Long range behavior of the van der Waals forces between a molecule and a perfectly conducting metallic plate based on joint work with Mariam Badalyan and Dirk Hundertmark Ioannis Anapolitanos special thanks to: Kurt Busch, Francesco Intravaia, Semjon Wugalter 21.10.2019, CIRM Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Van der Waals forces and heuristics Water-Water Water-Atom Atom-Atom dipole-dipole dipole-induced dipole induced dipole- interaction interaction induced dipole interaction Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Van der Waals forces in life and science ◮ Stabilize DNA, Influence boiling points ◮ Material sciences, Chemistry ◮ Molecule-Wall Interactions (i) Geckos climb vertical surfaces (ii) Can change the direction of atomic beams Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Modeling of the problem (for a hydrogen atom) ◮ nucleus at (0 , 0 , 0) ◮ position of the electron x = ( x 1 , x 2 , x 3 ) ◮ infinite plate at the plane x 1 = r Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Modeling of the problem (with perfectly conducting plate) ◮ ”anti-nucleus” at 2 re 1 ◮ ”anti-electron” with distance x ∗ = ( − x 1 , x 2 , x 3 ) from anti-nucleus. Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Hamiltonian of the system H ( r ) = H h + I ◮ Hamiltonian 2 H h = − ∆ x − 1 | x | 1 1 1 1 I = − − + + | − x + 2 re 1 + x ∗ | | 2 re 1 + x ∗ | 2 r | 2 re 1 − x | ���� � �� � � �� � � �� � Attraction Attraction Repulsion Repulsion K + / K − e − / e + e + / K + e − / K − � �� � 2 = | 2 re 1 − x | Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Hamiltonian of the system H ( r ) = H h + I ◮ Hamiltonian 2 H h = − ∆ x − 1 | x | 1 1 1 1 I = − − + + | − x + 2 re 1 + x ∗ | | 2 re 1 + x ∗ | 2 r | 2 re 1 − x | ���� � �� � � �� � � �� � Attraction Attraction Repulsion Repulsion K + / K − e − / e + e + / K + e − / K − � �� � 2 = | 2 re 1 − x | ◮ Always attractive : I < 0. Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Form core of the Hamiltonian ◮ Electron ’lives’ in halfspace A = { ( x 1 , x 2 , x 3 ) | x 1 < r } . ◮ In H 1 ( A ) (no boundary conditions): Hamiltonian not bounded from below, attraction of electron from the mirror charge is too singular . ◮ Form core C ∞ c ( A ): Electron can not touch the plane, or pass through it (Dirichlet b.c. rides to the rescue). Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Form domain and boundedness from below ◮ We use the ”Hardy inequality” � ∞ � ∞ | u ( x ) | 2 | u ′ ( x ) | 2 dx , in C ∞ dx ≤ c ( R + ) . 4 x 2 0 0 � ∞ � ∞ � ∞ � ′ � 1 | u ( x ) | 2 u ( x ) u ( x ) u ( x ) 2 x u ′ ( x ) dx dx = − dx = Re 4 x 2 4 x 0 0 0 � � ∞ � 1 2 � � ∞ � 1 � � 2 u ( x ) CS 2 � � | u ′ ( x ) | 2 dx ≤ dx � � � 4 x � 0 0 Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Form domain and boundedness from below ◮ We use the ”Hardy inequality” � ∞ � ∞ | u ( x ) | 2 | u ′ ( x ) | 2 dx , in C ∞ dx ≤ c ( R + ) . 4 x 2 0 0 � ∞ � ∞ � ∞ � ′ � 1 | u ( x ) | 2 u ( x ) u ( x ) u ( x ) 2 x u ′ ( x ) dx dx = − dx = Re 4 x 2 4 x 0 0 0 � � ∞ � 1 2 � � ∞ � 1 � � 2 u ( x ) CS 2 � � | u ′ ( x ) | 2 dx ≤ dx � � � 4 x � 0 0 ◮ Extension of the form on H 1 0 ( A ): Form closed and bounded from below. ◮ Thus H ( r ) can be realized as a self-adjoint operator. Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Plate and Electron ◮ z position of the electron with respect to a point on the plate. 