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Cold atoms Lecture 3. 17. October 2007 BEC for interacting particles Description of the interaction Mean field approximation: GP equation Variational properties of the GP equation Are the interactions important? In the dilute gaseous atomic clouds in the traps, the interactions are incomparably weaker than in liquid helium. That permits to develop a perturbative treatment and to study in a controlled manner many particle phenomena difficult to attack in HeII. Several roles of the interactions • the atomic collisions take care of thermalization • the mean field component of the interactions determines most of the deviations from the non-interacting case • beyond the mean field, the interactions change the quasi-particles and result into superfluidity even in these dilute systems 3 Fortunate properties of the interactions 1. Strange thing: the cloud lives for seconds, or even minutes at temperatures, at which the atoms should form a crystalline cluster. Why? For binding of two atoms, a third one is necessary to carry away the released binding energy and momentum. Such ternary collisions are very unlikely in the rare cloud, however. 2. The interactions are elastic and spin independent: they do not spoil the separation of the hyperfine atomic species and preserve thus the identity of the atoms. 3. At the very low energies in question, the effective interaction is typically weak and repulsive … which enhances the formation and stabilization of the condensate. 4 Fortunate properties of the interactions 1. Strange thing: the cloud lives for seconds, or even minutes at temperatures, at which the atoms should form a crystalline cluster. Why? For binding of two atoms, a third one is necessary to carry away the released binding energy and momentum. Such ternary collisions are very unlikely in the rare cloud, however. 2. The interactions are elastic and spin independent: they do not spoil the separation of the hyperfine atomic species and preserve thus the identity of the atoms. 3. At the very low energies in question, the effective interaction is typically weak and repulsive … which enhances the formation and stabilization of the condensate. 5 Interatomic interactions For neutral atoms, the pairwise interaction has two parts • van der Waals force 1 6 r • strong repulsion at shorter distances due to the Pauli principle for electrons Popular model is the 6-12 potential: 12 6 U TRUE (r ) 4 r r Example: Ar =1.6 10-22 J =0.34 nm corresponds to ~12 K!! Many bound states, too. 6 Interatomic interactions minimum 1 atoms, theradius pairwise6 vdW 2 interaction For neutral has two parts • van der Waals force 1 6 r • strong repulsion at shorter distances due to the Pauli principle for electrons Popular model is the 6-12 potential: 12 6 U TRUE (r ) 4 r r Example: Ar =1.6 10-22 J =0.34 nm corresponds to ~12 K!! Many bound states, too. 7 Interatomic interactions The repulsive part of the potential – not well known The attractive part of the potential can be measured with precision U TRUE (r ) repulsive part - C6 r6 Even this permits to define a characteristic length "local kinetic energy" "local potential energy" 2 1 2m 2 6 C6 66 6 2mC6 2 1/ 4 8 Interatomic interactions The repulsive part of the potential – not well known The attractive part of the potential can be measured with precision U TRUE (r ) repulsive part - C6 r6 Even this permits to define a characteristic length "local kinetic energy" "local potential energy" 2 1 2m 2 6 C6 66 6 2mC6 2 1/ 4 For the 6 - 12 potential C6 4 6 6 4 2m 2 2 1/ 4 9 Interatomic interactions The repulsive part of the potential – not well known The attractive part of the potential can be measured with precision U TRUE (r ) repulsive part - C6 r6 Even this permits to define a characteristic length "local kinetic