The WKB approximation in deformation quantization J. Tosiek, - PowerPoint PPT Presentation

1 The WKB approximation in deformation quantization J. Tosiek, R.Cordero and F. J. Turrubiates, J. Math. Phys. 57 , 062103 (2016) Jaromir Tosiek Lodz University of Technology, Poland in collaboration with Ruben Cordero and Francisco Turrubiates Instituto Politecnico Nacional, Mexico

THE ENERGY ∗ – EIGENVALUE EQUATION 1 2 The energy ∗ – eigenvalue equation 1 p 2 p ) = � For a Hamilton function H ( � r, � 2 M + V ( � r ) the ∗ – eigenvalue equa- tion is H ( � r, � p ) ∗ W E ( � r, � p ) = E W E ( � r, � p ) with an additional condition imposed on the Wigner energy eigen- function W E ( � r, � p ) { H ( � r, � p ) , W E ( � r, � p ) } M = 0 .

THE ENERGY ∗ – EIGENVALUE EQUATION 1 3 As the ∗ – product we use the Moyal product 1 � r ′ d� p ′ d� r ′′ d� p ′′ A ( � r ′ , � p ′ ) B ( � r ′′ , � p ′′ ) A ( � r, � p ) ∗ B ( � r, � p ) := R 12 d� ( π � ) 6 � 2 i �� � r ′′ − � p ′ − � r ′ − � p ′′ − � × exp ( � r ) · ( � p ) − ( � r ) · ( � p ) . � The dot ‘ · ’ stands for the scalar product. The above definition is valid for a wide class of tempered distribu- tions.

THE ENERGY ∗ – EIGENVALUE EQUATION 1 4 The Moyal product is closed i.e. � � R 6 A ( � r, � p ) ∗ B ( � r, � p ) d� r d� p = R 6 B ( � r, � p ) ∗ A ( � r, � p ) d� r d� p = � R 6 A ( � r, � p ) · B ( � r, � p ) d� r d� p. The mean value of a function A ( � r, � p ) in a state represented by a Wigner function W ( � r, � p ) equals � � � A ( � r, � p ) = R 6 A ( � r, � p ) · W ( � r, � p ) d� r d� p.

THE ENERGY ∗ – EIGENVALUE EQUATION 1 5 The Moyal bracket is defined as p ) } M := 1 � � { A ( � r, � p ) , B ( � r, � A ( � r, � p ) ∗ B ( � r, � p ) − B ( � r, � p ) ∗ A ( � r, � p ) . i � Not all of the solutions of the system of the ∗ - eigenvalue equations are physically acceptable. A Wigner eigenfunction W E ( � r, � p ) of the Hamilton function H ( � r, � p ) fulfills the following conditions: (i) is a real function, 1 (ii) the ∗ – square W E ( � r, � p ) ∗ W E ( � r, � p ) = 2 π � W E ( � r, � p ) and � R 6 W E ( � r, � p ) d� rd� p = 1 . (iii)

THE ENERGY ∗ – EIGENVALUE EQUATION 1 6 We restrict to the 1 – D case. But even in this simplest situation the system of eigenvalue equations leads to a pair of integral equation. This is why approximate methods are desirable. A naive treating of the ∗ – eigenvalue equation as a power series in the deformation parameter � does not work, because Wigner eigen- functions contain arbitrary large negative powers of � .

2 INGREDIENTS OF THE WKB CONSTRUCTION 7 2 Ingredients of the WKB construction Eigenstates and eigenvalues of energy can be found by solving the stationary Schroedinger equation − � 2 2 M ∆ ψ E ( � r ) + V ( � r ) ψ E ( � r ) = Eψ E ( � r ) . Every solution of the stationary Schroedinger equation can be writ- ten as a linear combination of two functions � i � i � � ψ E I ( � r ) = exp � σ I ( � r ) and ψ E II ( � r ) = exp � σ II ( � r ) , where σ I ( � r ) , σ II ( � r ) denote some complex valued functions.

2 INGREDIENTS OF THE WKB CONSTRUCTION 8 When we substitute functions ψ E I ( � r ) , ψ E II ( � r ) into the stationary Schroedinger equation, we obtain that phases σ I ( � r ) and σ II ( � r ) satisfy the partial nonlinear differential equation of the 2nd order 1 � 2 − i � � ∇ σ ( � r ) 2 M ∆ σ ( � r ) = E − V ( � r ) , (1) 2 M for σ ( � r ) = σ I ( � r ) and σ ( � r ) = σ II ( � r ) . In the classical limit � → 0 this equation reduces to the Hamilton – Jacobi stationary equation 1 � 2 = E − V ( � � ∇ σ ( � r ) r ) . (2) 2 M

2 INGREDIENTS OF THE WKB CONSTRUCTION 9 In the 1 –D case Eq. (1) is of the form � 2 d 2 σ ( x ) 1 � dσ ( x ) − i � = E − V ( x ) . dx 2 2 M dx 2 M In some part of its domain the solution can be written as a formal power series in the Planck constant ∞ � k � � � σ ( x ) = σ k ( x ) . i k =0

2 INGREDIENTS OF THE WKB CONSTRUCTION 10 Thus we receive an iterative system of equations � 2 � dσ 0 ( x ) 1 = E − V ( x ) , 2 M dx d 2 σ 0 ( x ) dσ 0 ( x ) dσ 1 ( x ) + 1 = 0 , dx 2 dx dx 2 � 2 d 2 σ 1 ( x ) � dσ 1 ( x ) dσ 0 ( x ) dσ 2 ( x ) + 1 + 1 = 0 , dx 2 dx dx 2 dx 2 . . . . . . . . . There are two solutions of these equations. They differ on the sign at even � power elements.

