KANTOROVICH AND CLOSE-COUPLING METHODS IN QUANTUM TUNNELING PROBLEM - PowerPoint PPT Presentation

KANTOROVICH AND CLOSE-COUPLING METHODS IN QUANTUM TUNNELING PROBLEM FOR A COUPLED PAIR OF IONS THROUGH LONG-RANGE POTENTIAL BARRIERS. Outline The problem statement A.A. Gusev, Close-coupling and Kantorovich (Adia- batic) methods O. Chuluunbaatar, BVP for fast subsystem V.P. Gerdt, BVP for slow subsystem The scattering problem (JINR, Dubna, Russia) Algorithm for asymptotic expansions Results Conclusions

The problem statement Let us consider a quantum system of two particles with masses m 1 , m 2 and radius-vectors ˜ x 1 , ˜ x 2 describing by the Hamiltonian H = − � 2 � 2 ˆ x 2 + ˜ x 2 ) + ˜ x 1 ) + ˜ ∇ 2 ∇ 2 x 1 − V (˜ x 1 − ˜ U 0 (˜ U 0 (˜ x 2 ) ˜ ˜ 2 m 1 2 m 2 We suppose that a pair of particles is coupled by a potential x 2 ) = µω 2 ˜ x 2 ) 2 , V (˜ x 1 − ˜ (˜ x 1 − ˜ 2 where µ = m 1 m 2 / ( m 1 + m 2 ) is a reduced mass and ω is a frequency of a three-dimensional harmonic oscillator, transmit through a potential barrier ˜ x 1 ) + ˜ U 0 (˜ U 0 (˜ x 2 ) like in heavy ion collisions.

The problem statement Hamiltonian written in the coordinates of the center of mass of the pair Y and the internal variable corresponding to the relative motion ˜ ˜ X , Y = m 1 ˜ x 1 + m 2 ˜ x 2 ˜ ˜ , X = ˜ x 1 − ˜ x 2 , M where M = m 1 + m 2 is the total mass, has the form H = − � 2 Y − � 2 ˆ X + ˜ V (˜ X) + ˜ x 1 ) + ˜ 2 M ∇ 2 2 µ ∇ 2 U 0 (˜ U 0 (˜ x 2 ) ˜ ˜ � x 2 � − ˜ Gaussian-type barrier ˜ A U 0 (˜ x i ) = 2 πσ exp i , at a = 5 , σ = 0 . 1 and √ 2 σ corresponding 2D potentials with m 1 = 1 , m 2 = 1 and m 1 = 1 , m 2 = 9

The problem statement Using the transformation to dimensionless variables � � ˜ ˜ Mω M Y � µω X ˜ ˜ y = Y = , x = X = , � µ x osc � x osc � � where x osc = µω is unit of length, we rewrite the Schr¨ odinger equation with Hamiltonian (1) as the following dimensionless equation: � � −∇ 2 x − ∇ 2 y + V (x) + U (x , y) − E Ψ(y , x) = 0 . Here the energy E = ˜ E/E osc and the potential functions V (x) = x 2 , U (x , y) = U 0 (˜ x 1 ) + U 0 (˜ x 2 ) are given in units of energy E osc = � ω/ 2 and dimensional variables ˜ x i are expressed via dimensionless ones x i � √ m 1 √ m 2 y + m 2 � x 1 = x osc x 1 = x osc ˜ M x , M � √ m 1 √ m 2 y − m 1 � ˜ x 2 = x osc x 2 = x osc M x . M

Barriers Gaussian-type x 2 A − ˜ � � ˜ i U 0 (˜ x i ) = √ exp 2 σ 2 πσ where σ = 0 . 1 , m 1 = 1 , m 2 = 9 , a = 5 . Truncated Coulomb potential ˆ ˆ  Z i Z i x min − x max , | ˜ x | ≤ ˜ x min ;  ˜ ˜      ˜ ˆ ˆ U 0 (˜ x i ) = Z i Z i . x | − x max , x min < | x | ≤ ˜ ˜ x max ; | ˜ ˜      0 | ˜ x | > ˜ x max  Coulomb-like potential ˜ x i ) = ˆ x s x s min ) − 1 /s U 0 (˜ Z i (˜ i + ˜

