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Quantum Mechanics 3: the Quantum Mechanics of Many-Particle Systems

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Quantum Mechanics 3: the Quantum Mechanics of Many-Particle Systems Quantum Mechanics 3: the quantum mechanics of many-particle systems W.J.P. Beenakker Academic year 2017–2018 Contents of the common part of the course: 1) Occupation-number representation 2) Quantum statistics (up to 2.5.3) § Module 1 (high-energy physics): 2) Quantum statistics (rest) 3) Relativistic 1-particle quantum mechanics 4) Quantization of the electromagnetic field 5) Many-particle interpretation of the relativistic quantum mechanics The following books have been used: F. Schwabl, “Advanced Quantum Mechanics”, third edition (Springer, 2005); David J. Griffiths, “Introduction to Quantum Mechanics”, second edition (Prentice Hall, Pearson Education Ltd, 2005); Eugen Merzbacher, “Quantum Mechanics”, third edition (John Wiley & Sons, 2003); B.H. Bransden and C.J. Joachain, “Quantum Mechanics”, second edition (Prentice Hall, Pearson Education Ltd, 2000). Contents 1 Occupation-number representation 1 1.1 Summary on identical particles in quantum mechanics . .. 1 1.2 Occupation-numberrepresentation . .. 5 1.2.1 ConstructionofFockspace. 5 1.3 Switching to a continuous 1-particle representation . .... 12 1.3.1 Position and momentum representation . 14 1.4 Additive many-particle quantities and particle conservation . ..... 15 1.4.1 Additive 1-particle quantities . 16 1.4.2 Additive 2-particle quantities . 19 1.5 Heisenberg picture and second quantization . .. 20 1.6 Examples and applications: bosonic systems . 23 1.6.1 The linear harmonic oscillator as identical-particle system . 23 1.6.2 Forced oscillators: coherent states and quasi particles . .... 26 1.6.3 Superfluidity .............................. 31 1.6.4 Superfluidity for weakly repulsive spin-0 bosons (part 1) . 33 1.6.5 Intermezzo: the Bogolyubov transformation for bosons . .... 38 1.6.6 Superfluidity for weakly repulsive spin-0 bosons (part 2) . 40 1.6.7 The wonderful world of superfluid 4He: the two-fluid model . 42 1.7 Examples and applications: fermionic systems . 44 1.7.1 Fermiseaandholetheory ....................... 44 1.7.2 The Bogolyubov transformation for fermions . 46 2 Quantum statistics 49 2.1 Thedensityoperator(J.vonNeumann,1927) . ... 52 2.2 Example: polarization of a spin-1/2 ensemble . 55 2.3 Theequationofmotionforthedensity operator . ... 58 2.4 Quantum mechanical ensembles in thermal equilibrium . 59 2.4.1 Thermal equilibrium (thermodynamic postulate) . 61 2.4.2 Canonical ensembles (J.W. Gibbs, 1902) . 61 2.4.3 Microcanonicalensembles . 65 2.4.4 Grand canonical ensembles (J.W. Gibbs, 1902) . 66 2.4.5 Summary ................................ 69 2.5 FermigasesatT=0 .............................. 69 2.5.1 GroundstateofaFermigas . 74 2.5.2 Fermi gas with periodic boundary conditions . 76 2.5.3 Fermi-gas model for conduction electrons in a metal . 77 2.5.4 Fermi-gasmodelforheavynuclei . 80 2.5.5 Stars in the final stage of stellar evolution . 84 2.6 Systems consisting of non-interacting particles . .... 87 2.7 Idealgasesinthreedimensions ........................ 90 2.7.1 Classicalgases.............................. 92 2.7.2 Bose gases and Bose–Einstein condensation (1925) . 93 2.7.3 Fermi gases at T =0(noexammaterial). 98 6 2.7.4 Experimental realization of Bose–Einstein condensates . 101 3 Relativistic 1-particle quantum mechanics 103 3.1 First attempt: the Klein–Gordon equation (1926) . 104 3.2 Secondattempt: theDiracequation(1928) . 109 3.2.1 The probability interpretation of the Dirac equation . 111 3.2.2 Covariant formulation of the Dirac equation . 112 3.2.3 Dirac spinors and Poincar´etransformations . 113 3.2.4 Solutions to the Dirac equation . 116 3.2.5 ThesuccessstoryoftheDiractheory . 118 3.2.6 Problems with the 1-particle Dirac theory . 122 4 Quantization of the electromagnetic field 123 4.1 Classical electromagnetic fields: covariant formulation . ..... 123 4.1.1 Solutions to the free electromagnetic wave equation . 125 4.1.2 Electromagnetic waves in terms of harmonic oscillations . 127 4.2 Quantizationandphotons ........................... 129 4.2.1 Particle interpretation belonging to the quantized electromagnetic field130 4.2.2 Photon states and density of states for photons . 131 4.2.3 Zero-point energy and the Casimir effect . 132 4.2.4 Continuum limit . 133 4.3 A new perspective on particle –wave duality . 134 4.4 Interactions with non-relativistic quantum systems . .... 135 4.4.1 Classical limit . 138 4.4.2 Spontaneous photon emission (no exam material) . 139 4.4.3 The photon gas: quantum statistics forphotons . 141 5 Many-particle interpretation of the relativistic QM 144 5.1 Quantization of the Klein–Gordon field . 145 5.2 Quantization of the Dirac field . 149 5.2.1 Extra: the 1-(quasi)particle approximation for electrons . .... 