By Luca Salasnich

The e-book offers an advent to the sphere quantization (second quantization) of sunshine and subject with functions to atomic physics. the 1st bankruptcy in brief experiences the origins of unique relativity and quantum mechanics and the fundamental notions of quantum details idea and quantum statistical mechanics. the second one bankruptcy is dedicated to the second one quantization of the electromagnetic box, whereas the 3rd bankruptcy exhibits the implications of the sunshine box quantization within the description of electromagnetic transitions. within the fourth bankruptcy it's analyzed the spin of the electron, and particularly its derivation from the Dirac equation, whereas the 5th bankruptcy investigates the results of exterior electrical and magnetic fields at the atomic spectra (Stark and Zeeman effects). The 6th bankruptcy describes the homes of platforms composed by means of many interacting exact debris through introducing the Hartree-Fock variational process, the density useful conception and the Born-Oppenheimer approximation. ultimately, within the 7th bankruptcy it's defined the second one quantization of the non-relativistic topic box, i.e. the Schrodinger box, which supplies a robust software for the research of many-body difficulties and in addition atomic quantum optics. on the finish of every bankruptcy there are a number of solved difficulties which could aid the scholars to place into perform the issues they learned.

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N ks + 1 | . . n ks + 1 . . ⊗. 53). 65) where 0 F is the zero of the Fock space (usually indicated with 0), and |ks⊗ is clearly the state of one photon with wavevector k and polarization s, such that eik·r ↑r|ks⊗ = ∇ εks . 66) From Eqs. e. 68) Nˆ ks | . . n ks . . ⊗ = n ks | . . n ks . . ⊗. 19). The differences are that Nˆ ks is a quantum number operator and that the energy E vac of the the vacuum state |0⊗ is not zero but is instead given by E vac = k s 1 βk . 70) A quantum harmonic oscillator of frequency βk has a finite minimal energy βk , which is called zero-point energy.

Z= {n ks } ↑ . . n ks . . |e−δ = βk Nˆ ks ks | . . n ks . . ⊗ {n ks } e−δ = βk n ks ks {n ks } e−δ = ks n ks ks 1 1 − e−δ = βk n ks e−δ = {n ks } ks ∈ = e−δ βk n ks βk n ks n=0 βk . 110) Quantum statistical mechanics dictates that the thermal average of any operator Aˆ is obtained as ˆ T = 1 T r [ Aˆ e−δ( Hˆ −μ Nˆ ) ]. 111) Z In our case the calculations are simplified because μ = 0. Let us suppose that Aˆ = Hˆ , it is then quite easy to show that 1 ν ν ˆ ˆ ln T r [e−δ H ] = − ln(Z). 112) By using Eq.

57) meaning that there are n ks photons with wavevector k and polarization s, n k∀ s ∀ photons with wavevector k∀ and polarization s ∀ , n k∀∀ s ∀∀ photons with wavevector k∀∀ and polarization s ∀∀ , et cetera. 59) is the Hilbert space of n identical photons, which is n times the tensor product ∓ of the single-photon Hilbert space H = H∓1 . Thus, F is the infinite direct sum ∞ of increasing n-photon Hilbert states Hn , and we can formally write ∈ F= H∓n . 60) n=0 Notice that in the definition of the Fock space F one must include the space H0 = H∓0 , which is the Hilbert space of 0 photons, containing only the vacuum state |0⊗ = | .