Close-coupling and Kantorovich (Adiabatic) methods The Schr¨ odinger equation reads as � 1 � H 2 ( x f ; x s )+ ˆ ˆ H 1 ( x s ) + ˆ V fs ( x f , x s ) − 2 E Ψ( x f , x s )=0 , g 3 s ( x s ) 1 ∂ g 2 f ( x f ) ∂ ˆ + ˆ H 2 = − V f ( x f ; x s ) , g 1 f ( x f ) ∂x f ∂x f 1 ∂ g 2 s ( x s ) ∂ ˆ + ˆ H 1 = − V s ( x s ) . g 1 s ( x s ) ∂x s ∂x s ˆ H 2 ( x f ; x s ) is the Hamiltonian of the fast subsystem, ˆ H 1 ( x s ) is the Hamiltonian of the slow subsystem, V fs ( x f , x s ) is interaction potential. The Kantorovich expansion of the desired solution of BVP: j max � Ψ( x f , x s ) = Φ j ( x f ; x s ) χ j ( x s ) . j =1

BVP for fast subsystem The equation for the basis functions of the fast variable x f and the potential curves, E i ( x s ) continuously depend on the slow variable x s as a parameter � � ˆ H 2 ( x f ; x s ) − E i ( x s ) Φ i ( x f ; x s ) = 0 , The boundary conditions N f ( x s ) g 2 f ( x s ) d Φ j ( x f ; x s ) � � lim + D f ( x s )Φ j ( x f ; x s ) = 0 . dx f x f → x t f ( x s ) The normalization condition x max ( x s ) f � � Φ i | Φ j � = Φ i ( x f ; x s )Φ j ( x f ; x s ) g 1 f ( x f ) dx f = δ ij . x min ( x s ) f

BVP for slow subsystem The effective potential matrices of dimension j max × j max : 1 E i ( x s ) δ ij + g 2 s ( x s ) ˆ U ij ( x s )= g 1 s ( x s ) W ij ( x s ) + V ij ( x s ) , g 3 s ( x s ) x max f � V ij ( x s ) = Φ i ( x f ; x s ) V fs ( x f , x s )Φ j ( x f ; x s ) g 1 f ( x f ) dx f , x min f x max f ∂ Φ i ( x f ; x s ) ∂ Φ j ( x f ; x s ) � W ij ( x s ) = g 1 f ( x f ) dx f , ∂x s ∂x s x min f x max f Φ i ( x f ; x s ) ∂ Φ j ( x f ; x s ) � Q ij ( x s ) = − g 1 f ( x f ) dx f . ∂x s x min f

BVP for slow subsystem The SDE for the slow subsystem (the adiabatic approximation is a diagonal approximation for the set of ODEs) H χ ( i ) ( x s ) = 2 E i I χ ( i ) ( x s ) , g 1 s ( x s )I d 1 g 2 s ( x s ) d + ˆ H= − V s ( x s )I+U( x s ) dx s dx s + g 2 s ( x s ) g 1 s ( x s )Q( x s ) d 1 dg 2 s ( x s )Q( z ) + , dx s g 1 s ( x s ) dx s with the boundary conditions N s g 2 s ( x s ) dχ ( x s ) � � lim + D s χ ( x s ) = 0 . dx s x s → x t s

The scattering problem is solved using the boundary conditions at d = 1 , z = z min and z = z max : d Φ( z ) � = R ( z min )Φ( z min ) , d Φ( z ) � � � = R ( z max )Φ( z max ) , � � dz dz � � z = z min z = z max where R ( z ) is a unknown N × N matrix-function, Φ( z ) = { χ ( j ) ( z ) } N o j =1 is the required N × N o matrix-solution and N o is the number of open channels, N o = max 2 E ≥ ǫ j j ≤ N .

