For a free particle

2. Suppose a particle of mass, m, has energy E, and wave function: WE (x, t = 0) = Aeikx + Be-ikx What is WE(x, t)? Calculate the probability density of the particle when it has the wavefunction, E (x, t). If you wish to simply your answer algebraically, use this information: let A = aeia and B = beif and A = a - B. where the variables a, b, a, ß are all real! WARNING: Simplifying the equation is somewhat time consuming. Show/explain why p(x,t) is not normalizable. According to quantum theory, all physical systems must have an associated wavefunction that is normalizable. Explain why plane wave solutions to Schrodinger's equation are used in quantum theory, despite not being normalizable?

1. A particle moving in the positive x-direction encounters a finite step-function potential of magnitude V, at x = 0 as illustrated. a. b. The energy of the particle is E > Vo- I. ii. III. iii. PHYS 2305 Homework 7 II. iii. The energy of the particle is E 0). Identify terms as Pincident, reflected and transmitted- Apply boundary conditions to determine relationships between the wavefunction coefficients. Determine P(T) the probability of transmission and P(R) the probability of reflection for the particle. Write the wavefunction for the particle in region 1 (x 0). Identify terms as incident, reflected and transmitted- Apply boundary conditions to determine relationships between the wavefunction coefficients. Determine P(T) the probability of transmission and P(R) the probability of reflection for the particle. X

Q3. Consider an infinite potential well of width d. In transitions between neighboring values of n, particles of mass that is in a position state as: 2nx f(x.t) = e-iwot + -iwit sin d (a) Proof that f(x.t) is still normalized for all value of t. (b) Find the probability distribution P(x. t) = \f(x.t)[²

For a one-dimensional box, we assume that the particle is confined between rigid, unyielding walls separated by a distance L. Beyond this allowed region, the potential energy V is enormously large and the wave function must therefore be? a. -1 b. 0 c. 1 d. None of the above

A particle in a 3-dimensional quadratic box with box length L has an energy given by h² E = (n+n+n). The degeneracies of the first, second, and 8mL² third level are a. e. 1, 2, 3 1, 3, 3 b. 1, 3, 1 c. 3, 3, 3 d. 1, 2, 2

Q3. Consider an infinite potential well of width d. In transitions between neighboring values of n, particles of mass that is in a position state as: 2πχ sin e-iwit d TX f(x. t) = -e-iwot + sin (a) Proof that f (x. t) is still normalized for all value of t. (b) Find the probability distribution P(x.t) = |f(x. t)|²

B7

Please solve and answer the question correctly also be sure to have the correct units on the final answer. Thank you!!

Question A3 Consider the energy eigenstates of a particle in a quantum harmonic oscillator with frequency w. a) Write down expressions for the energies of the three lowest states. b) c) Sketch the potential for this system, along with the position of the three lowest energy levels. Add to your sketch the form of the wavefunction and the probability density in the three lowest energy states. [10 marks]