1 1 ◮ H e − = − ∆ z − 2 | z − z ∗ | = − ∆ z − 4 z 1 rescaling E e − = 0 ( A ) , � ψ � L 2( A ) =1 ψ | H e − ψ inf < 0 ψ ∈ H 1 Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Plate and Electron ◮ z position of the electron with respect to a point on the plate. 1 1 ◮ H e − = − ∆ z − 2 | z − z ∗ | = − ∆ z − 4 z 1 rescaling E e − = 0 ( A ) , � ψ � L 2( A ) =1 ψ | H e − ψ inf < 0 ψ ∈ H 1 ◮ Lower bound: Because 2 ab ≤ a 2 + b 2 : 16 − ∂ 2 1 4(2 z 1 ) ≤ 1 1 1 ”Hardy” 1 = 2 16 + ≤ 4 z 2 ∂ z 2 4 z 1 1 1 4 z 1 ≥ − 1 1 16 = E h Thus − ∆ z − 4 > E h (hydrogen gs energy) ◮ Thus E e − > E h . The electron prefers to be close to the nucleus and not close to the plate. Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Existence of a ground state of the system ◮ HVZ type Theorem: − 1 inf σ ess ( H ( r )) = E e − 4 r ���� � �� � minimum energy of nucleus-plate electron at ’infinity’ attraction ◮ I < 0 r not too small = ⇒ inf σ ( H ( r )) ≤ E h . ◮ Recall E e − ≥ E h 4 > E h . ◮ Thus inf σ ( H ( r )) in discrete spectrum. Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Interaction energy and main result ◮ Ground state energy E ( r ) = inf ψ ∈ H 1 0 ( A ) , � ψ � L 2( A ) =1 ψ | H ( r ) ψ . ◮ Interaction energy W ( r ) = E ( r ) − lim s →∞ E ( s ) = E ( r ) − E h . Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Interaction energy and main result ◮ Ground state energy E ( r ) = inf ψ ∈ H 1 0 ( A ) , � ψ � L 2( A ) =1 ψ | H ( r ) ψ . ◮ Interaction energy W ( r ) = E ( r ) − lim s →∞ E ( s ) = E ( r ) − E h . Theorem (A., Badalyan, 51,32 e ) There exist a r 0 > 0 and D > 0 , so that for all r > r 0 : � � � W ( r ) + 1 � ≤ D � � � � r 3 r 5 holds. Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Main idea of the proof Write H = H ( r ) and E the ground state energy of H . ◮ P orthogonal projection P ⊥ = 1 − P , H ⊥ = P ⊥ HP ⊥ . Feshbach map F P ( λ ) = ( PHP − PHP ⊥ ( H ⊥ − λ ) − 1 P ⊥ HP ) | RanP . ◮ Thm: ( H ⊥ − E ) ≥ γ > 0 = ⇒ E is eigenvalue of F P ( E ). Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Main idea of the proof Write H = H ( r ) and E the ground state energy of H . ◮ P orthogonal projection P ⊥ = 1 − P , H ⊥ = P ⊥ HP ⊥ . Feshbach map F P ( λ ) = ( PHP − PHP ⊥ ( H ⊥ − λ ) − 1 P ⊥ HP ) | RanP . ◮ Thm: ( H ⊥ − E ) ≥ γ > 0 = ⇒ E is eigenvalue of F P ( E ). ◮ Choose P = | ψ �� ψ | cut off ground state of hydrogen atom. ◮ Step 1: ( H ⊥ − E ) ≥ γ , we need E e − > E h . 2 ψ � = E h − 1 r 3 + O ( 1 ◮ Step 2: PHP | RanP ≃ E h + � ψ, I r 5 ). � 1 I = − ( x · e 1 ) 2 − | x | 2 � + f odd ( x ) + O 8 r 3 8 r 4 r 5 ◮ Step 3: − PHP ⊥ ( H ⊥ − E ) − 1 P ⊥ HP = O ( 1 r 6 ). Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
Feshbach condition H ⊥ − E ≥ c > 0 ◮ IMS Localization: H = J 1 HJ 1 + J 2 HJ 2 − |∇ J 1 | 2 − |∇ J 2 | 2 for J 1 , J 2 : R 3 → R zwei C ∞ -Functions so that J 2 1 + J 2 2 = 1 ◮ For the term J 1 HJ 1 we need E e − > E h Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
The case of a general molecule ◮ Hamiltonian H N = H N ( r , v ), N number of electrons, r distance of molecule and plate, v relative orientation. Ioannis Anapolitanos van der Waals force between a molecule and a metallic plate special thanks to: Kurt Busch, Francesco Intravaia, Semjon
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