overview on the hilbert space harmonics oscillator

Proceedings of the 2nd International Conference on Natural and Environmental Sciences (ICONES)September 9-11, 2014 , Banda Aceh, Indonesia

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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 classicalmechanics 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 DeutschePhysikalischeGesellschaft [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 intellectualtriumph2. Based on this theory the microscopic behavior of nature can be explained satisfactorily and variousexperimental results can be predicted very accurately. In accordance with those set adage "Science is a way forhackers technology", then so science can begin to explore the realm of microscopic, technological developmentwas 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 sensitivitybecause it can manipulate electrons in atoms [2].

Harmonic motion occurs when a particular type of system vibrates around the balanced configuration. The systemconsists of objects that can be hung on a spring or floating on a liquid, molecular two atom, an atom in the crystallattice. There are many examples of harmonic motion in the worlds of microscopic and macroscopic. Requirementfor harmonic motion to occur is the presence of the restoring force that acts to restore to the balancedconfiguration if the system is disturbed, resulting in the corresponding mass inertia of objects beyond the balancedposition, so that the system oscillates continuously if there are no dissipative processes [3]. Harmonic oscillatingatom in the crystal has a wave function. Harmonic oscillation can be solved by using several methods, namely thesecond-order equations, generating functions, polynomial Hermitte and operators. Based on the wave functionand 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 notationfor each pair (u,v) with , ∈ is ⟨ | ⟩ ∈Applicable for all , , ∈ and , ∈ ∶

i. ⟨ | ⟩ ≥ 0 and ⟨ | ⟩ = 0 if and only if = 0ii. ⟨ | + ⟩ = ⟨ | ⟩ + ⟨ | ⟩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 becombined into


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⟨ + | ⟩ = ⟨ | ⟩ + ̅⟨ | ⟩for all , , ∈ , , ∈ .Let , ∈ . Then, is called orthogonal if and only if⟨ | ⟩ = 0It will be showed on the Heisenberg uncertainty relation of quantum mechanics with the Schwarz inequality.

Proposition 2.Let X is pre-Hilbert space. Then|⟨ | ⟩| ≤ ⟨ | ⟩ ⟨ | ⟩ , ∈In the form of norms, Schwarz inequality can be written:|⟨ | ⟩| ≤ ‖ ‖‖ ‖ , ∈ .Proof.Let ≠ 0. Then get 0 ≤ ⟨ − | − ⟩ = ⟨ | ⟩ − ⟨ | ⟩ − [⟨ | ⟩ − ⟨ | ⟩]Where =: ⟨ | ⟩⟨ | ⟩Proposition 3.Each 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 = 0. Then

‖ ‖ = ⟨ | ⟩ = ( ) ⟨ | ⟩ = | |‖ ‖for all ∈ , ∈ .Finally, the triangle inequality ‖ + ‖ ≤ ‖ ‖ + ‖ ‖ , ∈ ,taken from the Schwarz inequality. So for all , ∈‖ + ‖ = ⟨ + | + ⟩ = ⟨ | ⟩ + ⟨ | ⟩ + ⟨ | ⟩ + ⟨ | ⟩= ‖ ‖ + 2 ⟨ | ⟩ + ‖ ‖ ≤ ‖ ‖ + 2‖ ‖‖ ‖ + ‖ ‖ = (‖ ‖ + ‖ ‖) ,where Re z is real part from complex number z. From proposition 3 we can get notation and theorem to normspace on pre-Hilbert space is a norm space of equation 4.

Then, convergence → ℎ → ∞In the pre-Hilbert space X satisfies the equation‖ − ‖ → 0 ℎ → ∞.Proposition 4.Let X pre-Hilbert space. Then satisfactory

i. Continuous inner product, means that


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→ → ℎ → ∞corollary⟨ | ⟩ → ⟨ | ⟩ ℎ → ∞.

ii. Let M is dense of subset X. If⟨ | ⟩ = 0 ∈ ∈ ,ℎ = 0.

Proof: i. because limited, according to Schwarz inequality is|⟨ | ⟩ − ⟨ | ⟩| = |⟨ − | ⟩ + ⟨ | − ⟩| ≤ |⟨ − | ⟩| + |⟨ | − ⟩| ≤ ‖ − ‖‖ ‖ + ‖ ‖‖ − ‖→ 0 ℎ → ∞.Statement ii. Since M is dense of X, there is series in M so − in X with → ∞, satisfies⟨ | ⟩ = 0 for allSo ⟨ | ⟩ = 0. where = 0.

Definition 5.Hilbert 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, andii. Every Cauchy sequance with norm ‖u‖ is convergent.

If F = ℝor F = ℂ, then Xis called real Hilbert space orcomplex.

Proposition 6.Every 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 7.Let X is Hilbert space over the field F, and let L is linear subspace X. Then closurL from L is Hilbertspace with restriction inner product X on L.

Proof: First prove that is a linear space over F. Let , ∈ and , ∈ . Then there is sequences and in Lso → and → in with → ∞.

Let → ∞, if + ∈ for all , so + ∈ .

Restriction of inner product on X for subspace on X.

Let sequences Cauchy at . Then→ in with → ∞.

