Suppose we had a classical particle in a frictionless box, bouncing back and forth at constantspeed. The probability density of the position of the particle in soma box of length L is given by:0ans-fawr (7)p(x)=0x < 0A sin² (7) 0 La. Sketch the probability density as a function of positionb. What must A be in order for p(x) to be normalized? Remember that you are welcome touse resources to solve integrals such as Wolfram Alpha, a table of integrals etc.

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Answered: Suppose we had a classical particle in…

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Evaluate the E expressions for both the Classical (continuous, involves integration) and the Quantum (discrete, involves summation) models for the energy density u, (v).
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Consider the one-dimensional step-potential V (x) = {0 , x < 0; V0 , x > 0}(a) Calculate the probability R that an incoming particle propagating from the x < 0 region to the right will reflect from the step.(b) Calculate the probability T that the particle will be transmitted across the step.(c) Discuss the dependence of R and T on the energy E of the particle, and show that always R+T = 1.[Hints: Use the expression J = (-i*hbar / 2m)*(ψ*(x)ψ′(x) − ψ*'(x)ψ(x)) for the particle current to define current carried by the incoming wave Ji, reflected wave Jr, and transmitted wave Jt across the step.For a simple plane wave ψ(x) = eikx, the current J = hbar*k/m = p/m = v equals the classical particle velocity v. The reflection probability is R = |Jr/Ji|, and the transmission probability is T = |Jt/Ji|. You need to write and solve the Schrodinger equation in regions x < 0 and x > 0 separately, and connect the solutions via boundary conditions at x = 0 (ψ(x) and ψ′(x) must be…
A qubit is in state |) = o|0) +₁|1) at time t = 0. It then evolves according to the Schrödinger equation with the Hamiltonian Ĥ defined by its action on the basis vectors: Ĥ0) = 0|0) and Ĥ|1) = E|1), where E is a constant with units of energy. a) Solve for the state of the qubit at time t. b) Find the probability to observe the qubit in state 0 at time t. Explain the result by referring to the way that the time-evolution transforms the Bloch sphere.
(d) Prove that for a classical particle moving from left to right in a box with constant speed v, the average position = (1/T) ff x(t) dt = L/2, where T L/v is the time taken to move from left to right. And = : (1/T) S²x² (t) dt L²/3. Hint: Only consider a particle moving from left x = 0 to right x = L = and do not include the bouncing motion from right to left. The results for left to right are independent of the sense of motion and therefore the same results apply to all the bounces, so that we can prove it for just one sense of motion. Thus, the classical result is obtained from the Quantum solution when n >> 1. That is, for large energies compared to the minimum energy of the wave-particle system. This is usually referred to as the Classical Limit for Large Quantum Numbers.