On a translationally invariant nuclear single particle picture
aa r X i v : . [ nu c l - t h ] J un On a translationally invariant nuclear single particle picture
Walter
Gl¨ockle , ∗ ) Hiroyuki
Kamada , ∗∗ ) and Jacek Golak , Institut f¨ur Theoretische Physik II, Ruhr-Universit¨at Bochum, D-4 4780 Bochum,Germany Department of Physics, Faculty of Engineering, Kyushu Institute of Technology,Kitakyushu 804-8550, Japan M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30059 Krak´ow,Poland
If one assumes a translationally invariant motion of the nucleons relative to the c. m.position in single particle mean fields a correlated single particle picture of the nuclear wavefunction emerges. A single particle product ansatz leads for that Hamiltonian to nonlinearequations for the single particle wave functions. In contrast to a standard not translationallyinvariant shell model picture those single particle s-, p- etc states are coupled. The strengthof the resulting coupling is an open question. The Schr¨odinger equation for that Hamiltoniancan be solved by few- and many -body techniques, which will allow to check the validity ornon-validity of a single particle product ansatz.Realistic nuclear wave functions exhibit repulsive 2-body short range correlations. There-fore a translationally invariant single particle picture – if useful at all – can only be expectedbeyond those ranges. Since exact A = 3 and 4 nucleon ground state wave functions and be-yond based on modern nuclear forces are available, the translationally invariant shell modelpicture can be optimized by an adjustment to the exact wave function and its validity ornon-validity decided. §
1. Introduction
The shell model for the nucleus has a long tradition. However, in its standardform expressed in single particle variables it is plagued by violating translationalinvariance. Various methods have been suggested to remedy this situation, like forinstance the generator coordinate method.
Clearly, if the shell model isrealistic at all, the motion of the individual nucleons in a mean field happens in atranslationally invariant manner, namely as a function of ~u i ≡ ~x i − ~X , where ~x i arethe individual coordinates of particle i and ~X is the c. m. coordinate. However, thisset of coordinate vectors ~u i obeys the obvious condition P Ai =1 ~u i = 0, correlating themotion of all particles.It is the aim of the present investigation to work out the consequences of choosingthe coordinates ~u i for a shell model picture.In Section II we provide some formal basis for this specific choice of coordinates.We restrict ourselves in this first investigation to systems of three and four nucleons.Furthermore, our most simplistic ansatz for the wave function shifts the antisym-metry requirement to the spin-isospin space, which leads to a symmetric space part ∗ ) E-mail: [email protected] ∗∗ ) E-mail: [email protected] typeset using
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TEX.cls h Ver.0.9 i W. Gl¨ockle, H. Kamada and J. Golak under particle permutations. Then the very first ansatz for the space part is Φ ( ~u i ) = A Y i =1 R ( u i ) , (1.1)with A = 3 or 4. Here only s-wave states are assumed. We denote such a form acorrelated single particle picture.The nonlinear equations for 3 and 4 particles for the state R ( u ) assuming a sumof single particle potentials, V = P Ai =1 V ( u i ), are presented in Section III.In the case of the harmonic oscillator potential the nonlinear equations can besolved analytically and it is shown that the ansatz (1.1) is indeed the