energy" "local potential energy" 2 1 2m 2 6 C6 66 6 2mC6 2 1/ 4 rough estimate of the last bound state energy kBTC collision energy of the condensate atoms 10 Scattering length, pseudopotential Beyond the potential radius, say propagates in free space 3 , the scattered wave For small energies, the scattering is purely isotropic , the s-wave scattering. The outside wave is sin(kr 0 ) r For very small energies the radial part becomes just r as , as ... the scattering length This may be extrapolated also into the interaction sphere (we are not interested in the short range details) Equivalent potential ("pseudopotential") U (r ) g (r ) 4 as g m 2 11 Experimental data as 12 Experimental data 1 a.u. = 1 bohr 0.053 nm nm as 3.4 4.7 6.8 8.7 8.7 10.4 nm -1.4 4.1 -1.7 -1.9 5.6 127.2 13 Experimental data 1 a.u. = 1 bohr 0.053 nm nm as 3.4 4.7 6.8 8.7 8.7 10.4 nm -1.4 4.1 -1.7 -19.5 5.6 127.2 NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion "well behaved; monotonous increase seemingly erratic, very interesting physics of scattering resonances behind 14 Mean-field treatment of interacting atoms Many-body Hamiltonian and the Hartree approximation 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b We start from the mean field approximation. This is an educated way, similar to (almost identical with) the HARTREE APPROXIMATION we know for many electron systems. Most of the interactions is absorbed into the mean field and what remains are explicit quantum correlation corrections 1 2 ˆ H GP pa V (ra ) VH (ra ) a 2m VH (ra ) drbU (ra rb ) n(rb ) g n(ra ) n(r ) n r 2 ADDITIONAL NOTES self-consistent system 1 2 p V ( r ) V ( r ) H r E r 2m 16 Many-body Hamiltonian and the Hartree approximation 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b We start from the mean field approximation. This is an educated way, similar to (almost identical with) the HARTREE APPROXIMATION we know for many electron systems. Most of the interactions is absorbed into the mean field and what remains are explicit quantum correlation corrections ELECTRONS 1 2 ˆ H Hartree pa V (ra ) VH ( ra ) a 2m e '2 VH (ra ) drbU (ra rb )n(rb ), U ( r ) r n( r ) E r 2 1 2 p V ( r ) V ( r ) H r E r 2m 17 Hartree approximation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting This is a single self-consistent equation for a single orbital, theory (r ) ever. N 0 (r ) the simplest HF like we obtain a closed equation for the order parameter: 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. 18 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. 19 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: The lowest level coincides with the chemical potential 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. 20 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: The lowest level coincides with the chemical potential 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. 21 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: The lowest level coincides with the chemical potential 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. 22 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: The lowest level coincides with the chemical potential 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. • has the form of a simple non-linear Schrödinger equation • concerns a macroscopic quantity • suitable for numerical solution. 