(5) Work out the commutation relation:
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- Find the poisson brackets [F,G] for the functions: F = q1 + q2pỉ and G = q + 2p 3 Answer:(c) Consider a system of two qubits with canonical basis states {|0) , |1)}. Write down an example for a two- qubit density matrix corresponding to a separable pure state and an example for a two-qubit density matrix corresponding to an entangled pure state.2) ( Consider an electron trapped in a one-dimensional anharmonic potential. The Hamiltonian for this system is given as: H mw?£? + 2m + B&3 . Regarding the cubic term as being small, apply non- degenerate perturbation theory by letting the perturbation be: A, = ß2³. а) Calculate to first order the ground state energy of this anharmonic oscillator. b) ( | Calculate to second order the ground state energy of this anharmonic oscillator. c) ( Find the lowest order correction to the ground state wave function.
- 9. Prove the Potential V (r) and Momentum are Hermitians?5:A): Consider a system is in the state = 3 8x . Find the eigenvalue of £, ?2-) Hamiltonian operator 1 1 (pỉ + p²) +÷mw²(x? + x3) 2m Consider a system with two identical particles. Find the energy spectrum of the system and determine its degeneracy discuss.
- -ax (ii) Show that Y, = A,e¯* is an eigenfunction of the simple harmonic 1 ocillator Hamiltonian above when a = 2h Vkm. Find the corresponding eigenvalue. Interpret the result.(a) Using Dirac notation, write down the definition of a projection operator and that of a density operate and state the differences between the two.4. Show that the momentum operator in the position representation is given in the form: d er p=-ih- + f(x) dx Where f(x) is a real function which does not affect the value of the commutator