correct one.Obviously the question arises whether the ansatz (1.1) for the wave functionis at all valid for general single particle mean field potentials. To that effect theHamiltonian, including the sum of single particle potentials, can be expressed instandard Jacobi variables. This is displayed in Section IV.The resulting Sch¨rodinger equations for 3 and 4 particles (in this case bosonsfor the space part) can be solved exactly in the form of the Faddeev-Yakubovskyequations, which will be formulated in that Section. Having the exact wave functionat ones disposal, one can then investigate how well the above shell model ansatz (1.1)is realized, whether contributions beyond s-wave are needed, or whether that hopeis not realistic at all. An optimization algorithm relating the shell model ansatz tothe exact wave function is presented in Section V.The numerical investigations for solving the nonlinear equations for R ( u ), forsolving the Faddeev-Yakubovsky equations for the shell-model Hamiltonian and forthe optimal extraction of R ( u ) from the exact wave functions is left to forthcominginvestigations.The main task however remains. The realistic nuclear wave function is deter-mined by two- and three-nucleon forces. First estimates in the case of the α -particleindicate that even small contributions from proper 4N forces are needed. Basedon these forces numerically exact wave functions are nowadays routinely generatedfor three and four nucleons. The question however arises, howwell these wave functions for pair distances larger than a certain distance r can berepresented in the form of a correlated shell model ansatz like in (1.1), or whetherhigher partial waves and more complicated symmetries with respect to space-, spin-and isospin parts of the wave function are required. Clearly for pair distances smallerthan r short range repulsive features are present in the realistic wave functions whichcan not be represented in the shell model form. On the other hand it is known thatthe short pair distance behavior is essentially universal for light nuclei asidefrom proper normalization, which might allow an overall description: short rangedepletion and correlated shell model feature at larger distances. The value r isexpected to be somewhat smaller than 1 fm. Section VI provides some suggestionon how an optimal extraction of a correlated single particle picture can be obtainedfrom realistic three- and four-nucleon wave functions. We summarize in Section VII. n a translationally invariant nuclear single particle picture §
2. The Formal Basis
The translationally invariant single particle coordinates for n particles are de-fined as ~u i ≡ ~x i − n n X j =1 ~x j = n − n ~x i − n n X j = i ~x j (2.1)Since P ni =1 ~u i = 0, the mapping from the n ~x i to the n ~u i can not be inverted andwe choose the new variables as the first (n-1) ~u j together with the c.m. coordinate ~X ~X = 1 n n X j =1 ~x j . (2.2)It is a straightforward exercise to express the kinetic energy in terms of the newvariables T = − m n X k =1 ∇ x k = − m ( n − n n − X i =1 ∇ u i − n X i = j ~ ∇ u i · ~ ∇ u j ) − mn ∇ X (2.3)Clearly, the first part refers to the relative motion, the second part to the c. m.motion. While the choice of Jacobi coordinates avoids mixed terms in the kineticenergy, here they are unavoidable.Let us now restrict ourselves to three and four particles. If one chooses a Slaterdeterminant with equal space-dependent single particle wave functions, the symmet-ric part of the form (1.1) factors out and one is left with a totally