An electron is moving as a free particle in the $-x$-direction with momentum that has magnitude $4.50\times {10}^{-24}$ kg*m/s. Let $k_2=3k_1=3k$. At $t=0$, the probability distribution func­tion $\mid \Psi (x,t){\mid }^2$ has a maximum at $x=0$.
(a) What is the smallest positive value of $x$ for which the probability distribution function has a maximum at time $t=\frac{2\pi }{\omega }$, where $\omega =h{k}^2/2m$?
(b) From your result in part (a), what is the average speed with which the probability distribution is moving in the $+x$­-direction?

Verified step by step guidance

1. Step 1: Start by understanding the problem. The electron is described as a free particle, and its wavefunction Ψ(x, t) is a superposition of two plane waves with wave numbers k_1 = k and k_2 = 3k. The probability distribution |Ψ(x, t)|^2 exhibits interference, leading to maxima and minima. The goal is to find the smallest positive x where the probability distribution has a maximum at t = 2π/ω, and then calculate the average speed of the probability distribution in the +x-direction.
2. Step 2: Write the wavefunction Ψ(x, t) as a superposition of two plane waves: Ψ(x, t) = A[e^(i(kx - ω_1t)) + e^(i(3kx - ω_2t))], where ω_1 = ℏk^2/2m and ω_2 = ℏ(3k)^2/2m. The probability distribution is given by |Ψ(x, t)|^2, which depends on the interference between the two waves.
3. Step 3: To find the maxima of |Ψ(x, t)|^2, calculate the phase difference between the two waves. The phase difference is Δϕ = (3kx - ω_2t) - (kx - ω_1t) = 2kx - (ω_2 - ω_1)t. For maxima, Δϕ must be an integer multiple of 2π: 2kx - (ω_2 - ω_1)t = 2nπ, where n is an integer.
4. Step 4: At t = 2π/ω, substitute ω = ℏk^2/2m and calculate ω_2 - ω_1. Using ω_1 = ℏk^2/2m and ω_2 = ℏ(3k)^2/2m, we find ω_2 - ω_1 = ℏ(9k^2 - k^2)/2m = 4ℏk^2/m. Substitute this into the phase condition: 2kx - (4ℏk^2/m)(2π/ω) = 2nπ. Simplify to solve for x in terms of n, k, and other constants.
5. Step 5: For the smallest positive x, set n = 1 and solve for x. Once x is determined, calculate the average speed of the probability distribution. The average speed is given by v_avg = x/t, where t = 2π/ω. Substitute the expression for x and simplify to find v_avg in terms of k, ℏ, and m.

Key Concepts

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity. In this context, the electron's momentum is given as 4.50 x 10^-24 kg*m/s, indicating its motion in the -x-direction. Understanding momentum is crucial for analyzing the motion of particles and their interactions, especially in quantum mechanics.

Wave Function and Probability Distribution

In quantum mechanics, the wave function Ψ(x, t) describes the quantum state of a particle, and its squared magnitude |Ψ(x, t)|^2 represents the probability distribution of finding the particle at position x at time t. The maxima of this distribution indicate the most likely positions of the particle, which is essential for solving the problem regarding the position of the electron at a given time.

Angular Frequency (ω)

Angular frequency ω is a measure of how quickly an oscillating system changes its phase, defined as ω = 2π/T, where T is the period. In this problem, ω is related to the wave number k and the mass m of the electron, influencing the time evolution of the wave function. Understanding ω is key to determining the time-dependent behavior of the probability distribution.