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.

Q5: Consider a particle of mass m in a two-dimensional box having side length L and L₁ with L = 2L, and V=0 in the box, ∞ outside; Suppose V=10 J in the box. What effect has this on the eigenvalues? the eigen functions?

Normalize the following wavefunction and solve for the coefficient A. Assume that the quantum particle is in free-space, meaning that it is free to move from x € [-, ∞]. Show all work. a. Assume: the particle is free to move from x € [-0, 00] b. Wavefunction: 4(x) = A/Bxe¬ßx²

Consider the following wave function. TT X a = B sin(- (x) = E 2 π.χ. a −) + C · sin(² a. Does this function describes a particle-in-a-box acceptable wave function? Name the conditions to be fulfilled. b. Is this function an eigenfunction of the total energy operator H when H is the Hamilton operator.

Consider a finite potential step with V = V0 in the region x < 0, and V = 0 in the region x > 0 (image). For particles with energy E > V0, and coming into the system from the left, what would be the wavefunction used to describe the “transmitted” particles and the wavefunction used to describe the “reflected” particles?

Consider a particle in the one-dimensional box with the following wave function: (x,0) = Cx(a − x) 4. Normalize this wavefunction. 5. Express (x, 0) as a superposition of eigenfunctions (x). 6. What is the probability of each of these eigenfunctions? 7. Verify that the sum of all probabilities in (6) is unity. Hint: Use 8. When the system is at 9. When the system is at 10. When the system is at 11. When the system is at 12. When the system is at 13. When the system is at 14.What is (x, t) 15. What is (2) ? dt' 16. What is (d)? dt +∞ Σ n=0 1 (2n + 1)6 (x, 0), what is (x)? (x, 0), what is (²)? (x, 0), what is (p)? (x, 0), what is (p²)? (x, 0), what is Ax? (x, 0), what is Ap? = π6 960

Below is a figure that depicts the potential energy of an electron (a finite square well), as well as the energies associated with the first two wave-functions. a) Sketch the first two stationary wavefunctions (solutions to the Schrődinger equation) for an electron trapped in this fashion. Pay attention to detail! Use the two dashed lines as x-axes. U(x) E2 E1 b) If the potential energy were an infinite square well (not finite well as shown above), what would the energy of the first two allowed energy levels be (i.e., E1 and E2). Write the expressions in terms of constants and a (the width of the wellI) and then evaluate numerically for a = 6.0*1010 m. [If you don't remember the formula, you can derive it by using E = h²k²/2m, together with the condition on À = 2a/n.] c) Let's say I adjust the width of the well, a, such that E1 = 3.5 ev. In that case, calculate the wavelength (in nanometers) of a photon that would be emitted in the electron's transition from E2 to E1. [Remember: hc =…

Let Y(t) = X(t) - aX(t + s), where X(t) is a wide-sense stationary random process. a. Determine whether Y(t) is also a wide-sense stationary random process. b. Find the pdf of Y(t) if X(t) is also a Gaussian random process.