Part 1 a. Calculate the relative probability distribution, PR(X), for a 1-kg particle initially at rest and attached to a spring (a harmonic oscillator) of spring constant k = 2 N/m and an initial displacement from equilibrium of 1 m. b. Now enter in this function into the text box and set the proper classical limits by clicking the "change limits and evaluate" button. What is the normalized classical probability distribution, PN(X)? You must normalize (a) by doing the integral.

Calculate the probability that a particle will be found between 0.65L and 0.67L in a box of length L, for the case where the wavefunction is: a)ψ1 and b)ψ2. Assume that the wave function is constant in this range, so the probability is ψ2 dx. Show full and complete procedure in a clear way. DO NOT SKIP ANY STEP

Consider a dosed system Consisting closed beitha of N molocules with total evergy E. fach molocule has a nn.degenirate grund State uwith energy 3ero and a two fold degenorate encited state with energy E. a. what is The to tal number of microstates S2CE) arailable to The øys tem when E=ME where M i& an integer. b. Compute The entropy of The systim in the limit N>I and C. Calculate The heat Capaciiy of The System & sketch it D. what is the maximum energy of The SysTem for any posítiue tempreture!

The wave function of a particle in two dimensions in plane polar coordinates is given by: T ¥(r,0) = A. r. sin0exp[- where A and ao are positive real constants. 1. Find the constant A A using the normalization condition in the form SS |¥(r,0)|²rdrd0 = 1 2. Calculate the expectation values of r, and ². 3. Assuming that the momentum operator in plane polar coordinate is giving in the form ħa p = calculate the expectation values of p and p². 1 Ər' 4. Find the standard deviations of r and p and show that their product is consistent with the Heisenberg uncertainty principle.

Consider a single electron confined to a one-dimensional quantum well device of length L = 0.5 nm. The quantum well device acts as a “trap” for the electron 1.What are the boundary conditions for this system? Apply them to show that                           ψn(x) = Asin(nπx/L),           n = 1,2,3,...   (check image) 2.Normalize the wave function to find the constant A.   3. Sketch ψ1, ψ2, and ψ3, as well as |ψ1|2, |ψ2|2, and |ψ3|2, and evaluate the energy levels E1, E2, and E3 in eV.   4. Suppose the particle is in the first excited state. What is the probability of finding the particle between x = L/4 and x = 3L/4?   5. Suppose, instead of one electron, we trap five electrons in the quantum well. Draw an energy-level diagram to show the electron configuration of the ground state. What is the ground state energy?

PROBLEM 2 Calculate the probability distribution of momenta p for a ld oscillator in the ground state (n = 0). Calculate the mean square dispersions (x2), (p²), and the product (r2)(p²).

A particle in the ground state of the quantum harmonic oscillator, which is described by the following wavefunction for all values of x :  wave function found in image  1.Determine the energy of this particle, and show that its wavefunction satisfies the appropriate time-dependent Schro ̈dinger equation. 2.Determine the maximum displacement from x = 0 that a classical particle with the same en- ergy is allowed to have in this potential. Finally, write down an expression (without evaluating the integral) for the probability of finding the quantum particle outside of the classically allowed region.