Overview on the Hilbert Space Harmonics Oscillator

Transcription

1 Proceedings of the nd International Conference on Natural and Environmental Sciences (ICONES) September Banda Aceh Indonesia ISSN Overview on the Hilbert Space Harmonics Oscillator RintoAgustino and Saiman Natural Philosophy Laboratory UniversitasSamudra Meurandeh College Langsa Lama Langsa Aceh Abstract Have been calculated harmonics oscillator with Hilbert Space review Study of harmonics oscillator in classical mechanics usually uses Newtonian and Lagrange mechanicswhile in quantum mechanics using Hamiltonian Keywords: Hilbert Space harmonics oscillator INTRODUCTION Quantum theory was born when Max Planck's famous made lecture in front of the Deutsche PhysikalischeGesellschaft [1] In the next development stage under the touch of many figures (de Broglie Heisenberg Bohr Schrodinger Dirac Jordan Born etc) this theory developed to the top of human intellectual triumph Based on this theory the microscopic behavior of nature can be explained satisfactorily and various experimental results can be predicted very accurately In accordance with those set adage "Science is a way for hackers technology" then so science can begin to explore the realm of microscopic technological development was started on the sphere For example these technologies include solid state technology nuclear technology laser nanotechnology quantum dots nano-sized electronic device These technologies have very high sensitivity because it can manipulate electrons in atoms [] Harmonic motion occurs when a particular type of system vibrates around the balanced configuration The system consists of objects that can be hung on a spring or floating on a liquid molecular two atom an atom in the crystal lattice There are many examples of harmonic motion in the worlds of microscopic and macroscopic Requirement for harmonic motion to occur is the presence of the restoring force that acts to restore to the balanced configuration if the system is disturbed resulting in the corresponding mass inertia of objects beyond the balanced position so that the system oscillates continuously if there are no dissipative processes [3] Harmonic oscillating atom in the crystal has a wave function Harmonic oscillation can be solved by using several methods namely the second-order equations generating functions polynomial Hermitte and operators Based on the wave function and the probability momentum of atomic particles can be predicted HILBERT SPACE REVIEW Definition 1 Let X is a linear space over the field F A multiplication in the (inner product) on Xwith the notation for each pair (uv) with is Applicable for all and i ii 0 and 0 if and only if 0 iii Bar is conjugate complex Pre-Hilbert space over the field Fis a linear space X over F with inner product operation Shape (ii) and (iii) can be combined into 100

2 Proceedings of the nd International Conference on Natural and Environmental Sciences (ICONES) September Banda Aceh Indonesia ISSN for all Let Then is called orthogonal if and only if 0 It will be showed on the Heisenberg uncertainty relation of quantum mechanics with the Schwarz inequality Proposition Let X is pre-hilbert space Then In the form of norms Schwarz inequality can be written: Proof Let 0 Then get Where 0 : [ ] Proposition 3Each pre-hilbert space over the field F also norm space over F with the form: Proof: Have 0 for all ( and 0 if and only if ) for all Finally the triangle inequality 0 Then taken from the Schwarz inequality So for all ( ) where Re z is real part from complex number z From proposition 3 we can get notation and theorem to norm space on pre-hilbert space is a norm space of equation 4 Then convergence In the pre-hilbert space X satisfies the equation 0 Proposition 4Let X pre-hilbert space Then satisfactory i Continuous inner product means that 101

3 Proceedings of the nd International Conference on Natural and Environmental Sciences (ICONES) September Banda Aceh Indonesia ISSN corollary ii Let M is dense of subset X If 0 0 Proof: i because limited according to Schwarz inequality is 0 Statement ii Since M is dense of X there is series So 0 where 0 in M so 0 for all in X with satisfies Definition 5Hilbert space is defined as a pre-hilbert space so that the Banach space with norm u from equation In other words linear space X over field F is Hilbert space if and only if it satisfies: i There is inner product in X and ii Every Cauchy sequance with norm u is convergent If F ℝor F ℂ then Xis called real Hilbert space orcomplex Proposition 6Every finite dimensional Hilbert space is a pre-hilbert space It is derived from the fact that any finite-dimensional norm space is a Banach space Proposition 7Let X is Hilbert space over the field F and let L is linear subspace X Then closurl from L is Hilbert space with restriction inner product X on L Proof: First prove that is a linear space over F Let so and in with Let if for all so Restriction of inner product on X for subspace Let and Then there is sequences and in L on X sequences Cauchy at Then in with Since close in with then (Prugovecki 1971) In this regard there are two concepts of the sequence the first row is called a Cauchy sequence or fundamental sequence The second a sequence is said to converge to a vector member pre-hilbert space when the tribes came 10