23 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then ( r ) N 0 ( r ) n( r ) N [ n] N d 3 r ( r ) d 3 r n( r ) N 2 The Gross-Pitaevskii equation 2 1 2 p V (r ) g r r r 2m becomes 2 2m n( r ) n( r ) Bohm's quantum potential V ( r ) g n( r ) the effective mean-field potential 24 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then ( r ) N 0 ( r ) n( r ) N [ n] N d 3 r ( r ) d 3 r n( r ) N 2 The Gross-Pitaevskii equation 2 1 2 p V (r ) g r r r 2m becomes 2 2m n( r ) n( r ) Bohm's quantum potential V ( r ) g n( r ) the effective mean-field potential 25 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then ( r ) N 0 ( r ) n( r ) N [ n] N d 3 r ( r ) d 3 r n( r ) N 2 The Gross-Pitaevskii equation 2 1 2 p V (r ) g r r r 2m becomes 2 2m n( r ) n( r ) Bohm's quantum potential V ( r ) g n( r ) the effective mean-field potential 26 Gross-Pitaevskii equation – variational interpretation This equation results from a variational treatment of the Energy Functional 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition ADDITIONAL NOTES N [ n] N that is E[ n ] N [ n ] 0 which is the GP equation written for the particle density (previous slide). 27 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 which is the GP equation written for the particle density (previous slide). From there E[ n ] N [ n] 28 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 which is the GP equation written for the particle density (previous slide). From there chemical potential E[ n ] N [ n] by definition 29 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 which is the GP equation written for the particle density (previous slide). From there chemical potential E E[ n ] !!! N N [ n] by definition 30 Interacting atoms in a constant potential The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V 2 2m n( r ) n( r ) V (r ) V const. V ( r ) g n( r ) g n V 32 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V 2 2m n( r ) n( r ) V (r ) V const. V ( r ) g n( r ) g n V 33 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V 2 2m n( r ) n( r ) V (r ) V const. V ( r ) g n( r ) g n V The repulsive interaction increases the chemical potential The repulsive interaction increases the chemical potential If g 0, it would be n 0 and the gas would be thermodynamically unstable. 34 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V 2 n( r ) 2m n( r ) V (r ) V const. V ( r ) g n( r ) g n V The repulsive interaction increases the chemical potential total energy E 12 gn 2 V internal pressure E 1 2 P 2 gn V 35 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V 2 n( r ) 2m n( r ) V (r ) V const. V ( r ) g n( r ) g n V The repulsive interaction increases the chemical potential total energy E 12 gn 2 V internal pressure E 1 2 P 2 gn V 2 N E 12 V 36 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V 2 n( r ) 2m n( r ) V (r ) V const. V ( r ) g n( r ) g n V The repulsive interaction increases the chemical potential total energy E 12 gn 2 V If g 0, it would be internal pressure E 1 2 P 2 gn V 2 N E 12 V n 0 and the gas would be thermodynamically unstable. 37 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V 2 n( r ) 2m n( r ) V (r ) V const. V ( r ) g n( r ) g n V The repulsive interaction increases the chemical potential total energy E 12 gn 2 V If g 0, it would be internal pressure E 1 2 P 2 gn V 2 N E 12 V n 0 and the gas would be thermodynamically unstable. 38 Interacting atoms in a parabolic trap Reminescence: The trap potential and the ground state 400 nK level number 200 nK 87 Rb a0 1 m =10 nK x / a0 x 0 ( x, y, z ) 0 x x 0 y y 0 z z 0 (u ) 1 a0 e u2 2 a 02 , N ~ 106 at. 2 2 1 1 1 a0 , E0 2 m 2 2 ma0 2 Mum a02 u 1 1 2 2 V (u ) m u 2 2 a0 2 • characteristic energy • characteristic length40 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) 2 2 1 2 2 m r g r r r 2 2m Where is the particle number N? ( … a little reminder) 3 3 d r ( r ) d r n(r ) N 2 41 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) 2 2 1 2 2 m r g r r r 2 2m Where is the particle number N? ( … a little reminder) 3 3 d r ( r ) d r n(r ) N 2 Dimensionless GP equation for the trap r r a0 energy = energy 2 2 2 4 as 1 2 2 2 m a0 r r a0 r a0 r a0 2 2 m 2ma0 r a0 r N a03 d r (r ) 3 2 1 2 8 as N 2 r r r r a0 42 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) 2 2 1 2 2 