2 INGREDIENTS OF THE WKB CONSTRUCTION 11 In the case when the phase σ ( x ) is the power series in the Planck constant, the wave function ∞ � � � k i � � � ψ E ( x ) = ψ E k ( x ) , ψ E k ( x ) = exp σ k ( x ) . i � k =0 Each function ψ E k ( x ) need not be an element of L 2 ( R ) but as it is smooth and, due to physical requirements, bounded, the product � � � � x + ξ x − ξ ψ E k ψ E k is a tempered generalised function. 2 2

2 INGREDIENTS OF THE WKB CONSTRUCTION 12 The analysed approximation can be realised as an iterative proce- dure, in which the n -th approximation ψ E ( n ) ( x ) of the wave function ψ E ( x ) equals ψ E (0) ( x ) := ψ E 0 ( x ) ψ E ( n ) ( x ) = ψ E ( n − 1) ( x ) · ψ E n ( x ) , n � 1 .

2 INGREDIENTS OF THE WKB CONSTRUCTION 13 Applying the Weyl correspondence W − 1 to an energy eigenstate � i � � � ψ E ( x ) := x | ψ E = exp � σ ( x ) we see that its Wigner function is of the form � + ∞ � x + ξ � � x − ξ � � − iξp � 1 W E ( x, p ) = dξ ψ E ψ E = exp 2 π � 2 2 � −∞ � i � + ∞ 1 � � x − ξ � � x + ξ � �� = dξ exp σ − σ − ξp . 2 π � 2 2 � −∞

2 INGREDIENTS OF THE WKB CONSTRUCTION 14 Thus � + ∞ W E ( n − 1) ( x, p ′ ) W E n ( x, p − p ′ ) dp ′ = W E ( n ) ( x, p ) = −∞ � + ∞ W E ( n − 1) ( x, p − p ′′ ) W E n ( x, p ′′ ) dp ′′ . = −∞

2 INGREDIENTS OF THE WKB CONSTRUCTION 15 The semiclassical approximation cannot be applied everywhere. Thus the wave function is a sum of spatially separable functions k � ψ E ( x ) = ψ Ea l b l ( x ) l =1 −∞ � a 1 < b 1 = a 2 < b 2 = a 3 < . . . < b k − 1 = a k < b k � ∞ . It is a vital question about a phase space counterpart of a state being the superposition of wave functions.

2 INGREDIENTS OF THE WKB CONSTRUCTION 16 Let us consider a Wigner function originating from a wave function ψ Ea l b l ( x ) . � Min . [2( x − a l ) , 2( b l − x )] � � 1 x + ξ W Ea l b l ( x, p ) = dξ ψ Ea l b l × 2 π � 2 Max . [2( a l − x ) , 2( x − b l )] � x − ξ � � − iξp � × ψ Ea l b l . exp 2 � (i) The Wigner function W Ea l b l ( x, p ) vanishes outside the set ( a l , b l ) × R . (ii) As the function ψ Ea l b l ( x ) itself can be a sum of functions, we see that every Wigner function of a superposition of wave functions with supports from an interval [ a l , b l ] is still limited to the strip a l � x � b l .

2 INGREDIENTS OF THE WKB CONSTRUCTION 17 One can deduce that if an operator ˆ A in the position representation satisfies the condition a < x, x ′ < b, x | ˆ A | x ′ � � � = 0 only for then the function W − 1 ( ˆ A )( x, p ) may be different from 0 only for x contained in the interval ( a, b ) . Moreover, the function W − 1 ( ˆ A )( x, p ) is a smooth function respect to the momentum p. For every ˜ x ∈ ( a, b ) and every positive number Λ > 0 there exists a value of momentum ˜ p p | > Λ and W − 1 ( ˆ such that | ˜ A )(˜ x, ˜ p ) � = 0 .

2 INGREDIENTS OF THE WKB CONSTRUCTION 18 Consider a two-component linear combination of functions Y ( x − a l ) ψ Ea l b l ( x ) Y ( b l − x ) + Y ( x − a r ) ψ Ea r b r ( x ) Y ( b r − x ) , −∞ � a l < b l � a r < b r � ∞ . Its Wigner function W E ( x, p ) = W − 1 � 1 + W − 1 � 1 � � �� �� 2 π � | ψ Ea l b l ψ Ea l b l | 2 π � | ψ Ea r b r ψ Ea r b r | + + W − 1 � 1 1 � �� �� 2 π � | ψ Ea l b l ψ Ea r b r | + 2 π � | ψ Ea r b r ψ Ea l b l | .

2 INGREDIENTS OF THE WKB CONSTRUCTION 19 The interference operator ˆ �� �� Int := | ψ Ea l b l ψ Ea r b r | + | ψ Ea r b r ψ Ea l b l | (i) is self-adjoint. (ii) It is not a projector. (iii) Its trace vanishes and it has three possible eigenvalues λ : = 1 � 1 1 �� � � √ λ = −|| ψ Ea l b l ||·|| ψ Ea r b r || , |− || ψ Ea l b l ||| ψ Ea l b l − || ψ Ea r b r ||| ψ Ea r b r 2 � � λ = 0 , its eigenvector is every vector orthogonal to | ψ Ea l b l and | ψ Ea r b r , = 1 � 1 1 �� � � √ λ = || ψ Ea l b l ||·|| ψ Ea r b r || , | + || ψ Ea l b l ||| ψ Ea l b l + || ψ Ea r b r ||| ψ Ea r b r . 2