antisymmetricspin-isospin part. For a proton-proton-neutron ( ppn ) system this has the form χ = | ( t = 0 12 ) T = 12 > | ( s = 1 12 ) S = 12 > − | ( t = 1 12 ) T = 12 > | ( s = 0 12 ) S = 12 >, (2.4)where the two-body spin or isospin state is coupled with the spin or isospin of thethird particle to total spin S = or total isospin T = . This state together with asymmetric space part is known as the principal S-state for realistic He wavefunctions and contributes with about 90% to the norm. This result by itself clearlyindicates that this choice of the one Slater determinant can not exhaust the full wavefunction but at least a very large portion of it.For the ppnn system the totally antisymmetric spin-isospin part of the wavefunction has the form χ = (1 − P − P ) | ( 12 12 )0( 12 12 )0 S = 0 > (1 + P P ) | ( 12 12 )11 > | ( 12 12 )1 − > , (2.5) W. Gl¨ockle, H. Kamada and J. Golak where P ij is a transposition of particles i and j . That state has total spin S = 0and total isospin T = 0. Again in relation to the norm of a realistic α -particle wavefunction it accounts for about 90%. Now we provide some formal properties, whose verification is left to the reader.The Heisenberg commutation relations[ w kα , u jβ ] = δ kj i δ αβ (2.6)are obeyed, where w kα ≡ ∂T∂ ˙ u kα are components of the conjugate momenta.The relative orbital angular momentum has the form ~L rel = ~u × i ~ ∇ u + ~u × i ~ ∇ u , (2.7)which justifies that standard Clebsch-Gordon coupling in the variables ~u and ~u can be used.Using (2.3) for n = 3, the translationally invariant shell model Hamiltonian isgiven by H = − m ( ∇ u + ∇ u − ~ ∇ u · ~ ∇ u ) + V ( u ) + V ( u ) + V ( u ) , (2.8)where u = | ~u + ~u | . Obviously, a separation of variables is not possible. However,the symmetry of the kinetic energy under particle exchanges is valid: ∇ u + ∇ u − ~ ∇ u · ~ ∇ u = ∇ u + ∇ u − ~ ∇ u · ~ ∇ u = ∇ u + ∇ u − ~ ∇ u · ~ ∇ u . (2.9)In the case of four particles the translationally invariant shell model Hamiltonianis given as H = − m ( ∇ u + ∇ u + ∇ u −
23 ( ~ ∇ u · ~ ∇ u + ~ ∇ u · ~ ∇ u + ~ ∇ u · ~ ∇ u ))+ V ( u ) + V ( u ) + V ( u ) + V ( u ) , (2.10)with u = | ~u + ~u + ~u | . All the formal relations corresponding to (2.6),( 2.7), and(2.9) are valid for four particles as is expected. §
3. Nonlinear Equations for the Translationally Invariant Shell ModelStates
For three particles the simplest ansatz for a symmetrical space part wave functionis Φ ( u , u , u ) = R ( u ) R ( u ) R ( u ) , (3.1)with u = | ~u + ~u | .It is straightforward, though tedious, to evaluate the action of the kinetic energyin (2.8) onto (3.1). If we put R ( u ) = r ( u ) u , the Schr¨odinger equation based on H n a translationally invariant nuclear single particle picture − m [ r ′′ ( u ) r ( u ) r ( u ) + r ( u ) r ′′ ( u ) r ( u ) + r ( u ) r ( u ) r ′′ ( u ) − ( r ′ ( u ) − r ( u ) u ) r ( u )( r ′ ( u ) − r ( u ) u )ˆ u · ˆ u − r ( u )( r ′ ( u ) − r ( u ) u )( r ′ ( u ) − r ( u ) u )ˆ u · ˆ u − ( r ′ ( u ) − r ( u ) u )( r ′ ( u ) − r ( u ) u ) r ( u )ˆ u · ˆ u ]+ ( V ( u ) + V ( u ) + V ( u ) − E ) r ( u ) r ( u r ( u ) = 0 , (3.2)where ˆ u · ˆ u = − u + ˆ u · ~u u ˆ u · ˆ u = − u + ˆ u · ~u u (3.3)Here the independent variables are u , u and x = ˆ u · ˆ u .We can not exclude that higher partial waves should be included. The simplestansatz for a p-wave admixture is given by Φ ( ~u , ~u , ~u )= R ( u ) R ( u ) R ( u )ˆ u · ˆ u + R ( u ) R ( u ) R ( u )ˆ u · ˆ u + R ( u ) R ( u ) R ( u )ˆ u · ˆ u , (3.4)where for the sake of simplicity we assumed that the ‘third’ state, which is notinvolved