The van der Waals Interaction Derek Stampone The van der Waals Interaction Derek Stampone Binghamton University 5/9/2013
Outline The van der Waals Interaction 1 Introduction Derek Stampone 2 The Hamiltonian Outline Introduction The 3 Semi-Classical Approach Hamiltonian Semi-Classical Approach 4 Second-Order Perturbation Second-Order Perturbation 5 Van der Waals Interactions: Evaluations by use of a Theory Paper statistical mechanical method Conclusion 6 Conclusion
The van der Waals Interaction The van der Waals Interaction Derek Stampone Interaction between neutral objects Outline 1 Keeson Force - Force between Introduction two permanent dipoles The Hamiltonian 2 Debeye Force - Force between Semi-Classical Approach permanent dipole and induced Second-Order dipole Perturbation 3 London Dispersion Force - Force Theory Paper Conclusion between two induced dipoles.
System Arrangement The van der Waals Interaction Derek Stampone +e! "e! +e! "e! Outline Introduction The x 1! x 2! Hamiltonian Semi-Classical Approach R ! Second-Order Perturbation Figure: Two nearby polarizable atoms. Theory Paper Conclusion
Unperturbed Hamiltonian The van der Waals Interaction Derek Stampone Outline Introduction The H 0 = p 2 1 + p 2 2 m + 1 2 m + 1 Hamiltonian 1 2 kx 2 2 2 kx 2 2 Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion
Coulomb Interaction The van der Waals Interaction Derek Stampone Outline Introduction The � e 2 e 2 e 2 e 2 1 � Hamiltonian H ′ = R − − + Semi-Classical 4 πǫ 0 R − x 1 R + x 2 R − x 1 − x 2 Approach Second-Order Perturbation Theory Paper Conclusion
Perturbed Hamiltonian The van der Waals Interaction Derek Stampone Outline Take the limit as x 1 ≪ R and x ≪ R we get Introduction The H ′ ≈ − e 2 x 1 x 2 Hamiltonian Semi-Classical 2 πǫ 0 R 3 Approach Second-Order Perturbation Theory Paper Conclusion
Change of Variables The van der Waals Interaction Derek Stampone 1 1 Outline x ± = √ 2( x 1 ± x 2 ) p ± √ 2( p 1 ± p 2 ) Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion
Change of Variables The van der Waals Interaction Derek Stampone 1 1 Outline x ± = √ 2( x 1 ± x 2 ) p ± √ 2( p 1 ± p 2 ) Introduction � 1 � 1 The e 2 e 2 � � � � � � + + 1 − + 1 Hamiltonian 2 mp 2 x 2 2 mp 2 x 2 H = k − + k + + 2 πǫ 0 R 3 2 πǫ 0 R 3 2 2 − Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion
Change of Variables The van der Waals Interaction Derek Stampone 1 1 Outline x ± = √ 2( x 1 ± x 2 ) p ± √ 2( p 1 ± p 2 ) Introduction � 1 � 1 The e 2 e 2 � � � � � � + + 1 − + 1 Hamiltonian 2 mp 2 x 2 2 mp 2 x 2 H = k − + k + + 2 πǫ 0 R 3 2 πǫ 0 R 3 2 2 − Semi-Classical Approach � Second-Order k ∓ ( e 2 / 2 πǫ 0 R 3 ) E = 1 Perturbation 2 � ( ω + + ω − ) ω ± = m Theory Paper Conclusion
The van der Waals Interaction Without H ′ , Derek Stampone � E 0 = � ω ω = k/m Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion
The van der Waals Interaction Without H ′ , Derek Stampone � E 0 = � ω ω = k/m Outline Introduction The Hamiltonian Semi-Classical We’ll assume that k ≫ ( e 2 / 2 πǫ 0 R 3 ) , and get Approach Second-Order � e 2 Perturbation � 2 1 � Theory Paper ∆ V ≡ E − E 0 ≈ − 8 m 2 ω 3 2 πǫ 0 R 6 Conclusion 0
Second-Order Perturbation The van der Waals Interaction Derek ∞ Stampone | ψ n | H ′ | ψ 0 | 2 � E 2 0 = E 0 − E n Outline n =1 Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion
Second-Order Perturbation The van der Waals Interaction Derek ∞ Stampone | ψ n | H ′ | ψ 0 | 2 � E 2 0 = E 0 − E n Outline n =1 Introduction The Hamiltonian Semi-Classical where Approach | ψ 0 � = | 0 �| 0 � | ψ n � = | n 1 �| n 2 � Second-Order Perturbation and Theory Paper H ′ ≈ − e 2 x 1 x 2 Conclusion 2 πǫ 0 R 3
Second-Order Perturbation (Cont.) The van der Waals Interaction Write x in terms of raising and lowering operators. Only Derek non-zero term will be n 1 = n 2 = 1 Stampone Outline Introduction The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion
Second-Order Perturbation (Cont.) The van der Waals Interaction Write x in terms of raising and lowering operators. Only Derek non-zero term will be n 1 = n 2 = 1 Stampone Outline Introduction � 2 e 2 |� 1 | x | 0 �| 2 |� 1 | x | 0 �| 2 � E 2 The 0 = � 1 � 3 Hamiltonian 2 πǫ 0 R 3 2 � ω 0 + 1 � 2 � ω 0 + 3 � 2 � ω 0 − 2 � ω 0 Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion
Second-Order Perturbation (Cont.) The van der Waals Interaction Write x in terms of raising and lowering operators. Only Derek non-zero term will be n 1 = n 2 = 1 Stampone Outline Introduction � 2 e 2 |� 1 | x | 0 �| 2 |� 1 | x | 0 �| 2 � E 2 The 0 = � 1 � 3 Hamiltonian 2 πǫ 0 R 3 2 � ω 0 + 1 � 2 � ω 0 + 3 � 2 � ω 0 − 2 � ω 0 Semi-Classical Approach Second-Order Perturbation � e 2 � 2 1 Theory Paper � E 2 0 = − Conclusion 8 m 2 ω 3 R 6 2 πǫ 0 0
Van der Waals Interactions: Evaluations by use of a statistical mechanical method The van der Waals Interaction Derek Stampone Johan S. Høye’s paper is about using statistical mechanical Outline methods to show an equivalence between the Casimir effect Introduction and second-order perturbation to Schr¨ odinger’s equation. The Hamiltonian Semi-Classical Approach Second-Order Perturbation Theory Paper Conclusion
Van der Waals Interactions: Evaluations by use of a statistical mechanical method The van der Waals Interaction Derek Stampone Johan S. Høye’s paper is about using statistical mechanical Outline methods to show an equivalence between the Casimir effect Introduction and second-order perturbation to Schr¨ odinger’s equation. The Hamiltonian Semi-Classical Approach Second-Order Høye reduces the problem between interacting dipoles to a Perturbation classical polymer problem in four dimensions with imaginary Theory Paper time. He also avoids quantization of the electromagnetic field. Conclusion
Conclusion The van der Waals Interaction Derek Stampone Outline The van der Waals interaction is a weak but ubiquitous force Introduction that can exist in a variety of situations. The weak attraction The Hamiltonian between dipoles is an important result. The van der Waals Semi-Classical interaction is a framework to describe a variety of phenomenon Approach including the Casimir effect. Second-Order Perturbation Theory Paper Conclusion
Conclusion The van der Waals Interaction Derek Stampone Outline Research has been done on how Introduction geckos use the van der Waals The Hamiltonian interaction to cling to surfaces Semi-Classical and how to apply similar Approach methods to produce sticky Second-Order Perturbation tape. Theory Paper Conclusion
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