Since close, ∈ , then→ in with → ∞. (Prugovecki, 1971)

In this regard, there are two concepts of the sequence, the first row is called a Cauchy sequence or fundamentalsequence. The second, a sequence is said to converge to a vector member pre-Hilbert space when the tribes came


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closer to the ranks of the vector. A pre-Hilbert space is called a Hilbert space if every Cauchy sequence in the spaceis a convergent sequence. Each pre-Hilbert space with scalar product there is a Hilbert space with scalar productsuch 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 locationin 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 calledobservable. 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 quantumphysics with classical physics, in quantum physics that there is only a prediction, so that the statisticalmeasurement results only. ≔ ⟨ | ⟩, ∈ ( )and (∆ ) ≔ ‖ − ‖ , ∈ ( )Relations average value and dispersion (∆ ) on observable A on state . Since operator A is symmetry,then the average value of is real. ∆ = ‖ − ‖ ≥ 0.

iv. Dynamics . according equations ( ) = ℏ , ∈ 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 ( ℏ ) is a one-parametric unitary group generated by skew-adjointoperator− ℏ . If ∈ ( ), thenℏ ( ) = ( ) ∈ ℝ .

APPLICATION OF HARMONIC OSCILLATOR IN QUANTUM MECHANICS

In classical mechanics, a harmonic oscillator is expressed by a point mass, > 0, with motion = ( ) on ℝ whichis usually expressed in differential equations ( ) = − ( )For > 0. The total energy is given by = 2 + 2where ( ) ≔ ( ) is the momentum of the particle at time t. So that the harmonic oscillator is represented bythe Schrodinger equation, namely ℏ = − ℏ2 + 2 .Substitution to following equation → − ℏ → ℏ .


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Then look for the solution of equation = ( , ) with

| ( , )| = 1 ∈ ℝ.Using ( , ) = ( ) ℏ , stationary Schrodinger equation is obtained

= − ℏ2 + 2 ℝ.Let Hilbert space is defined as ≔ ℂ(ℝ) with inner product

⟨ | ⟩ = ( ) ( ) .Every unit vector ∈ is called particle state (harmonic oscillator), namely

⟨ | ⟩ = | ( , )| = 1Let −∞ ≤ < ≤ ∞. Define

| ( , )| ≔ probability to particle ininterval [a, b].Definition 8.Formalism Hamiltonianℋ:D(ℋ) ⊆ X → X harmonic oscillator is given by the equation

ℋϕ ≔= − ℏ2mϕ +mω x2 ϕwhereD(ℋ) ≔ S. S space is Hermitian function u defined

( ) ≔ (−1) , = 0,1,2, …,where

= 12 ( !) .Proposition 9.i. Operatorℋ simetricii. For all n = 0,1,2, …, ℋϕ = E ϕ ,

with ϕ (x) ≔ u x ,

x ≔ ℏmω , and


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E = ℏω n + 12 , n = 0,1,2, …,iii. Eigen function form {ϕ }complete orthonormal system in X.

In other word, a simple oscillation energy is quantized.

Proof: statement i. For all , ∈ , integrate probabilistic to be

′′ = lim→ ( ) ( ) − − ′ = − ′ = ′with ⟨ |ℋ ⟩ = ⟨ ℋ| ⟩ , ∈ (ℋ).Statement ii, = x polynomial (x). Obtained ∈ .

Definition 10.Operator H: D(H) ⊆ X → X defined by

≔ ⟨ | ⟩called Hamiltonian harmonic oscillator. Here, ϕ ∈ D(ℋ) if and only if

| ⟨ | ⟩ | < ∞.Proposition 11.i. Hamiltonian H: D(H) ⊆ X → X is self-adjointii. Operator H is extension formalism Hamiltonianℋ.

Proof:statement i. already proven

Statement ii. Let ∈ (ℋ)is ∈ . Where definition space S,ℋ ∈ . Since { } complete orthonormal systemin X, ℋ = ⟨ |ℋ ⟩ = ⟨ | ⟩ ,by symmetry in proposition 9. Then, over series convergence ∈ ( ). Sinceℋ ⊆ , then= , = 0,1,2, …,Suppose particle in state , then |⟨ | ⟩| = have probabilitas energy .Remark 12.(Dynamics harmonic)oscillator. If given∈ ℎ ⟨ | ⟩ = 1


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Suppose harmonic oscillator in state ψ at time t = 0.

( ) = ℏ , ∈ ℝ,Explicitly, expressed by

( ) = ℏ ⟨ | ⟩ ∈ ℝ.Convergence series in Hilbert space ≔ ℂ(ℝ).

Definition 13.Operator A: D(A) ⊆ X → X with( )( ) ≔ − ℏ ( )for all ∈ ℝcalledmomentum operator. Here ( ) ≔ { ∈ : ∈ }, with the same differential, generally into( )( ) ≔ ( ) ∈ ℝcalled operator position. Here ( ) ≔ { ∈ : ∈ }Remark 14.(Heisenberg Uncertainty Principle). If given the circumstances∈ ℎ ⟨ | ⟩ = 1 .with position average and disperse (∆ ) particle in state is given

= ⟨ | ⟩ = | ( )|and (∆ ) = ‖ − ‖ = ( − ) | ( )|Momentum average and disperse (∆ ) at state is= ⟨ | ⟩ (∆ ) = ‖ − ‖ .So, ∆ ∆ ≥ ℏ2

ACKNOWLEDGMENTS

Authors express sincere thanks to UniversitasSamudra, Langsa for funding to attend this conference. Authors alsothanks to Dean of Engineering Faculty for allowing to use laboratory.


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REFERENCES

2. van der Waerden, B.L., Sources of Quantum Mechanics, Dover Publications, Inc., New York (1967)3. Rosyid, M.F, MekanikaKuantum : Model Matematis Gejala Alam Mikroskopik Tinjauan Tak Relativistik, Jurusan Fisika

Universitas Gadjah Mada (2009).4. Beiser, A., Modern Physics, Inc., New York (1992).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), 529-575, (physics/9811050 v1)

overview on the hilbert space harmonics oscillator

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