4 Proceedings of the nd International Conference on Natural and Environmental Sciences (ICONES) September Banda Aceh Indonesia ISSN closer to the ranks of the vector A pre-hilbert space is called a Hilbert space if every Cauchy sequence in the space is a convergent sequence Each pre-hilbert space with scalar product there is a Hilbert space with scalar product such that the set M dominates X X is called a Hilbert space for the improvement of the pre-hilbert space Sociologically it is analogous to saying that the Javanese dominatesgampongsidodadi meaning that each location in Sidodadi village Javanese can easily be found Due to space dominates M (dense) space X ABSTRACTION FOR QUANTUM MECHANICS Quantum system is a depiction of an atom or molecule described by a complex Hilbert space X i State of physics Unit vector in X called state if 1 Two vectors and called equivalent if and only if for some complex number with 1 Intuitively each state physics at the quantum system is expressed by the state (Sauer 1999) ii Quantity physics Self-adjoint (Hermitian) operator : ( ) on a Hilbert space X is called observable Usually the quantum quantity expressed with energy which is the Hermit operator : ( ) which is called the Hamiltonian of the quantum system iii Measurement Suppose measuring observable A at position Basically different measurement in quantum physics with classical physics in quantum physics that there is only a prediction so that the statistical measurement results only ( ) and ( ) ( ) Relations average value and dispersion ( ) on observable A on state Since operator A is symmetry 0 then the average value of is real iv Dynamics according equations ( ) ℝ Description of the time evolution in quantum mechanics is if the system state on 0 then ( ) is the state of the system at time t Here ( ) is a one-parametric unitary group generated by skewadjointoperator If ( ) then ( ) ( ) ℝ APPLICATION OF HARMONIC OSCILLATOR IN QUANTUM MECHANICS In classical mechanics a harmonic oscillator is expressed by a point mass is usually expressed in differential equations For ( ) > 0 The total energy is given by > 0 with motion ( ) on ℝ which ( ) where ( ) ( ) is the momentum of the particle at time t So that the harmonic oscillator is represented by the Schrodinger equation namely Substitution to following equation 103

5 Proceedings of the nd International Conference on Natural and Environmental Sciences (ICONES) September Banda Aceh Indonesia ISSN Then look for the solution of equation Using ( ) ( ) ( ) with ( ) 1 stationary Schrodinger equation is obtained Let Hilbert space is defined as ℂ Let < ℝ (ℝ) with inner product Every unit vector ℝ ( ) ( ) is called particle state (harmonic oscillator) namely Define ( ) interval [a b] ( ) 1 probability to particle in Definition 8Formalism Hamiltonian ℋ: D(ℋ) X X harmonic oscillator is given by the equation ℋϕ mω x ϕ ϕ m whered(ℋ) S S space is Hermitian function u defined ( ) where (1) Proposition 9 i Operator ℋ simetric ii For all n 01 with ϕ (x) u x 1 (!) 01 ℋϕ E ϕ x mω 104 and

6 Proceedings of the nd International Conference on Natural and Environmental Sciences (ICONES) September Banda Aceh Indonesia ISSN E ω n 1 n 01 iii Eigen function form {ϕ }complete orthonormal system in X In other word a simple oscillation energy is quantized Proof: statement i For all with Statement ii integrate probabilistic to be ( ) ( ) lim ℋ ℋ x polynomial (x) Obtained Definition 10Operator H: D(H) X X defined by (ℋ) called Hamiltonian harmonic oscillator Here ϕ D(ℋ) if and only if < Proposition 11 i Hamiltonian H: D(H) X X is self-adjoint ii Operator H is extension formalism Hamiltonian ℋ Proof: statement i already proven Statement ii Let in X (ℋ)is Where definition space S ℋ Since { ℋ ℋ by symmetry in proposition 9 Then over series convergence Suppose particle in state then ( ) Since ℋ 01 have probabilitas energy Remark 1(Dynamics harmonic)oscillator If given 105 } complete orthonormal system 1 then

7 Proceedings of the nd International Conference on Natural and Environmental Sciences (ICONES) September Banda Aceh Indonesia ISSN Suppose harmonic oscillator in state ψ at time t 0 ( ) Explicitly expressed by Convergence series in Hilbert space ( ) ℂ (ℝ) )( ) ( )( ) calledmomentum operator Here ( ) { : called operator position Here ( ) { : ℝ Definition 13Operator A: D(A) X X with ( ℝ ( )for all ℝ } with the same differential generally into ( ) ℝ } Remark 14(Heisenberg Uncertainty Principle) If given the circumstances with position average and disperse ( ) particle in state and ( ) Momentum average So 1 and disperse ( ) at state is given ( ) ( ) ( ) is ( ) ACKNOWLEDGMENTS Authors express sincere thanks to UniversitasSamudra Langsa for funding to attend this conference Authors also thanks to Dean of Engineering Faculty for allowing to use laboratory 106

8 Proceedings of the nd International Conference on Natural and Environmental Sciences (ICONES) September Banda Aceh Indonesia ISSN REFERENCES van der Waerden BL Sources of Quantum Mechanics Dover Publications Inc New York (1967) 3 Rosyid MF MekanikaKuantum : Model Matematis Gejala Alam Mikroskopik Tinjauan Tak Relativistik Jurusan Fisika Universitas Gadjah Mada (009) 4 Beiser A Modern Physics Inc New York (199) 5 Prugovecki E 1971 Quantum Mechanics in Hilbert Spaces Academics Press New York 5 Sauer T The Relativity of Discovery : Hilbert s First Note on Foundations of Physics Arch Hist Exact Sci 53(1999) (physics/ v1) 107