**Problem 4: Quantum Mechanics and Uncertainty Principle**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:**The problem involves a wavefunction that describes the quantum state of a particle. The function is defined over the entire real line for \(x\), suggesting it's a Gaussian function in form. The task is to check if the relationship between the standard deviations of position (\(\Delta x\)) and momentum (\(\Delta p\)) aligns with the uncertainty principle, which states that the product \(\Delta p \Delta x\) should be greater than or equal to \(\hbar/2\), where \(\hbar\) is the reduced Planck's constant. This problem requires knowledge of how to calculate the standard deviations of position and momentum from the wavefunction, potentially involving integrals over probability densities.

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4. A particle is in state described by the a wavefunction y=(at)1/4e-(ax2)/2 where -o

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Concept explainers

Rutherford's scattering experiment

Rutherford’s scattering experiment is also known as the gold foil experiment. This experiment disproved the plum pudding model of J.J. Thomson. In a gold foil experiment, high-energy alpha particles are made to strike a thin gold foil. The scattered alpha particles are detected using a detector. This experiment showed that the positive charge of the atom is at the center. Most of the atom’s mass is also confined to a small volume at the center. This region is known as the nucleus of the atom.

Question

**Problem 4: Quantum Mechanics and Uncertainty Principle**

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:**

The problem involves a wavefunction that describes the quantum state of a particle. The function is defined over the entire real line for \(x\), suggesting it's a Gaussian function in form. The task is to check if the relationship between the standard deviations of position (\(\Delta x\)) and momentum (\(\Delta p\)) aligns with the uncertainty principle, which states that the product \(\Delta p \Delta x\) should be greater than or equal to \(\hbar/2\), where \(\hbar\) is the reduced Planck's constant.

This problem requires knowledge of how to calculate the standard deviations of position and momentum from the wavefunction, potentially involving integrals over probability densities.

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