in the p-wave admixture, remains unchanged. We leave it to the reader toderive the resulting equation.For four particles the most simple ansatz is Φ ( u , u , u , u ) = R ( u ) R ( u ) R ( u ) R ( u ) , (3.5)with | u = | ~u + ~u + ~u | . The resulting equation based on H is − m [ r ′′ ( u ) r ( u ) r ( u ) r ( u )+ r ( u ) r ( u ) r ( u ) r ′′ ( u ) + r ( u ) r ′′ ( u ) r ( u ) r ( u )+ r ( u ) r ( u ) r ( u ) r ′′ ( u ) + r ( u ) r ( u ) r ′′ ( u ) r ( u )+ r ( u ) r ( u ) r ( u ) r ′′ ( u ) − r ′ ( u ) − r ( u ) u ) r ( u ) r ( u )( r ′ ( u ) − r ( u ) u )ˆ u · ˆ u − r ( u )( r ′ ( u ) − r ( u ) u ) r ( u )( r ′ ( u ) − r ( u ) u )ˆ u · ˆ u − r ( u ) r ( u )( r ′ ( u ) − r ( u ) u )( r ′ ( u ) − r ( u ) u )ˆ u · ˆ u ] − m [3 r ( u ) r ( u ) r ( u ) r ′′ ( u ) W. Gl¨ockle, H. Kamada and J. Golak + ( r ′ ( u ) − r ( u ) u )( r ′ ( u ) − r ( u ) u ) r ( u ) r ( u )ˆ u · ˆ u − (( r ( u )( r ′ ( u ) − r ( u ) u ) r ( u )( r ′ ( u ) − r ( u ) u )+ r ( u ) r ( u )( r ′ ( u ) − r ( u ) u )( r ′ ( u ) − r ( u ) u )+ r ( u )( r ′ ( u ) − r ( u ) u ) r ( u )( r ′ ( u ) − r ( u ) u )))ˆ u · ˆ u − ((( r ′ ( u ) − r ( u ) u ) r ( u ) r ( u )( r ′ ( u ) − r ( u ) u )+ ( r ′ ( u ) − r ( u ) u ) r ( u ) r ( u )( r ′ ( u ) − r ( u ) u )))ˆ u · ˆ u + ( r ′ ( u ) − r ( u ) u ) r ( u )( r ′ ( u ) − r ( u ) u ) r ( u )ˆ u · ˆ u − r ( u ) r ( u )( r ′ ( u ) − r ( u ) u )( r ′ ( u ) − r ( u ) u )ˆ u · ˆ u + r ( u )( r ′ ( u ) − r ( u ) u )( r ′ ( u ) − r ( u ) u ) r ( u )ˆ u · ˆ u + ( V ( u ) + V ( u ) + V ( u ) + V ( u )) r ( u ) r ( u ) r ( u ) r ( u )= Er ( u ) r ( u ) r ( u ) r ( u ) . (3.6)Again, extensions to the ansatz (3.5) are obvious. Already the presence of theexplicit angular dependence in (3.2) and (3.6) suggest that higher orbital angularmomentum admixtures are likely and that the most simple ansatz for the groundstate may be poor.Choosing the mean field potential V ( u i ) to be a harmonic oscillator, V ( u i ) = mω u i , the nonlinear equations, (3.2) and (3.6), can be solved analytically. As exam-ple we consider four particles and introduce standard Jacobi coordinates ~x = ~x − ~x ~y = ~x −
12 ( ~x + ~x ) ~z = ~x −
13 ( ~x + ~x + ~x ) ~X = 14 ( ~x + ~x + ~x + ~x ) . (3.7)Then the potential energy V = mω ( x + y + z ) as well as the kinetic energy T rel = − m (2 ∇ x + 32 ∇ y + 43 ∇ z ) (3.8)allow for a separation of the variables with the result Φ ( x, y, z ) = e − mω ( x + y + z ) = Y i =1 R ( u i ) . (3.9)where R ( u ) = e − mω u . (3.10) n a translationally invariant nuclear single particle picture E = ω . The corresponding result for threeparticles, now for E = 3 ω , is Φ ( x, y ) = Y i =1 R ( u i ) , (3.11)with the same function R ( u ). It is straightforward to verify that (3.9) and (3.11)fulfill the nonlinear equations (3.6) and (3.2). §
4. The Faddeev-Yakubovsky Equations for Three and Four Particles
The shell model Hamiltonians H and H , Eqs. (2.8) and (2.10), can be rewrittenin terms of standard Jacobi coordinates. This allows one to solve the two Schr¨odingerequations exactly in the form of the Faddeev-Yakubovsky equations and thereforeto test the quality of shell model ansatz. For three particles one defines the Jacobicoordinates as ~x = ~u − ~u = 2 ~u + ~u ~y = ~u −