153 A Fourier series and Fourier integrals i A.1 Fourierseries .................................. i A.2 Fourier integrals . ii A.2.1 Definition of the δ function . iii B Properties of the Pauli spin matrices iv C Lagrange-multiplier method v D Special relativity: conventions and definitions v E Electromagnetic orthonormality relations ix 1 Occupation-number representation In this chapter the quantum mechanics of identical-particle systems will be worked out in detail. The corresponding space of quantum states will be con- structed in the occupation-number representation by employing creation and annihilation operators. This will involve the introduction of the notion of quasi particles and the concept of second quantization. Similar material can be found in Schwabl (Ch.1,2 and 3) and Merzbacher (Ch.21,22 and the oscillator part of Ch.14). 1.1 Summary on identical particles in quantum mechanics Particles are called identical if they cannot be distinguished by means of specific intrinsic properties (such as spin, charge, mass, ). ··· This indistinguishability has important quantum mechanical implications in situations where the wave functions of the identical particles overlap, causing the particles to be ob- servable simultaneously in the same spatial region. Examples are the interaction region of a scattering experiment or a gas container. If the particles are effectively localized, such as metal ions in a solid piece of metal, then the identity of the particles will not play a role. In those situations the particles are effectively distinguishable by means of their spatial coordinates and their wave functions have a negligible overlap. For systems consisting of identical particles two additional constraints have to be imposed while setting up quantum mechanics (QM). Exchanging the particles of a system of identical particles should have no observ- • able effect, otherwise the particles would actually be distinguishable. This gives rise to the concept of permutation degeneracy, i.e. for such a system the expecta- tion value for an arbitrary many-particle observable should not change upon inter- changing the identical particles in the state function. As a consequence, quantum mechanical observables for identical-particle systems should be symmetric functions of the separate 1-particle observables. Due to the permutation degeneracy, it seems to be impossible to fix the quantum • state of an identical-particle system by means of a complete measurement. Nature has bypassed this quantum mechanical obstruction through the symmetrization postulate: identical-particle systems can be described by means of either totally symmetric state functions if the particles are bosons or totally antisymmetric state functions if the particles are fermions. In the non-relativistic QM it is an empirical fact that no mixed symmetry occurs in nature. 1 In this chapter the symmetrization postulate will be reformulated in an alternative way, giving rise to the conclusion that only totally symmetric/antisymmetric state functions will fit the bill. Totally symmetric state functions can be represented in the so-called q-representation by ψ (q , , q , t), where q , , q are the “coordinates” of the N separate identical par- S 1 ··· N 1 ··· N ticles. These coordinates are the eigenvalues belonging to a complete set of commuting 1-particle observablesq ˆ. Obviously there are many ways to choose these coordinates. A popular choice is for instance qj = (spatial coordinate ~rj , magnetic spin quantum number m σ , ), where the dots represent other possible internal (intrinsic) degrees of free- sj ≡ j ··· dom of particle j . For a symmetric state function we have ˆ P ψS(q1, , qN , t) = ψS (qP (1) , , qP (N) , t) = ψS(q1, , qN , t) , (1) P∀ˆ ··· ··· ··· with Pˆ a permutation operator that permutes the sets of coordinates of the identical particles according to q q , q q , , q q . (2) 1 → P (1) 2 → P (2) ··· N → P (N) Particles whose quantum states are described by totally symmetric state functions are called bosons. They have the following properties: - bosons have integer spin (see Ch. 4 and 5); - bosons obey so-called Bose–Einstein statistics (see Ch. 2); - bosons prefer to be in the same quantum state. Totally antisymmetric state functions can be represented in the q-representation by ψ (q , , q , t), with A 1 ··· N ˆ + ψA(q1, , qN , t) even P ˆ ··· P ψA(q1, , qN , t) = ψA(qP (1) , , qP (N) , t) = . (3) ∀ˆ ··· ··· P ψ (q , , q , t) odd Pˆ − A 1 ··· N A permutation Pˆ is called even/odd if it consists of an even/odd number of two-particle interchanges. Particles whose quantum states are described by totally antisymmetric state functions are called fermions. They have the following properties: - fermions have half integer spin (see Ch. 5); - fermions obey so-called Fermi–Dirac statistics (see Ch. 2); - fermions are not allowed to be in the same quantum state. 2 If the above (anti)symmetrization procedure results in spatial symmetrization, the particles have an increased probability to be in each other’s vicinity. Note that this can apply to identical fermions if they happen to be in an antisymmtric spin state. On the other hand, spatial antisymmetrization gives rise to a decreased probability for the particles to be in each other’s vicinity. Note that this can apply to identical spin-1 bosons if they happen to be in an antisymmtric spin state. Isolated non-interacting many-particle systems: many-particle systems with negi- gible interactions among the particles are called non-interacting many-particle systems.
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