Chapter 40: Problem 2
A free particle moving in one dimension has wave function $$\Psi(x, t) = A[e^{i(kx-vt)} - e^{i(2kx-4vt)}]$$ where \(k\) and \(\omega\) 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 \(t = 2\pi/\omega\). (c) Calculate \(v_{av}\) as the distance the maxima have moved divided by the elapsed time. Compare your result to the expression \(v_{av} = (\omega_2 - \omega_1)/(k_2 - k_1)\) from Example 40.1.
Short Answer
Step by step solution
Simplify the Wave Function at t = 0
Find the Probability Density Function at t = 0
Determine x for Maximum Probability at t = 0
Simplify the Wave Function at t = 2\pi/\omega
Find the Probability Density Function at t = 2\pi/\omega
Calculate Average Velocity of Maximum Positions
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wave Function
Probability Density
Average Velocity
Complex Exponentials
- **Concisely encode information**: Amplitude and phase are represented compactly.
- **Facilitate Calculations**: Operations like differentiation and integration are simplified.