12 ( ~u + ~u ) = 32 ~u , (4.1)or ~u = 23 ~y~u = 12 ~x − ~y. (4.2)This gives for the Hamiltonian H = − m ∇ x − m ∇ y + V ( 23 y ) + V ( | ~x − ~y | ) + V ( | ~x + 13 ~y | ) (4.3)The above expression has a formal similarity to a three-body Hamiltonian composedof two-body forces: H , b = − m ∇ x − m ∇ y + V b ( x ) + V ( | ~x + ~y | ) + V ( | ~x − ~y | ) . (4.4)However, Eqs. (4.2) and (4.4) are different. Nevertheless the formal structure of theFaddeev equation can be used. The three-body bound state obeys Ψ = G X i =1 V i Ψ ≡ X i =1 ψ i , (4.5)where G represents the free three-body propagator, and V i ≡ V ( u i ). Then onearrives in a standard manner at ψ i = G T i X j = i ψ j , (4.6) W. Gl¨ockle, H. Kamada and J. Golak where T i obeys the Lippmann Schwinger equation T i = V i + V i G T i (4.7)Because of the identity of the particles one arrives at the well known form for thetotal state Ψ = (1 + P ) ψ , (4.8)with P ≡ P P + P P , which is a sum of a cyclical and an anticyclical permu-tation of three particles. One Faddeev equation is sufficient, namely ψ = G T P ψ (4.9)The Faddeev equation can be solved in configuration space as an integro-differentialequation or, what we prefer, in momentum space as an integral equation. In the lat-ter case one needs the momentum space representation of the shell model potentialas well as of the Lippmann Schwinger equation in terms of the conjugate momenta ~p x and ~p y of the Jacobi momenta ~x and ~y .With standard (unit) normalizations it results in < ~x~y | ~u ~u > = ( 13 ) δ ( ~u − ~y ) δ ( ~u − ~x + 13 ~y ) (4.10)Furthermore, as consequence of the locality assumption < ~u ′ ~u ′ | V ( u ) | ~u ~u > = δ ( ~u − ~u ′ ) δ ( ~u − ~u ′ ) V ( u ) (4.11)and using (4.10) one obtains < ~p ′ x ~p ′ y | V ( u ) | ~p x ~p y > = δ ( ~p x − ~p ′ x ) 1(2 π ) Z d ye i ( ~p y − ~p ′ y ) · ~y V ( 23 y ) . (4.12)Due to that structure the T-matrix element in (4.7) must have the form < ~p ′ x ~p ′ y | T | ~p x ~p y > = δ ( ~p ′ x − ~p x ) t ( ~p ′ y , ~p y , z = E − p x m ) , (4.13)where t obeys t ( ~p ′ y , ~p y , z ) = V ( ~p ′ y , ~p y ) + Z d p ′′ y V ( ~p ′ y , ~p ′′ y ) 1 E − p x m − m p ′′ y t ( ~p ′′ y , ~p y , z ) . (4.14)For two-body forces the δ -function in (4.13) would have been for the spectatormomentum ~p y . We assume that the mean field forces are spin-independent andrequire symmetry in the spatial part.In such a system has been shown to be easily solvable using directly momen-tum vectors and thus avoiding any partial wave decomposition. We follow the sameapproach. Then (4.9), using (4.13) has the form < ~p x ~p y | ψ > = 1 E − p x m − m p y Z d p y ′ t ( ~p y , ~p ′ y , z = E − p x m ) n a translationally invariant nuclear single particle picture Z d p x ′′ d p y ′′ < ~p x ~p y ′ | P | ~p x ′′ ~p y ′′ >< ~p x ′′ ~p y ′′ | ψ > . (4.15)The permutation matrix element is well known and is given as < ~p x ~p y ′ | P | ~p x ′′ ~p y ′′ > = ( 83 ) ( δ ( ~p y ′ + 23 ~p x + 43 ~p x ′′ ) δ ( ~p y ′′ − ~p x − ~p x ′′ )+ δ ( ~p y ′ − ~p x − ~p x ′′ ) δ ( ~p y ′′ + 43 ~p x + 23 ~p x ′′ )) (4.16)Therefore, Eq. (4.15) turns into < ~p x ~p y | ψ > = 1 E − p x m − m p y ( 83 ) Z d p x ′′ ( t ( ~p y , − ~p x − ~p x ′′ , z = E − p x m ) < ~p x ′′ , ~p x + 23 ~p x ′′ | ψ > + t ( ~p y , ~p x + 43 ~p x ′′ , z = E − p x m ) < ~p x ′′ , − ~p x − ~p x ′′ | ψ > . (4.17)Because of the uniqueness of the solution, any solution of (4.17) has the property < − ~p x , ~p y | ψ > = < ~p x , ~p y | ψ > . This equation is can then be solved by iterationusing a Lanczos type algorithm. As follows from (4.16) the total state given by (4.8) has the form < ~p x , ~p y | Ψ > = < ~p x , ~p y | ψ > +( 34 ) < − ~p x − ~p y , ~p x + 12 ~p y | ψ > + ( 34 ) < − ~p x + 34 ~p y , − ~p x + 12 ~p y | ψ > . (4.18)In the case of four particles we use the Yakubovsky equations. For four bosonsand two-body forces this has been solved rigorously the first time in.
