Question 1. In a one-dimensional problem, consider a particle whose wave function is:

ψ(x) = N(e^ipox/h)/√(x² + a²)

where a and P_o are real constants and N is a normalization coefficient.

a. Determine N so that ψ(x) is normalized.

b. The position of the particle is measured. What is the probability of finding a result between -a/√3 and +a/√3 ?

c. Calculate the mean value of the momentum of a particle which has ψ(x) for its wave function.

Question 2. Consider, in a one-dimensional problem, a particle of mass m whose wave function at time t is ψ/(x, t).

a. At time 1, the distance d of this particle from the origin is measured. Write, as a function of ψ(x, t), the probability p(d_o ) of finding a result greater than a given length d_o. What are the limits of P(d_o) when d_o → 0 and d_o → ∞?

b. Instead of performing the measurement of question a, one measures the velocity v of the particle at time t. Express, as a function of ψ(x, t), the probability of finding a result greater than a given value v_o.

Question 3. The wave function of a free particle, in a one-dimensional problem, is given at time t = 0 by:

ψ(x, 0) - N _-∞∫^+∞dk e^-|k|k_oe^ikx

where k_o and N are constants.

a. What is the probability p(p₁, 0) that a measurement of the momentum, performed at time t = 0, will yield a result included between - p₁ and + p₁? Sketch the function p(p₁, 0).

b. What happens to this probability p(p₁, t) if the measurement is performed at time t? Interpret.

c. What is the form of the wave packet at time t = 0? Calculate for this time the product ΔX.ΔP; what is your conclusion? Describe qualitatively the subsequent evolution of the wave packet.

Question 4. Spreading of a free wave packet

Consider a free particle.

a. Show, applying Ehrenfest's theorem, that < X > is a linear function of time, the mean value < P > remaining constant.

b. Write the equations of motion for the mean values < X² > and (XP + PX>. Integrate these equations.

c. Show that, with a suitable choice of the time origin, the root-mean¬square deviation ΔX is given by:

(ΔX)² = 1/m² (ΔP)₀²t² + (ΔX)₀²

where (ΔX)_o and (ΔP)_o are the root-mean-square deviations at the initial time. How does the width of the wave packet vary as a function of time (see 3-c of complement G₁)? Give a physical interpretation.

Question 5. Particle subject to a constant force

In a one-dimensional problem, consider a particle of potential energy V(X) = - fX, where f is a positive constant [V(X) arises, for example, from a gravity field or a uniform electric field].

a. Write Ehrenfest's theorem for the mean values of the position X and the momentum P of the particle. Integrate these equations; compare with the classical motion.

b. Show that the root-mean-square deviation ΔP does not vary over time.

c. Write the Schrodinger equation in the { |p> } representation. Deduce from it a relation between ∂/∂t|< p|ψ(t)>|² and ∂/∂t|< p|ψ(t)>|². Integrate the equation thus obtained; give a physical interpretation.