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- What do you know about the difference between Newtonian, Lagrangian and Hamiltonian in describing the motion of particles !Demonstrate that in an electromagnetic field, the gauge transformation transfers the L to an equivalent Lagrangian L', where L' = L+ dFG,1 and F(q, t) is a function of generalised coordinates (q:) and time t. Calculate the generalised momentum and Hamiltonian of the charged particle travelling in an electromagnetic field using the aforementioned Lagrangian.Evaluate the commutator [Â,B̂] of the following operators.
- Consider an ideal gas system composed of Krypton atoms at a temperature of 302.15 K and 116.64 kPa in pressure. The atomic mass of Krypton is 83.8 Da. Given that the total volume of the system is 1 Liter: a. Calculate the total Hamiltonian of the system. b. Calculate the relative fluctuation (coefficient of variation) in the HamiltonianConsider a particle of spin s = 3/2. (a) Find the matrices representing the operators S^ x , S^ y ,S^ z , ^ Sx 2 and ^ S y 2 within the basis of ^ S 2 and S^ z (b) Find the energy levels of this particle when its Hamiltonian is given by ^H= ϵ 0 h 2 ( Sx 2−S y 2 )− ϵ 0 h ( S^ Z ) where ϵ 0 is a constant having the dimensions of energy. Are these levels degenerate? (c) If the system was initially in an eigenstate Ψ0=( 1 0 0 0) , find the state of the system at timeFind the real component if the complex number a + bi is raised to m if a = 7.4, b = 4, and m = 5.
- In Poincare transformation if scalar field is invariant under translation, then prove that generator of translation is momentum 4 vector.According to Ehrenfest's theorem, the time evolution of an expectation value <A>(t) follows the Ehrenfest equations of motion (d/dt)<A>(t) = (i/[hbar])<[H,A]>(t). For the harmonic oscillator, the Hamiltonian is given by H = p2/2m + m[omega]2x2/2. a) Determine the Ehrenfest equations of motion for <x> and <p>. b) Solve these equations for the initial conditions <x> = x0, <p> = p0, where x0 and p0 are real constants. Better formatted version of the question attached.Prove the following: if the Hamiltonian is independent of time, then ∆E doesn't change in time. Show work and be explicit to prove the statement.