7. One electron is trapped in a one-dimensional square well potential with infinitely high sides. a. If you have a probe that has a width for electron detection Ax = 0.00350L in the x direction, for the first excited state ( n =2), what is the probability that the electron is found in the probe when it is centered at x = L/4,(hint: you can use an approximation for this - you do not need to do an integral)? b. What is the average number of electrons that you would detect using the probe described in part "b." centered at x= L/4, if the electron is in the first excited state (n = 2 ) for each experiment and you repeat the experiment N, = 100,000 times?
7. One electron is trapped in a one-dimensional square well potential with infinitely high sides. a. If you have a probe that has a width for electron detection Ax = 0.00350L in the x direction, for the first excited state ( n =2), what is the probability that the electron is found in the probe when it is centered at x = L/4,(hint: you can use an approximation for this - you do not need to do an integral)? b. What is the average number of electrons that you would detect using the probe described in part "b." centered at x= L/4, if the electron is in the first excited state (n = 2 ) for each experiment and you repeat the experiment N, = 100,000 times?
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![**Quantum Mechanics Problem: Electron in a One-Dimensional Square Well**
**Problem Statement:**
One electron is trapped in a one-dimensional square well potential with infinitely high sides.
**Questions:**
a. If you have a probe that has a width for electron detection \(\Delta x = 0.00350L\) in the x direction, for the first excited state \((n = 2)\), what is the probability that the electron is found in the probe when it is centered at \(x = L/4\)? (Hint: you can use an approximation for this - you do not need to do an integral).
b. What is the average number of electrons that you would detect using the probe described in part "a." centered at \(x = L/4\), if the electron is in the first excited state \((n = 2)\) for each experiment and you repeat the experiment \(N_p = 100,000\) times?
**Solution Explanation:**
**Part (a):**
To determine the probability that the electron is found in the probe when it is centered at \(x = L/4\), we need to evaluate the wave function of the electron in the first excited state \((n = 2)\) at \(x = L/4\). The wave function for a particle in a one-dimensional infinite potential well is given by:
\[
\Psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)
\]
For the first excited state \((n = 2)\), the wave function becomes:
\[
\Psi_2(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{2\pi x}{L}\right)
\]
To find the probability of locating the electron within the small region \(\Delta x\) around \(x = L/4\), we use the probability density \(|\Psi_2(x)|^2\):
\[
\text{Probability density} = |\Psi_2(x)|^2 = \left(\sqrt{\frac{2}{L}} \sin\left(\frac{2\pi x}{L}\right)\right)^2 = \frac{2}{L} \sin^2\left(\frac{2\pi x}{L}\right)
\]
Substituting \(x = L/](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb067f6a6-d6d0-4a81-90c2-26a75bd86046%2F1e950094-fa77-452e-bf04-6f5ee393bfeb%2Fod0r6eo_processed.png&w=3840&q=75)
Transcribed Image Text:**Quantum Mechanics Problem: Electron in a One-Dimensional Square Well**
**Problem Statement:**
One electron is trapped in a one-dimensional square well potential with infinitely high sides.
**Questions:**
a. If you have a probe that has a width for electron detection \(\Delta x = 0.00350L\) in the x direction, for the first excited state \((n = 2)\), what is the probability that the electron is found in the probe when it is centered at \(x = L/4\)? (Hint: you can use an approximation for this - you do not need to do an integral).
b. What is the average number of electrons that you would detect using the probe described in part "a." centered at \(x = L/4\), if the electron is in the first excited state \((n = 2)\) for each experiment and you repeat the experiment \(N_p = 100,000\) times?
**Solution Explanation:**
**Part (a):**
To determine the probability that the electron is found in the probe when it is centered at \(x = L/4\), we need to evaluate the wave function of the electron in the first excited state \((n = 2)\) at \(x = L/4\). The wave function for a particle in a one-dimensional infinite potential well is given by:
\[
\Psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)
\]
For the first excited state \((n = 2)\), the wave function becomes:
\[
\Psi_2(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{2\pi x}{L}\right)
\]
To find the probability of locating the electron within the small region \(\Delta x\) around \(x = L/4\), we use the probability density \(|\Psi_2(x)|^2\):
\[
\text{Probability density} = |\Psi_2(x)|^2 = \left(\sqrt{\frac{2}{L}} \sin\left(\frac{2\pi x}{L}\right)\right)^2 = \frac{2}{L} \sin^2\left(\frac{2\pi x}{L}\right)
\]
Substituting \(x = L/
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