Now we havedifferent potentials depending on the relative coordinates ~u i , which require a renewedderivation. Starting from Ψ = G X i =1 V ( u i ) Ψ ≡ X i =1 ψ i (4.19)one arrives in a standard first step at ψ = G T ( ψ + ψ + ψ ) , (4.20)where T obeys the Lippmann Schwinger equation (4.14). (Note however, the mod-ified free four-body propagator.)In the spirit of the Yakubovsky scheme one regards a three- body subsystem bydefining ψ ≡ G T ( ψ + ψ ) (4.21)and a remaining component ψ , ≡ G T ψ . (4.22)0 W. Gl¨ockle, H. Kamada and J. Golak
Then ψ = ψ + ψ , . (4.23)Correspondingly one defines ψ = G T ( ψ + ψ ) ψ = G T ( ψ + ψ ) ψ , = G T ψ ψ , = G T ψ (4.24)where ψ = ψ + ψ , ψ = ψ + ψ , . (4.25)Then (4.21) and ( 4.25) yield ψ = G T ( ψ + ψ + ψ , + ψ , ) . (4.26)Due to the identity of the particles one has ψ + ψ = P ψ ψ , = P ψ , , (4.27)and (4.26) can be rewritten as(1 − G T P ) ψ = G T (1 + P ) ψ , . (4.28)The left hand side by itself defines a three-body problem. After inversion one obtains ψ = G ˆ T (1 + P ) ψ , , (4.29)where ˆ T obeys ˆ T = T + T P G ˆ T . (4.30)It remains to consider (4.22), which in analogy to (4.25) has the form ψ , = G T ( ψ + ψ , ) . (4.31)Using now ψ , = P ψ , (4.32)we rewrite (4.31) as (1 − G T P ) ψ , = G T ψ . (4.33)Inversion yields ψ , = G ˜ T ψ , (4.34) n a translationally invariant nuclear single particle picture T obeys ˜ T = T + T P G ˜ T (4.35)Finally permutation symmetry yields ψ = P P ψ (4.36)and one ends up with two coupled equations ψ = G ˆ T (1 + P ) ψ , ψ , = G ˜ T P P ψ . (4.37)The total wave function is now given as Ψ = ψ + ψ + ψ + ψ = (1 + P ) ψ + P P ( ψ + ψ , ) + (1 + P + P ) ψ , . (4.38)While a corresponding coupled set based on two- body forces has been rigorouslysolved in a partial wave representation, it is also possible to directly use momentumvectors as has been demonstrated in. We would propose to follow that second option. We leave it to the reader towork out the explicit momentum space representation of (4.37) and (4.38) in termsof appropriate Jacobi momentum vectors. §
5. Shell Model Ansatz versus Exact Wave Function
The solution of the Faddeev equation (4.17) yields the full three- dimensionalthree-boson Faddeev component in momentum space. This is the input for the fullwave function given in (4.18). Since we search for the lowest energy state, Ψ is ascalar and therefore depends only on 3 variables < ~p x , ~p y | Ψ > → Ψ ( p x , p y , ˆ p x · ˆ p y ) (5.1)As a consequence, the dependence of the configuration space wave function < ~x~y | Ψ > will also reduce to a three-variable dependence Ψ ( x, y, ˆ x · ˆ y ): < ~x~y | Ψ > = 1(2 π ) Z d p x d p y e i ( ~p x · ~x + ~p y · ~y ) Ψ ( p x , p y , ˆ p x · ˆ p y )= 1(2 π ) Z d p x d p y cos ( ~p x · ~x + ~p y · ~y ) Ψ ( p x , p y , ˆ p x · ˆ p y ) ≡ Ψ ( x, y, ˆ x · ˆ y ) . (5.2)We used the reality property of Ψ to replace the exponential by the cosine.The expectation is now that Ψ SM ( ~x, ~y ) ≡ R ( u ) R ( u ) R ( u ) , (5.3)with u = | ~u + ~u | being a good approximation to Ψ ( x, y, ˆ x · ˆ y ). In the case of theharmonic oscillator this is exactly fulfilled.2 W. Gl¨ockle, H. Kamada and J. Golak
In general one faces the task to minimize | Ψ ( x, y, ˆ x · ˆ y ) − R ( u ) R ( u ) R ( u ) | for all x, y, ˆ x · ˆ y or u , u , ˆ u · ˆ u . Explicitly this requirement is | Ψ ( x, y, ˆ x · ˆ y ) − R ( y ) R ( | ~x − ~y | ) R ( | ~x + ~y | ) | or | Ψ ( | ~u + 2 ~u | , u , u +2 ~u · ˆ u | ~u +2 ~u | − R ( u ) R ( u ) R ( | ~u + ~u | ) to beminimal.Instead of an optimized pointwise adjustment one can try an average adjustmentminimizing Z du du d ˆ u · ˆ u ( Ψ ( | ~u + 2 ~u | , u , u + 2 ~u · ˆ u | ~u + 2 ~u | ) − R ( u ) R ( u ) R ( | ~u + ~u | )) (5.4)in relation to the choice of R ( u ). For instance, one can expand R ( u ) into harmonicoscillator wave functions Φ m ( u ), where mω is optimally adjusted to the given meanfield potential V ( u ).Thus R ( u ) = X m φ m ( u ) C m , (5.5)and the set C m is to be varied minimizing the above integral. Differentiating withrespect to C k and putting the result to zero yields a nonlinear relation for the