m r g r r r 2 2m Where is the particle number N? ( … a little reminder) 3 3 d r ( r ) d r n(r ) N 2 Dimensionless GP equation for the trap r r a0 energy = energy 2 2 2 4 as 1 2 2 2 m a0 r r a0 r a0 r a0 2 2 m 2ma0 2 2 N r a0 r 3 d3r (r ) 1 a single dimensonless a0 parameter 2 8 as N 2 r r r r a0 43 Importance of the interaction – synopsis Without interaction, the condensate would occupy the ground state of the oscillator (dashed - - - - -) In fact, there is a significant broadening of the condensate of 80 000 sodium atoms in the experiment by Hau et al. (1998), perfectly reproduced by the solution of the GP equation 44 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction compete, the condensate shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is EINT EKIN gNn N N 2 as a03 Na02 Nas a0 45 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction compete, the condensate shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is E INT E KIN gNn N N 2 as a03 Na02 Nas a0 4 46 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction compete, the condensate shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes can vary in a wide range Quantitative The decisive parameter for the "importance" of interactions is E INT E KIN gNn N collective effect weak or strong depending on N N 2 as a03 Na02 Nas a0 4 1 weak individual collisions 47 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction compete, the condensate shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes can vary in a wide range Quantitative The decisive parameter for the "importance" of interactions is E INT E KIN gNn N N 2 as a03 Na02 Nas a0 4 0 realistic value collective effect weak or strong depending on N 1 weak individual collisions 48 The end ADDITIONAL NOTES On the way to the mean-field Hamiltonian 50 ADDITIONAL NOTES On the way to the mean-field Hamiltonian First, the following exact transformations are performed ˆ ˆ W V 1 1 2 ˆ H pa V (ra ) 2 a 2m a Uˆ U (ra rb ) a b Vˆ V (ra ) d 3 r V (r ) (r ra ) d 3r V (r ) nˆ (r ) a 1 ˆ U 2 a U (ra rb ) a b 1 2 3 3 d r d r ' U r r ' particle density operator r ra r ' rb a b TRICK!! d 3 r d 3 r 'U r r ' r ra r ' rb r r ' 2 a b SI eliminates nˆ (r' ) nˆ (r ) 1 (self-interaction) 1 Hˆ Wˆ d3 r V (r ) nˆ (r ) d 3 r d 3 r 'U r r ' nˆ (r ) nˆ (r' ) r r ' 2 51 ADDITIONAL NOTES On the way to the mean-field Hamiltonian Second, a specific many-body state is chosen, which defines the mean field: n(r ) nˆ (r ) nˆ (r ) Then, the operator of the (quantum) density fluctuation is defined: nˆ r n r nˆ r nˆ r nˆ r' nˆ r n r' n r nˆ r' nˆ r nˆ r' n r n r' The Hamiltonian, still exactly, becomes Hˆ Wˆ d 3 r V (r ) d 3 r' U r r ' n(r' ) nˆ (r ) 1 2 1 2 3 3 d r d r 'U r r ' n( r ) n( r' ) 3 3 d r d r 'U r r ' nˆ (r ) nˆ ( r' ) nˆ ( r ) r r ' 52 ADDITIONAL NOTES On the way to the mean-field Hamiltonian In the last step, the third line containing exchange, correlation and the self-interaction correction is neglected. The mean-field Hamiltonian of the main lecture results: Hˆ Wˆ d 3 r V (r ) d 3r' U r r ' n(r' ) nˆ (r ) 1 d r d r 'U r r ' n( r ) n( r' ) 2 1 2 3 3 VH r substitute back nˆ (r ) (r ra ) a and integrate 3 3 d r d r 'U r r ' nˆ (r ) nˆ (r' ) nˆ (r ) r r ' REMARKS • Second line … an additive constant compensation for doublecounting of the Hartree interaction energy • In the original (variational) Hartree approximation, the self-interaction is not left out, leading to non-orthogonal Hartree orbitals BACK 53 ADDITIONAL NOTES Variational approach to the condensate ground state 54 ADDITIONAL NOTES Variational estimate of the condensate properties VARIATIONAL PRINCIPLE OF QUANTUM MECHANICS The ground state and energy are uniquely defined by E Hˆ ' Hˆ ' for all ' H NS , ' ' 1 In words, ' is a normalized symmetrical wave function of N particles. The minimum condition in