Introduction Path to quantization 1 Operators in Hilbert

Introduction • Path to quantization: 1. Operators in Hilbert space (Heisenberg, Schrödinger, Dirac, …) 2. Path integrals (Feynman) 3. Phase space formulation (Wigner, Weyl, Moyal, …) A state of classical identical particle system can be described by a phase-space distribution. Time evolution of obeys Boltzmann equation. Quasi probability distributions not a true probability density as it takes on negative values for some quantum states! Quantum version Description of state of EM field in terms of coherent states rather than photon number states Coherent states are not orthogonal and are over complete

• Classical probability density (integral over phase space of delta functions) ? Average value of is the Fourier transform of the average value of • Quantum Fock states, coherent states, position eigenstates, … • Baker-Hausdorff relation

The momentum operator is the generator of translation Phase space distribution is the Fourier transform of this average Wigner function (1932) Complete information about state of the system!

Some properties Prob. distribution in position space • The integral on momentum gives the marginal probability in real space • The integral on position gives the marginal probability in momentum space • Coherent state basis (instead of position basis) Complex amplitude position momentum Wigner function Symmetric characteristic function

Examples of Wigner functions 1. Coherent state 2. Number state (tutorial problem T 15) Circular symmetric gaussian with standard deviation Laguerre polynomial Oscillates from positive to negative Highly non classical! effect of the nonlocality of the quantum state associated to the quantum interference phenomenon Quantum coherent versus classical coherent light Optical and Quantum Electronics 33: 239 -252, 2001

Glauber Sudarshan P function • Density operator of a mixture of coherent state • Normalized Prob. to be in the state “Probability” distribution (generalized functions…) Can be negative also!