* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Atomic theory wikipedia, lookup

Ferromagnetism wikipedia, lookup

Noether's theorem wikipedia, lookup

Coupled cluster wikipedia, lookup

Hydrogen atom wikipedia, lookup

Quantum electrodynamics wikipedia, lookup

Dirac bracket wikipedia, lookup

Hidden variable theory wikipedia, lookup

Bra–ket notation wikipedia, lookup

Aharonov–Bohm effect wikipedia, lookup

Quantum state wikipedia, lookup

Density matrix wikipedia, lookup

Zero-point energy wikipedia, lookup

Second quantization wikipedia, lookup

Perturbation theory (quantum mechanics) wikipedia, lookup

Topological quantum field theory wikipedia, lookup

Particle in a box wikipedia, lookup

Path integral formulation wikipedia, lookup

Wave–particle duality wikipedia, lookup

Casimir effect wikipedia, lookup

Renormalization group wikipedia, lookup

Compact operator on Hilbert space wikipedia, lookup

Self-adjoint operator wikipedia, lookup

Coherent states wikipedia, lookup

Renormalization wikipedia, lookup

Quantum field theory wikipedia, lookup

Theoretical and experimental justification for the Schrödinger equation wikipedia, lookup

History of quantum field theory wikipedia, lookup

Relativistic quantum mechanics wikipedia, lookup

Symmetry in quantum mechanics wikipedia, lookup

Molecular Hamiltonian wikipedia, lookup

Transcript

Chapter II Klein Gordan Field Lecture 2 Books Recommended: Lectures on Quantum Field Theory by Ashok Das A First Book of QFT by A Lahiri and P B Pal Creation and Annihilation operators Consider -----(1) Conjugate Momentum --(2) We can also write -----(3) ----(4) Check that right side is independent of t. We can write following commutation relations between the operators and , similar to harmonic oscillator: -------(5) We now calculate following commutation relation and use (5) ----(6) Also, Similarly, ---(7) We can also prove the commutators -----(8) By inserting the expressions of above Operators in terms of field operators i.e., Using(3) and (4). Perform above exercise. Now we calculate Hamiltonian using field expansions (1) and (2) ---(9) Where, 1st term of (9) ----(10) Where we used -(11) 2nd term of (9) -----(12) 3rd term of (9) -----(13) Using (10) , (12) and (13) in (9) ---(14) Which is H for KG field. Note the following ---(15) ----(16) Considering analogy between SHO and above discuss field theory, can be interpreted as annihilation operator and as creation operator. Positive energy component of the field Annihilate the quantum whereas the negative energy component create the quantum. This quantum is called a particle of positive energy. Normal Ordering We can defined the ground state ----(27) The state is normalized such that Consider Hamiltonian ----(28) Using commutator Eq (27) become --(29) Using (27) and (29), ground state energy will be ----(30) Which is infinite. Infinite number of oscillators are contributing to energy. Need to redefine Hamiltonian Energy differences are physical quantities not absolute energy. We can subtract the infinities and can Redefine the Hamiltonian. A consistent approach for this is known as normal ordering. When we have the expression involving the Product of annihilation and creation operators , we defined the normal ordered product by Moving all annihilation operators to the right Of all creation operators as if commutators Were zero For example: ------(31) Normal ordered Hamiltonian -----(32) Where number operator N(k) ---(33) Total number operator ---(34) We can find -------(35) Also ---------(36) Exercise: (i) Find the linear momentum operator for the Klein Gordon field using the Noehter Theorem i.e. find (ii) Use the field expansions in above and show That (iii) What will normal ordered form for P? Do we need Normal ordering for momentum? Note: For a gneral field as KG field is scalar i.e., And therefore, = 0 i.e., spin of particles Described by KG eq is zero.