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Mathematical Formulation of Quantum Mechanics


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short seminar on application of linear algebra

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Mathematical Formulation of Quantum Mechanics

  2. 2. Contents• Quantum Mechanics• Basic Principles of Quantum Mechanics• Postulates of Quantum Mechanics : Analogy with Linear Algebra• Schrödinger picture of quantum mechanics• Summary
  3. 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. 4. BASIC PRINCIPLES OF QUANTUM MECHANICS1. Physical States2. Physical Quantities3. Composition4. Dynamics
  5. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 15. THANKSThe orbitals of an electron in a hydrogen atom areeigenfunctions of the energy.