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A particle is in state described by the a wavefunction y=(at)1/4e-(ax2)/2 where -∞< x <∞. Verify that the value of the product ApAx for this wavelength is consistent with the predictions from the uncertainty principle.

A particle is in state described by the a wavefunction y=(at)1/4e-(ax2)/2 where -∞< x <∞. Verify that the value of the product ApAx for this wavelength is consistent with the predictions from the uncertainty principle.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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**Problem 4: Quantum Mechanics Uncertainty Verification** A particle is in a state described by the wavefunction \( \psi = (\alpha \pi)^{1/4} e^{-(\alpha x^2)/2} \) where \( -\infty \leq x \leq \infty \). Verify that the value of the product \( \Delta p \Delta x \) for this wavelength is consistent with the predictions from the uncertainty principle. **Explanation:** This problem involves confirming that the uncertainty in momentum (\(\Delta p\)) and position (\(\Delta x\)) matches the quantum mechanical Heisenberg Uncertainty Principle. The given wavefunction is a Gaussian, which often represents the ground state of a quantum harmonic oscillator. The task is to calculate these uncertainties from the wavefunction and show their product aligns with theoretical predictions.
Transcribed Image Text:**Problem 4: Quantum Mechanics Uncertainty Verification** A particle is in a state described by the wavefunction \( \psi = (\alpha \pi)^{1/4} e^{-(\alpha x^2)/2} \) where \( -\infty \leq x \leq \infty \). Verify that the value of the product \( \Delta p \Delta x \) for this wavelength is consistent with the predictions from the uncertainty principle. **Explanation:** This problem involves confirming that the uncertainty in momentum (\(\Delta p\)) and position (\(\Delta x\)) matches the quantum mechanical Heisenberg Uncertainty Principle. The given wavefunction is a Gaussian, which often represents the ground state of a quantum harmonic oscillator. The task is to calculate these uncertainties from the wavefunction and show their product aligns with theoretical predictions.
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