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At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image.
Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find <E> for the particle at time t. (Hint: <E> can be obtained by inspection, without an integral)
![The image shows a mathematical expression for a potential energy function \( V(x) \). The function is defined piecewise as follows:
\[
V(x) =
\begin{cases}
0, & 0 < x < L \\
\infty, & \text{elsewhere}
\end{cases}
\]
Explanation:
- The function \( V(x) \) represents a potential barrier.
- It is equal to 0 within the interval \( 0 < x < L \).
- Outside of this interval, the potential function \( V(x) \) is infinite (\(\infty\)), indicating an impenetrable barrier. This type of potential is often used in quantum mechanics to describe a particle in an infinite potential well, where the particle is free to move within the interval \( 0 < x < L \) but cannot exist outside of it.](https://content.bartleby.com/qna-images/question/918af633-b9fe-4531-accf-614f64c7f1b6/fdeebe55-1ed6-494b-a3c8-0dfa74a83c30/820mkh_processed.png)
Transcribed Image Text:The image shows a mathematical expression for a potential energy function \( V(x) \). The function is defined piecewise as follows:
\[
V(x) =
\begin{cases}
0, & 0 < x < L \\
\infty, & \text{elsewhere}
\end{cases}
\]
Explanation:
- The function \( V(x) \) represents a potential barrier.
- It is equal to 0 within the interval \( 0 < x < L \).
- Outside of this interval, the potential function \( V(x) \) is infinite (\(\infty\)), indicating an impenetrable barrier. This type of potential is often used in quantum mechanics to describe a particle in an infinite potential well, where the particle is free to move within the interval \( 0 < x < L \) but cannot exist outside of it.
![The function is defined as follows:
\[
\Psi(x) =
\begin{cases}
\left( \frac{1+i}{2} \right) \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right) + \frac{1}{\sqrt{2L}} \sin\left(\frac{2\pi x}{L}\right), & 0 < x < L \\
0, & \text{elsewhere}
\end{cases}
\]
This describes a piecewise function, where \(\Psi(x)\) takes a specific form involving trigonometric functions and complex numbers for \(0 < x < L\), and is zero elsewhere.](https://content.bartleby.com/qna-images/question/918af633-b9fe-4531-accf-614f64c7f1b6/fdeebe55-1ed6-494b-a3c8-0dfa74a83c30/rond95l_processed.png)
Transcribed Image Text:The function is defined as follows:
\[
\Psi(x) =
\begin{cases}
\left( \frac{1+i}{2} \right) \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right) + \frac{1}{\sqrt{2L}} \sin\left(\frac{2\pi x}{L}\right), & 0 < x < L \\
0, & \text{elsewhere}
\end{cases}
\]
This describes a piecewise function, where \(\Psi(x)\) takes a specific form involving trigonometric functions and complex numbers for \(0 < x < L\), and is zero elsewhere.
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