the variational form is Hˆ 0 equivalent with the SR Hˆ E HARTREE VARIATIONAL ANSATZ FOR THE CONDENSATE WAVE F. For our many-particle Hamiltonian, 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ), U (r ) g (r ) a b the true ground state is approximated by the condensate for non-interacting particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock) r1 , r2 , , rp , , rN 0 r1 0 r2 0 rp 0 rN 55 ADDITIONAL NOTES Variational estimate of the condensate properties Here, 0 is a normalized real spinless orbital. It is a functional variable to be found from the variational condition E 0 ] 0 ] Hˆ 0 ] 0 with 0 ] 0 ] 1 0 0 1 Explicit calculation yields E 0 ] 2 2m N d 3 r 0 r N d 3 r V r 0 r 2 2 4 1 N N 1 g d 3r 0 r 2 Variation of energy with the use of a Lagrange multiplier: N 1E 0 ] 0 0 0 0 r , 0 0 r BY PARTS 2 2 3 4 3 3 3 d r 2 d r V r N 1 g d r 0 0 0 0 0 0 2m 2 This results into the GP equation derived here in the variational way: 2 1 2 p V (r ) N -1 g 0 r 0 r 0 r 2m eliminates self-interaction BACK 56 ADDITIONAL NOTES Variational estimate of the condensate properties ANNEX Interpretation of the Lagrange multiplier The idea is to identify it with the chemical potential. First, we modify the notation to express the particle number dependence 1 2 1 E N ] N p V N 1 g d3r 4 2 2m 2 1 2 EN E N 0 N ], p V ( r ) N - 1 g 0 N r 0 r N 0 N r 2m The first result is that is not the average energy per particle: 1 2 1 p 0 N 0 N V 0 N N 1 g d 3r 0 N 4 2m 2 1 2 N 0 N p 0 N 0 N V 0 N N 1 g d3r 0 N 4 2m EN / N E N 0 N ]/ N 0 N from the GPE 57 ADDITIONAL NOTES Variational estimate of the condensate properties Compare now systems with N and N -1 particles: EN E N 0 N ] E N 1 0 N ] N E N 1 0, N 1 ] N EN 1 N N … energy to remove a particle use of the variational principle for GPE without relaxation of the condensate In the "thermodynamic"asymptotics of large N, the inequality tends to equality. This only makes sense, and can be proved, for g > 0. BACK Reminescent of the Koopmans’ theorem in the HF theory of atoms. Derivation: E N ] N V 2 N N 1 g d3 r 4 N 1 2m p2 E N 1 ] N 1 1 2m p 2 N 1 V 2 N 1 N 2 g d 3r 4 E N E N 1 1 2m 1 1 p 2 V 2 N N 1 N 1 N 2 g d 3r 4 1 N for 0 N 58 ADDITIONAL NOTES Variational estimate of the condensate properties SCALING ANSATZ FOR A SPHERICAL PARABOLIC TRAP The potential energy has the form V r 12 m02 r 2 12 m02 x 2 y 2 z 2 Without interactions, the GPE reduces to the SE for isotropic oscillator 1 2 1 2 2 3 r p m r r 0 0 0 0 2 2 2m The solution (for the ground state orbital) is 1 r2 2 2 a0 3 00 r A0 e , a0 m0 , 0 2 A0 ma02 1/ 4 2 a0 We (have used and) will need two integrals: I1 du e u2 2 , I 2 du e u2 2 u 2 12 3 59 ADDITIONAL NOTES Variational estimate of the condensate properties SCALING ANSATZ The condensate orbital will be taken in the form 1 r2 2 2b 3 0 r A e , A b 2 1/ 4 It is just like the ground state orbital for the isotropic oscillator, but with a rescaled size. This is reminescent of the well-known scaling for the ground state of the helium atom. Next, the total energy is calculated for this orbital E 0 ] 2 2m N d 3 r 0 r N d 3 r V r 0 r The solution (for) 2 is 1 6 a0 0 NA 4 d3 r e 2 b 2 r2 2 b r2 4 1 N N 1 g d 3r 0 r 2 2r 2 2 2 ma r 2 N 1 A6 20 g d3 r e b 2 1 3 d re 2 a0 r2 2 b 60 ADDITIONAL NOTES Variational estimate of the condensate properties For an explicit evaluation, we (have used and) will use the identities: 2 m E 0 ] 0 a02 , m02 0 4 2 as 1 1 2 , A , g I1 b b m a0 2 The integrals, by the Fubini theorem, are a product of three: 2 I b 3I 2 b 1 0 N 2 2 b I b 1 a02 4 1 2 N 1 a0 2b b ma02 1 3 3/ 2 2 4 2 m as I1 b / 2 I1 b 3 3 3 a02 b 2 N 1 as a03 E 0 ] 0 N 2 2 3 0 N E a0 2 a0 b 4 b 3 1 1 b dimension-less dimension-less E 2 2 3 orbital size energy per particle 4 a0 Finally, This expression is plotted in the figures in the main lecture. BACK 61 The end