A free particle moving in one dimension has wave function $\psi (x,t)=A[e^{i(kx-\omega t)}-e^{i(2kx-4\omega t)}]$ where $k$ and $v$ are positive real constants.
(a) At $t=0$, what are the two smallest positive values of $x$ for which the probability function $\mid \psi (x,t){\mid }^2$ is a maximum?
(b) Repeat part (a) for time $t=\frac{2\pi }{\omega }$.

Verified step by step guidance

1. Step 1: Start by understanding the given wave function ψ(x,t) = A[e^(i(kx-ωt)) - e^(i(2kx-4ωt))]. The probability density function is given by |ψ(x,t)|^2, which is the square of the magnitude of the wave function. To find the maxima of |ψ(x,t)|^2, we need to analyze the interference pattern created by the two exponential terms.
2. Step 2: At t = 0, substitute t = 0 into the wave function. This simplifies ψ(x,t) to ψ(x,0) = A[e^(i(kx)) - e^(i(2kx))]. The probability density function becomes |ψ(x,0)|^2 = |A|^2 |e^(i(kx)) - e^(i(2kx))|^2. Expand the magnitude squared using the formula |a - b|^2 = (a - b)(a* - b*), where * denotes the complex conjugate.
3. Step 3: Simplify the expression for |ψ(x,0)|^2. Use Euler's formula e^(iθ) = cos(θ) + i*sin(θ) to rewrite the exponential terms. After simplification, the probability density function will depend on trigonometric terms involving kx. Identify the conditions for maxima by setting the derivative of |ψ(x,0)|^2 with respect to x equal to zero and solving for x.
4. Step 4: For part (b), repeat the process for t = 2π/ω. Substitute t = 2π/ω into the wave function ψ(x,t) = A[e^(i(kx-ωt)) - e^(i(2kx-4ωt))]. This simplifies ψ(x,t) to ψ(x,2π/ω) = A[e^(i(kx - 2π)) - e^(i(2kx - 8π))]. Follow the same steps as in part (a) to find the probability density function |ψ(x,2π/ω)|^2 and determine the conditions for maxima.
5. Step 5: Solve the equations obtained in steps 3 and 4 to find the two smallest positive values of x for which |ψ(x,t)|^2 is a maximum at t = 0 and t = 2π/ω. These values will depend on the periodicity of the trigonometric terms and the interference pattern created by the wave function.

Key Concepts

Wave Function

In quantum mechanics, a wave function, denoted as ψ(x,t), describes the quantum state of a particle. It contains all the information about the system and is used to calculate probabilities of finding a particle in a particular state. The square of the absolute value of the wave function, |ψ(x,t)|^2, gives the probability density, indicating where the particle is likely to be found.

Probability Density

Probability density is a measure that describes the likelihood of finding a particle in a specific position at a given time. For a wave function ψ(x,t), the probability density is calculated as |ψ(x,t)|^2. Peaks in this function correspond to positions where the particle is most likely to be detected, which is crucial for solving the given problem.

Phase and Interference

The wave function in the question consists of two components that can interfere with each other. The terms e^i(kx-ωt) and e^i(2kx-4ωt) represent waves with different wavelengths and frequencies. The interference between these waves leads to a modulation of the probability density, creating regions of constructive and destructive interference, which are essential for determining the positions of maximum probability.