Matrix-solution Φ v ( z ) = Φ( z ) describing the incidence of the particle and its scattering, which has the asymptotic form “incident wave + outgoing waves”, is X (+) ( z )T v ,  � z > 0 , v = → ,   X (+) ( z ) + X ( − ) ( z )R v , z < 0 ,  Φ v ( z → ±∞ ) = X ( − ) ( z ) + X (+) ( z )R v , � z > 0 , v = ← ,   X ( − ) ( z )T v , z < 0 ,  where R v and T v are the reflection and transmission N o × N o matrices, v = → and v = ← denote the initial direction of the particle motion along the z axis. Φ Φ � ( → ±∞ → ±∞ Φ � ( Φ → ±∞ → ±∞ ( ) ( ) ) ) z z → ← → ← z z + � � � − † − † † − � † ( ) ( ) ( ) ( ) (+) ( ) ( ) ( ) (+) ( ) ( ) ( ) X X X T → X R → X R ← X T ← z z z z z z + − − − ( ) ( ) (+) ( ) ( ) ( ) (+) ( ) ( ) ( ) ( ) ( ) X R → X T X T X R ← X X z z z z z z → ← z < z > z < z > z < z > z < z > 0 0 0 0 0 0 0 0 (a) (b) Schematic diagrams of the continuum spectrum waves having the asymptotic form: (a) “incident wave + outgoing waves”, (b) “incident waves + ingoing wave”.

Here the leading term of the asymptotic rectangle-matrix functions X ( ± ) ( z ) has the form � � p j z − Z j �� ( z ) → ( p j | z | d − 1 ) − 1 / 2 exp X ( ± ) ± ı ln(2 p j | z | ) δ ij , ij p j � p j = 2 E − ǫ j i = 1 , . . . , N, j = 1 , . . . , N o , where Z j = Z + j at z > 0 and Z j = Z − j at z < 0 .

The matrix-solution Φ v ( z, E ) is normalized by � ∞ v ′ ( z, E ′ )Φ v ( z, E ) z d − 1 dz = 2 πδ ( E ′ − E ) δ v ′ v I oo , Φ † z 0 where I oo is the unit N o × N o matrix and z 0 = −∞ if d = 1 or z 0 > 0 if d ≥ 2 . Let us rewrite Eq. (1) in the matrix form at z + → + ∞ and z − → −∞ as � � Φ → ( z + ) Φ ← ( z + ) Φ → ( z − ) Φ ← ( z − ) � X ( − ) ( z + ) � � X (+) ( z + ) � 0 0 = + S , X (+) ( z − ) X ( − ) ( z − ) 0 0 where the unitary and symmetric scattering matrix S � R → � T ← S † S = SS † = I , S = S T S = , T → R ← is composed of the reflection and transmission matrices.

In addition, it should be noted that functions X ( ± ) ( z ) satisfy relations Wr(Q( z ); X ( ∓ ) ( z ) , X ( ± ) ( z )) = ± 2 ı I oo , Wr(Q( z ); X ( ± ) ( z ) , X ( ± ) ( z )) = 0 , where Wr(Q( z ); a( z ) , b( z )) is a generalized Wronskian with a long derivative defined as � � d b( z ) � a T ( z ) Wr(Q( z ); a( z ) , b( z )) = z d − 1 − Q( z )b( z ) dz � T � � d a( z ) − − Q( z )a( z ) b( z ) . dz This Wronskian is used to estimate a desirable accuracy of the above expansion.

From Wronskian conditions, we obtain the following properties of the reflection and transmission matrices: T † → T → + R † → R → = T † ← T ← + R † ← R ← = I oo , T † → R ← + R † → T ← = R † ← T → + T † ← R → = 0 , T T R T R T → = T ← , → = R → , ← = R ← . This means that the scattering matrix is symmetric and unitary.

Asymptotic expansions of regular and irregular solutions in longitudinal coordinates We seek the solution of SDE in the form: χ i ′ ( x s ) = φ i ′ ( x s ) R i ′ ( x s ) + ψ i ′ ( x s ) dR i ′ ( x s ) , dx s where φ i ′ ( x s ) and ψ i ′ ( x s ) are unknown functions, while R i ′ ( x s ) is known function and dR i ′ ( x s ) is derivative of R i ′ ( x s ) with respect to x s . dx s We choose R i ′ ( x s ) as solutions of auxiliary problem   Z ( l ) 1 d d � x d − 1 i ′ − k 2  R i ′ ( x s ) = 0 .  − + s i ′ x d − 1 dx s dx s x l s s l ≥ 1 Note, if Z ( l ≥ 3) = 0 then solutions of last equation are presented via i hypergeometric functions, in particular, via exponential, trigonometric, Bessel, Coulomb functions, etc.