Planck's constant = 6.626068 × 10-34 m2 kg / s
A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured
Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose – that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j[3 2+i][ 2-i 1 ]
Given the state of a system at t and the forces and constraints to which it is subject, there is an equation, ‘Schrödinger's equation’, that gives the state at any other time U|vt> -> |vt′>. The important properties of U for our purposes are that it is deterministic, which is to say that it takes the state of a system at one time into a unique state at any other, and it is linear, which is to say that if it takes a state |A> onto the state |A′>, and it takes the state |B> onto the state |B′>, then it takes any state of the form α|A> + β|B> onto the state α|A′> + β|B′>.
Carrying out a "measurement" of an observable B on a system in a state |A> has the effect of collapsing the system into a B-eigenstate corresponding to the eigenvalue observed. This is known as the Collapse Postulate. Which particular B-eigenstate it collapses into is a matter of probability, and the probabilities are given by a rule known as Born'sRule:prob(bi) = |<A|B=bi>|2.
First few hydrogen atom orbitals; cross section showing color-coded probability density for different n=1,2,3 and l="s","p","d"; note: m=0The picture shows the first few hydrogen atom orbitals (energy eigenfunctions). These are cross-sections of the probability density that are color-coded (black=zero density, white=highest density). The angular momentum quantum number l is denoted in each column, using the usual spectroscopic letter code ("s" means l=0; "p": l=1; "d": l=2). The main quantum number n (=1,2,3,...) is marked to the right of each row. For all pictures the magnetic quantum number m has been set to 0, and the cross-sectional plane is the x-z plane (z is the vertical axis). The probability density in three-dimensional space is obtained by rotating the one shown here around the z-axis.
Mathematical Formulation of Quantum Mechanics
MATHEMETICAL FORMULATION OFQUANTUM MECHANICS
Contents• Quantum Mechanics• Basic Principles of Quantum Mechanics• Postulates of Quantum Mechanics : Analogy with Linear Algebra• Schrödinger picture of quantum mechanics• Summary
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.
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.
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.
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
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′>.
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.
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.
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.
• 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| >
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.
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.
THANKSThe orbitals of an electron in a hydrogen atom areeigenfunctions of the energy.