Q2) [5pts] Use the Gaussian trial function e-bx². to obtain the lowest upper bound on the ground-state energy of the linear potential: V(x) = a |x|, and show that this Emin is 1/3 E min - 3 ħ²a² 2\ 2mm (1) Find A. (2) Find and , and then (3) Find Emin
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- 2i+1 i+1 |- +> + 3 [recall, |+ -> means that particle #1 is in the |+> state (usual Z basis) and #2 is in the |-> state.] A) Show that this state is already normalized. B) Is this state separable or entangled? C) A measurement of S, is made on particle #1. What are the possible results and with what probabilities? D) A measurement of Sz is made on particle #2. What are the possible results and with what probabilities? E) Calculate the expectation value of the correlation function between these two measurements . (Don't use matrices -- use probabilities!)please answer c) only 2. a) A spinless particle, mass m, is confined to a two-dimensional box of length L. The stationary Schrödinger equation is - +a) v(x, y) = Ev(x, y), for 0 < r, y < L. The bound- ary conditions on ý are that it vanishes at the edges of the box. Verify that solutions are given by 2 v(1, y) sin L where n., ny = 1,2..., and find the corresponding energy. Let L and m be such that h'n?/(2mL²) = 1 eV. How many states of the system have energies between 9 eV and 24 eV? b) We now consider a macroscopic box (L of order cm) so that h'n?/(2mL?) ~ 10-20 eV. If we define the wave vector k as ("", ""), show that the density of states g(k), defined such that the number of states with |k| between k and k +dk is given by g(k)dk, is Ak 9(k) = 27 c) Use the expression for g(k) to show that at room temperature the partition function for the translational energy of a particle in a macroscopic 2-dimensional box is Z1 = Aoq, where 2/3 oq = ng = mk„T/2nh?. Hence show that the average…Q 3: What is the value of the commutator [X, Px]?
- c) How does the classical kinetic energy of the free electron compare in magnitude with the result you obtained in the previous part?0 A physical system is described by a two-dimensional vector space with Hamiltonian operator Ĥ given by Ĥ = (_) where a is a constant. At time t = 0, the system is prepared in state (t = 0)) = -i2.5 0 determine the expectation value (Ŝ) at time t = πħ/(4x). O a. 2.17 O b. -2.50 O c. -1.25 O d. 2.50 O e. 5.00 0 (¹). For operator $ = (2 i2.5