Say that the wave function for some particle is given by the symmetric "tent" wave function as in the provided image. Show that ψ(x) is properly normalized. What is for the particle? Calculate the uncertainty △x in the particle's position.
Say that the wave function for some particle is given by the symmetric "tent" wave function as in the provided image. Show that ψ(x) is properly normalized. What is for the particle? Calculate the uncertainty △x in the particle's position.
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Chapter1: Units, Trigonometry. And Vectors
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Say that the wave function for some particle is given by the symmetric "tent" wave function as in the provided image.
Show that ψ(x) is properly normalized. What is <x> for the particle? Calculate the uncertainty △x in the particle's position.
(Note: the wave function is an even function)
![The function \(\Psi(x)\) is defined as follows:
\[
\Psi(x) =
\begin{cases}
\sqrt{\frac{12}{a^3} \left(\frac{a}{2} - |x|\right)} & \text{if } |x| \leq \frac{a}{2} \\
0 & \text{if } |x| > \frac{a}{2}
\end{cases}
\]
This describes a piecewise function with two conditions based on the absolute value of \(x\).
1. For \(|x| \leq \frac{a}{2}\), \(\Psi(x)\) is a square root function that includes the parameters \(a\) and \(x\).
2. For \(|x| > \frac{a}{2}\), \(\Psi(x)\) is zero.
There are no graphs or diagrams to describe.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F772ba77e-5b00-472d-814c-750947c36afa%2F098fd039-3024-48fe-a063-e5e48982b482%2Fofr2ycq_processed.png&w=3840&q=75)
Transcribed Image Text:The function \(\Psi(x)\) is defined as follows:
\[
\Psi(x) =
\begin{cases}
\sqrt{\frac{12}{a^3} \left(\frac{a}{2} - |x|\right)} & \text{if } |x| \leq \frac{a}{2} \\
0 & \text{if } |x| > \frac{a}{2}
\end{cases}
\]
This describes a piecewise function with two conditions based on the absolute value of \(x\).
1. For \(|x| \leq \frac{a}{2}\), \(\Psi(x)\) is a square root function that includes the parameters \(a\) and \(x\).
2. For \(|x| > \frac{a}{2}\), \(\Psi(x)\) is zero.
There are no graphs or diagrams to describe.
Transcribed Image Text:**Figure 2.27: A "tent" wave function.**
This diagram represents a triangular or "tent" wave function. The horizontal axis is labeled as \( x \), while the vertical axis is labeled as \( \Psi \). The graph is symmetrical around the vertical axis (\( x = 0 \)) and extends from \( -a/2 \) to \( a/2 \) on the horizontal axis. The peak of the tent is at the origin, where \( x = 0 \), indicating the maximum value of the wave function. The shape resembles an isosceles triangle with its base on the x-axis and its peak pointing upwards.
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