coef-ficients C m . This might be solved by an iterative procedure allowing first C = 0.Then keeping also C = 0 in addition one might start with C from the previousstep and determine C . Finally one can iterate the nonlinear equation for C and C starting with the values found before; etc. Very likely, however, one has to allowin addition for p-wave admixtures as given in (3.4) and possibly even higher orbitalangular momentum values.The direct solution of the nonlinear equation (3.2) poses a severe problem. More-over, very likely p-wave and possibly higher order admixtures have to be taken intoaccount, which requires an extension of the nonlinear equation (3.2) as mentionedabove. Discretization in the u , u , ˆ x · ˆ y - values is necessary and iterative proceduresappear unavoidable. Thereby each run is of course an eigenvalue problem for theenergy E .In the case of four nucleons the symmetric state of lowest energy is again a scalarand thus depends on 5 variables: Ψ = Ψ ( x, y, z, ˆ x · ˆ y, ˆ x · ˆ z, ˆ y · ˆ z ) , (5.6)where ~x, ~y, ~z are one choice of standard Jacobi coordinates. The optimal extractionof R ( u ) in Ψ SM ( ~x, ~y, ~z ) ≡ R ( u ) R ( u ) R ( u ) R ( u ) (5.7)and possibly higher angular momentum admixture follows analogous strategies asfor three nucleons. §
6. Realistic Three- and Four-Nucleon Wave Functions
Based on modern nuclear forces like combined with three-nucleon (3N)forces of the Tucson-Melbourne type or based on the most recent consistent two- n a translationally invariant nuclear single particle picture numericallyexact solutions of the Faddeev - Yakubovsky equations are available. If a correlatedsingle particle picture applies at all it can only be valid beyond a certain value r of the pair distances. The two-body correlation function to find two nucleons at adistance r has its maximum around r = 1 fm universally for all light nuclei. Thus r has to be smaller than 1 fm. For the most simple correlated shell modelansatz of Eq. (3.1) or symmetric extensions beyond s-wave and (3.5) the exact wavefunction for He and He is to be projected onto the totally antisymmetric spin-isospin states χ and χ , Eqs. (2.4) and (2.5), respectively: Ψ exact , ≡ < χ , | Ψ exact , > . (6.1)For a global adjustment one has to minimize Z dV ( Ψ exact , − , Y i =1 R ( u i )) , Y i 7. Summary In nature a nuclear wave function is translationally invariant. Therefore, if ashell model picture is a good representation of a nuclear wave function, the singleparticle states have to depend on translationally invariant coordinates. Our choiceof coordinates ~u i ≡ ~x i − ~X relating the individual position vectors ~x i to the c. m.coordinate ~X fulfills this condition with the additional constraint that they have tosum up to zero: P ni =1 ~u i = 0. Choosing the first n − W. Gl¨ockle, H. Kamada and J. Golak c. m. coordinate one can formulate a shell model Hamiltonian composed of kineticenergy containing now also mixed terms ~ ∇ u i · ~ ∇ u j and single particle potentialsdepending on the coordinates | u i | . Assuming the energetically lowest energy stateto that Hamiltonian to be a Slater determinant with equal space dependent singleparticle wave functions, R ( u i ), which is the most simple choice, one obtains nonlinearequations for R ( u i ). They have been worked out for nucleon numbers A = 3 and 4.For the special choice of harmonic oscillator potentials the nonlinear equations canbe analytically solved and that most simple ansatz for the wave function turns outto be correct. In the case of general mean field potentials partial wave contributionsbeyond s-states might be necessary.In order to shed light on the question how well such a shell model ansatz isjustified we regarded in some detail three and four nucleons. The correspondingshell model Hamiltonian can be written in terms of standard Jacobi coordinatesand numerically exact solutions can be generated based on the Faddeev-Yakubovskyequations. Knowing the exact wave functions one can check the validity of theSlater determinant ansatz. 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