6. The radial position probability is obtained by angular integration. The functions Rnl r and the radial probability densities r2 Rnl 2 are shown graphically in Fig. This is shown for hydrogen in Fig. The energy level diagram for the Coulomb po- tential Table 6. To what degree does this depend on the special features of the physical situation investigated here? In this case, H, L2 , and Lz are simultaneously diagonalizable, i. The degree of degeneracy of the energy eigenvalues was determined ear- lier as n2 ; if one additionally takes into account the double occupancy of each state by two electrons of opposite spin, this becomes 2 n2.
The position expectation values and uncertainties in the eigenstates of the Coulomb Hamiltonian are also of interest.
In contrast, according to 6. In addition, we calculated in 6. The Central Potential I orbital angular momentum L are circular. With the help of the uncertainty relation, one can estimate the ground state energy without further knowledge of what the state looks like. In our treatment of the Coulomb potential in the above sections, we have occasionally referred to the hydrogen atom. These include the relativis- tic correction to the electron mass, the Darwin term and the spin—orbit coupling Chap.
Time dependence: circular Keplerian orbits. We now discuss the classical limit mentioned on p. Considering our experience with the classical limit of the harmonic oscillator at the end of Sect. We restrict consideration to circular orbits. In classical mechanics, circular orbits have the maximal allowed angular momentum corresponding to a given energy.
Moreover, we know from 5. In particular, we obtain for the energy eigen- values 6. From 6. We can summarize the result as follows. This result is of interest for Rydberg states of the hydrogen atom. One can apply the result to planetary motion and compute, for example, n0 for the earth or for a satellite orbiting the earth and estimate the aforementioned uncertainties.
We have restricted consideration here to circular orbits; the calculation of quantum mechanical wave packets in elliptical orbits turns out to be more complicated. Nauenberg: Phys Rev. A 40, ; J. Gay, D. Delande, A. Bommier: Phys. A 39, The Central Potential I 6. The Central Potential I Problems 6. Note concerning a—e : Consider the recursion formula proven in the preceding exercise.
You then need only compute one expectation value directly. Problems 6. Be sure to take the nuclear potential into account, and use the Poisson equation. Hint: Separate variables with respect to the coordinate x and the coordinates par- allel to the surface. Motion in an Electromagnetic Field 7. By the correspondence principle Sect.
With 7. However, there do exist situations in which the diamagnetic and paramagnetic terms can be of com- parable magnitude. See also the end of Sect. Because of 7. This will lead us in Chap. In Chap. The gauge transfor- mation introduces an additional space and time dependent phase factor into the wave function.
Wave packets constructed from such stationary states with and without A are related by a combined gauge and Galilei transformation Problem To this end, we now consider the interference experiment shown in Fig.
Similarly, for the wave function when only slit 2 is open, we have Fig. The Aharonov—Bohm interference experiment. If both slits are open, we superimpose 7. Remark: If we consider in detail the cylindrical waves leaving the slits in 7. Aharonov, D. Bohm: Phys. A related phenomenon Fig. Summarizing: Classically, E and B are the physically relevant quantities, since they determine the Lorentz force. Chambers: Phys. Hamisch, K. Grohmann, D. Wohlleben: Z. Zimmerman, J. Mercereau: Phys. Jaklevic, J.
Lambe, J. Mercereau, E. Silver: Phys. The electrons form Cooper pairs. A closed path about the cylinder starting at the point x0 Fig. Motion in an Electromagnetic Field This quantization has also been observed experimentally7. The vector potential 7. Doll, M. Deaver, Jr. Fairbank: Phys. Consequently, the energy eigenvalues of 7. These play an important role in solid state physics. The problem is not yet completely solved, since for example we have not yet determined the degeneracy and the wave function of our particles.
In Sect. Problems 7. Compare the result with Sect. Problems d Find the wave functions and the energy eigenvalues for A. Show that the latter in general no longer obey periodic boundary conditions.
Operators, Matrices, State Vectors 8. We now list a few properties of the matrix Anm. Summarizing: Operators can be represented by matrices and states by vectors. To illustrate this, we give three examples: 1. Energy eigenfunctions of the harmonic oscillator.
