- (b) One can see on the above expression that ¯ n is a rapidly increasing func- tion of the temperature. - hω c (or T ∼ 7.1 K for this experiment), the mean excitation number is of the order of (e − 1. - Below this temperature, the occupation of the level n l = 0 becomes predominant, as can be seen on the curves of Fig. - temperatures of 1.6 K, 2 K, 3 K and 3.9 K, respectively.. - (c) In order to measure a temperature with such a device, one must use a statistical sample which is significantly populated in the level n l = 1. - Increase significantly the total time of measurement in order to detect occupation probabilities of the level n r = 1 significantly less than 10 − 2. - Reduce the value of the magnetic field B, in order to reduce the cyclotron frequency ω c , and to increase (for a given temperature) the occupation probability of the level n l = 1. - Exact Results for the Three-Body Problem. - The three-body problem is a famous question of mechanics. - The purpose of this chapter is to derive some rigorous results for the three- body problem in quantum mechanics. - Here we are interested in obtaining rigorous lower bounds on three-body ground state energies. - 18.1 The Two-Body Problem. - Write the Hamiltonian ˆ H of the system. - (18.1) where M = 2m is the total mass of the system. - Give the value of the reduced mass µ in terms of m.. - Give the ex- pression for E (2) (µ) in the two cases V (r. - 186 18 Exact Results for the Three-Body Problem. - 18.2 The Variational Method. - Show that n | H ˆ | n = E n. - Consider an arbitrary vector | ψ of the Hilbert space of the system.. - Show that the previous result remains valid if ˆ H is the Hamiltonian of a two-body subsystem and | ψ a three-body state. - In order to do so, one can denote by ˆ H 12 the Hamiltonian of the (1, 2) subsystem in the three-body system of wave function ψ( r 1 , r 2 , r 3. - 18.3 Relating the Three-Body and Two-Body Sectors. - p 3 − p 1 ) 2 and show that the three-body Hamiltonian ˆ H (3) can be written as. - where ˆ P = ˆ p 1 + ˆ p 2 + ˆ p 3 is the total three-body momentum, and where the relative Hamiltonian ˆ H rel (3) is a sum of two-particle Hamiltonians of the type defined in (18.1),. - with a new value µ of the reduced mass. - Do the two-body Hamiltonians ˆ H ij commute in general? What would be the result if they did?. - Show that the three-body ground state energy is related to the ground state energy of each two-body subsystem by the inequality:. - 18.5 From Mesons to Baryons in the Quark Model 187 18.3.4. - Which lower bounds on the three-body ground-state energy E (3) does one obtain in the two cases V (r. - In the first case, the exact result, which can be obtained numerically, is E (3). - 18.4 The Three-Body Harmonic Oscillator. - The three-body problem can be solved exactly in the case of harmonic inter- actions V (r. - Rewrite the three-body Hamiltonian in terms of these variables for a harmonic two-body interaction V (r. - Derive the three-body ground state energy from the result. - Show that the inequality (18.3) is saturated, i.e.. - the bound (18.3) coincides with the exact result in that case.. - Do you think that the bound (18.3), which is valid for any potential, can be improved without further specifying the potential?. - 18.5 From Mesons to Baryons in the Quark Model. - 188 18 Exact Results for the Three-Body Problem V qq (r. - and the two-body Hamil- tonians ˆ H (2) (µ) and ˆ H (2. - A striking characteristic of the level spacings in quark–antiquark sys- tems is that these spacings are approximately independent of the nature of the quarks under consideration, therefore independent of the quark masses.. - Why does this justify the form of the above potential V q¯ q (r. - and express the constant a in terms of the coupling constant g.. - 18.6 Solutions. - Section 18.1: The Two-Body Problem 18.1.1. - The two-body Hamiltonian is. - where M = 2m and µ = m/2 are respectively the total mass and the reduced mass of the system.. - 18.6 Solutions 189. - Section 18.2: The Variational Method 18.2.1. - n } is a basis of the Hilbert space. - If ˆ H = ˆ H 12 , for fixed r 3 , ψ( r 1 , r 2 , r 3 ) can be considered as a non- normalized two-body wave function. - Section 18.3: Relating the Three-Body and Two-Body Sectors 18.3.1. - 2µ + ˆ V (r ij ) (18.4) with a reduced mass µ = 3m/4.. - If they did, the three-body energies would just be the sum of two-body energies as calculated with a reduced mass µ = 3m/4, and the solution of the three-body problem would be simple.. - 190 18 Exact Results for the Three-Body Problem 18.3.3. - However, owing to the results of questions 2.2 and 2.3, we have Ω | H ˆ ij | Ω ≥ E (2) (µ. - In the harmonic case, we obtain:. - Section 18.4: The Three-Body Harmonic Oscillator. - The three Hamiltonians ˆ H 1 , ˆ H 2 , and ˆ H cm commute. - 18.6 Solutions 191. - Section 18.5: From Mesons to Baryons in the Quark Model 18.5.1. - In a logarithmic potential, the level spacing is independent of the mass. - This is a remarkable feature of the observed spectra, at least for heavy quarks, and justifies the investigation of the logarithmic potential. - 192 18 Exact Results for the Three-Body Problem. - The generalization of such inequalities can be found in the literature quoted below
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