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short seminar on application of linear algebra

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- 1. MATHEMETICAL FORMULATION OFQUANTUM MECHANICS
- 2. Contents• Quantum Mechanics• Basic Principles of Quantum Mechanics• Postulates of Quantum Mechanics : Analogy with Linear Algebra• Schrödinger picture of quantum mechanics• Summary
- 3. QUANTUM MECHANICS Branch of physics dealing with physical phenomena at microscopic scales, where the action is on the order of the Planck constant.• The dual - particle-like and wave-like behaviour and interactions of energy and matter.
- 4. BASIC PRINCIPLES OF QUANTUM MECHANICS1. Physical States2. Physical Quantities3. Composition4. Dynamics
- 5. BASIC PRINCIPLES OF QUANTUM MECHANICS• Physical States Every physical system is associated with a HilbertSpace H. Every unit vector in the space corresponds to apossible pure state of the system. The vector is represented by a function known asthe wave-function, or ψ-function.
- 6. BASIC PRINCIPLES OF QUANTUM MECHANICS• Physical Quantities Hermitian operators in the Hilbert spaceassociated with a system. Their eigenvalues represent the possibleresults of measurements of these quantities.
- 7. BASIC PRINCIPLES OF QUANTUM MECHANICS• Composition The Hilbert space associated with a complexsystem is the tensor product of those associatedwith the simple system. H1⊗H2
- 8. BASIC PRINCIPLES OF QUANTUM MECHANICS• Dynamics Contexts of type 1: ‘Schrödingers equation’ : gives the state at any othertime U|vt> → |vt′> U is deterministic U is linearIf U takes a state |A> onto the state |A′>, and it takes thestate |B> onto the state |B′>,then it takes any state of the form α|A> + β|B> onto thestate α|A′> + β|B′>.
- 9. BASIC PRINCIPLES OF QUANTUM MECHANICS• Dynamics Contexts of type 2 ("Measurement Contexts"):Collapse Postulate.The eigenstate getting collapsed is a matter ofprobability, given by a rule known asBorns Rule: prob(bi) = |<A|B=bi>|2.
- 10. POSTULATES OF QUANTUM MECHANICS : ANALOGY WITH LINEAR ALGEBRA• Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product . < ᵩᵩ | >• Rays (one-dimensional subspaces) in H are associated with states of the system.
- 11. POSTULATES OF QUANTUM MECHANICS : ANALOGY WITH LINEAR ALGEBRA• Physical observables are represented by Hermitian matrices on H.The expected value (in the sense of probabilitytheory) of the observable A for the system instate represented by the unit vector |ᵩ Є H > is < ᵩ |ᵩ |A >• A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues.
- 12. • More generally, a state can be represented by a so-called density operator, ᵩwhich is a trace class, nonnegative self-adjoint operator normalized to be of trace 1.• The expected value of A in the state is tr(A ᵩ)• If is the orthogonal projector onto the one- dimensional subspace of H spanned by , then• tr(A ᵩ = < ᵩ ᵩ ) |A| >
- 13. Schrödinger picture of quantum mechanicsthe dynamics is given as follows:• The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If | ᵩ > denotes the state of the system at any one (t)time t,where H is a densely-defined self-adjoint operator, called thesystem Hamiltonian , i is the imaginary unit and h isthe reduced Planck constant. As an observable, H correspondsto the total Energy of the system.
- 14. SummaryQuantum system --- Mathematical FormulationPossible states --- Unit VectorsState Space --- Hilbert SpaceObservable --- Self- adjoint Linear OperatorEach eigenstate of an observable corresponds to aneigenvector of the operator, and the associatedeigenvalue corresponds to the value of the observablein that eigenstate.
- 15. THANKSThe orbitals of an electron in a hydrogen atom areeigenfunctions of the energy.

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