In this basis, the position operator x for example takes the form see 3. Operators, Matrices, State Vectors 2. Momentum eigenfunctions.
Position eigenfunctions. Remark: As always, the index on the wave functions indicates the eigenvalue as well as the operator to which it belongs. Note the similarity between 8.
As emphasized in Sect. This is a consequence of the form of the position and momentum operators and the general results for the probability densities of observables in Sect. This can be illustrated in the case of three- dimensional vectors v in the space IR3 generally IRn. Instead of characterizing a vector v by its components vi with respect to a particular coordinate system, it is often more convenient to use the coordinate indepen- dent vector notation v.
Dennery, A. The observables are represented by Hermitian operators A, with func- tions of observables being represented by the corresponding functions of the operators. If in a measurement of A the value an is found, then the state of the system changes to the corresponding eigenstate n. See Sect. See also Remark i , Sect. Arbitrary states are obtained by superposition of the position eigenstates 8. N2 be elements of 1 2 an N2 -dimensional vector space, then their direct product vi vj spans an N1 N2 - dimensional space.
The generalization to a time dependent H will be treated in Chap. The state vectors depend on time, while the operators corresponding to physical observables are independent of time, aside from explicit time dependence. In particular, operators such as x, p, L, etc. Conservation laws As in classical mechanics, if appropriate symmetries are present, then con- servation laws hold for the Hamiltonian, the angular momentum, and the momentum.
In Table 8. Operators, Matrices, State Vectors Table 8. The state vectors evolve due to the perturbative part of the Hamiltonian, the operators due to the free part H0. We omit the index H. This led us in Sect. Problems Problems 8. What is the commu- tation relation for the corresponding operators in the Heisenberg representation?
Determine the wave function for energy E in the momentum representation. The transformation to coordinate space yields the integral repre- sentation of the Airy functions. Spin 9. Moreover, in contrast to 9. The experiment was carried out by O. Stern and W. Gerlach in with silver atoms. Silver has a spherically symmetric charge distribution plus one 5s-electron. Thus, the total angular momentum of silver is zero, i. The experiment gives a splitting into two beams.
For this value of the angular momentum, there are only two orientations of the angular momentum vector. The gyromagnetic ratio is twice as large for the rotation about its own axis as for the orbital motion.
We will return to the magnetic mo- ment in Sect. The other elementary particles also have spin. Fermions possess half- integral spin, bosons integral spin including zero see Sect. Pauli: Z. Compton: J. Franklin Inst. Uhlenbeck, S. Goudsmit: Naturwiss. Now, if e is a unit vector, then according to Sect.
It follows from 9. Equation 9. Adding 9. From 9. The relations 9. See, e. It arises from the internal charge distribution, which can be understood in light of the fact that the fundamental constituents of hadrons are quarks. The spins of the proton and the neutron are parallel in the deuteron, i. Spin and position or momentum can assume precise values simul- taneously and independently of one another, i. Spin As in Sect. Addition of Angular Momenta We postpone the general problem until Sect.
It is reasonable to suppose that the total spin S assumes the values 1 and 0. By Thus, we have found all four eigenstates of S 2 and Sz. The states We seek eigenstates of J 2 , L2 , S 2 , Jz. For the following, recall 5.
With In order to determine the remaining states, we remark that, beginning with And of course the states Equation Consequently, the states We now wish to determine the values of j for given j1 and j2. This is presented in Table Table These values of m are thus already used up. Evidently, one obtains a half- integral angular momentum upon adding an integral and a half-integral an- gular momentum.
The two limiting values in Determine the energy eigenvalues. Approximation Methods for Stationary States Although we have succeeded in solving some important and interesting quan- tum mechanical problems, an exact solution is not possible in complicated situations, and we must then resort to approximation methods.
For the calcu- lation of stationary states and energy eigenvalues, these include perturbation theory, the variational method , and the WKB approximation. However, in many cases it is an asymptotic expansion1 , i. The bound states of a potential cannot be obtained from the continuum states by means of perturbation theory. To this end, we multiply Approximation Methods for Stationary States In general, one must solve an eigenvalue problem, but often the correct states can be guessed.
From Here, an additional remark may be of use. In any nontrivial problem, H0 and H1 do not commute. These two operators cannot therefore be diago- nalized simultaneously. In this procedure, an error in the wave function manifests itself at quadratic order in the energy; i. Aside from its function as an approximation method, the variational prin- ciple is also an important tool in mathematical physics in the proof of exact inequalities.
One can then express For the energy E, let the classical turning points be b and a Fig. The two solutions must be com- bined in such a way that only the decreasing part remains.
According to our original supposition, In the transition from Using Remark: Using Thus, the two conditions are compatible.
If En is known — e. Problems Include second- order perturbation theory with respect to the contribution ax3. Note the degeneracy which occurs. Derive from According to our estimate in Sect. We will discuss each of the terms in turn by explaining their physical origin heuristically and by taking their precise form from the theory of the Dirac equation. In order to understand The discrepancy occurs because the rest frame of the electron is not an inertial frame.
Jackson: Classical Electrodynamics, 2nd edn. Wiley, New York Instead, the operators H, J 2 , Jz , L2 , S 2 now form a complete set, and the eigenfunctions can be characterized by their eigenvalues. Formally identical formulae were given by Sommerfeld on the basis of the old quantum theory with only the p4 -term. Figure Relativistic Corrections Fig. These could be calculated directly using the Laguerre polynomials, but the calculation is tedious.
Here, we instead make use of algebraic meth- ods3. Specializing to the Coulomb potential, one obtains from Becker, F. The observed shift is The complete quantum electrodynamic theory of radiative corrections gives Lamb, Jr. Retherford: Phys. Itzykson, J. It remains to evaluate the expectation value with respect to the spin degrees of freedom.
The cm radiation is very important in astronomy. From its intensity, Doppler broadening, and Doppler shift, one obtains information concerning the density, temperature, and motion of interstellar and intergalactic hydro- gen clouds. The latter is represented by the last two terms of Eq. The Hamiltonian Bethe, E. Further reading: The relativistic Dirac equation, which is outside the scope of this book, is treated in: J. Bjorken, S. Discuss the degeneracy of the energy levels without and with the spin—orbit interaction.
Several-Electron Atoms Remark: Every element P can be represented as the product of transpositions Pij. An element P is called even odd if there are an even odd number of the Pij.
For any symmetrical operator S 1,. The question then arises whether all these states are realized in nature. On aesthetic grounds, one might suspect that the completely symmetric states and the completely antisymmetric states occupy a privileged position.
Quite generally Eq. Fermions have half-integral spin, whereas bosons have integral spin. The requirement that a permutation of identical particles must not have any observable consequences implies that observables O are symmetric permutation invariant. The funda- mental particles at this level are see also Table This is the Pauli exclusion principle 4.
Remark: The completely symmetric and the completely antisymmetric N -particle states form the basis of two one-dimensional representations of the permutation group SN.
This is seen from Since the Pij do not all commute with each other for more than two particles, there are also wave functions for which not all the Pij are diagonal. These are basis functions of higher-dimensional representations of the permutation group. These states do not occur in nature. Messiah, O. Greenberg: Phys. We now construct states of these two types.
The antisymmetry of No state may be multiply occupied Pauli exclusion principle. The ground state is obtained by putting the N fermions one after another into the low- est available states. Composite particles Example — the H-atom: An H-atom consists of two fermions: a proton p and an electron e. Hence, the H-atom is a boson. In general, if the number of fermions in a composite particle is odd, then it is a fermion, otherwise a boson; e.
Several-Electron Atoms Table Energy values of the He atom for various n1 and n2 In the bound, excited states, one of the electrons has the principal quantum number 1. This service is more advanced with JavaScript available. Advanced Quantum Mechanics. Authors view affiliations Franz Schwabl. Essential for advanced undergraduates and graduate students All mathematical steps are given Numerous applications Numerous exercises Includes supplementary material: sn.
Front Matter Pages Second Quantization. Pages Correlation Functions, Scattering, and Response. Relativistic Wave Equations and their Derivation.
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