Problem 13 - Fundamentals of Physics

Question

A one-dimensional infinite well of length \(200 \mathrm{pm}\) con- tains an electron in its third excited state. We position an electrondetector probe of width \(2.00 \mathrm{pm}\) so that it is centered on a point of maximum probability density. (a) What is the probability of detection by the probe? (b) If we insert the probe as described 1000 times, how many times should we expect the electron to materialize on the end of the probe (and thus be detected)?

Step-by-Step Solution

Answer

The probability of detection is approximately 2%, and we expect about 20 detections in 1000 attempts.

Step 1: Understand the Setup

We have an infinite potential well with a length of 200 picometers containing an electron. The electron is in its third excited state, which is the fourth energy level (since the ground state is considered the first state). We need to find the probability that an electron detector probe, centered at a maximum probability density, detects the electron.

Step 2: Determine Electron Wavefunction

The wavefunction \[\psi_n(x) = \sqrt{\frac{2}{L}} \sin \left(\frac{n\pi x}{L}\right)\]is the solution for a particle in a one-dimensional infinite well of length \(L\). Here, \(n = 4\) since the electron is in the third excited state.

Step 3: Find the Probability Density

The probability density function is given by:\[\lvert \psi_4(x) \rvert^2 = \left( \frac{2}{L} \right) \sin^2 \left( \frac{4\pi x}{L} \right)\]The maximum probability density occurs at \(x = \frac{L}{8}, \frac{3L}{8}, \frac{5L}{8}, \frac{7L}{8}\).

Step 4: Determine the Probe Position

The probe is centered at one of the maximums of the probability density. Let's choose \(x = \frac{L}{8}\), which is 25 picometers from the left wall of the well.

Step 5: Calculate the Detection Probability

To find the probability of detecting the electron with the probe of width 2 pm centered at \(x = \frac{L}{8}\):\[P = \int_{24}^{26} \lvert \psi_4(x) \rvert^2 \, dx = \int_{24}^{26} \left( \frac{2}{L} \right) \sin^2 \left( \frac{4\pi x}{L} \right) \, dx\]Evaluate the integral using a calculator or software to find the probability.

Step 6: Calculate Expected Number of Detections

Multiply the probability by the number of attempts (1000) to find the expected number of detections:\[E = 1000 \times P\]

Step 7: Conclusion

After evaluating the integral and computations, substitute the numerical value of \(P\) to find the expected number of detections, \(E\).

Key Concepts

Infinite Potential Well

In quantum mechanics, the infinite potential well is a fundamental concept used to explore how particles behave in confined spaces. Imagine a particle, like an electron, trapped in a narrow box where the walls are insurmountable barriers. This box restricts the electron's movement entirely within its confines.

Key characteristics include:

• The potential energy inside the box is zero, while outside, it is infinite. This setup means the electron cannot escape the box.
• The box, or well, has a definite length, denoted by "L," which in this example is 200 picometers.

Understanding this scenario is crucial for studying quantum states because it offers a simplified model to analyze particle behavior under quantum constraints.

Probability Density

Probability density provides information about where an electron is likely to be located within a defined space. In quantum mechanics, it is not possible to determine the exact location of an electron. Instead, we assess the likelihood of finding it in particular regions.

The probability density is derived from the wavefunction \(\psi_n(x)\), and for the infinite potential well, the formula is given by \(|\psi_n(x)|^2\). This mathematical expression helps us:

• Identify points where the electron is most likely to be found (maxima of probability density).
• Understand that these probabilities are maxima because of the sin squared term in the formula, which shows periodic increases and decreases.

In this scenario, placing the probe at maxima points maximizes the chances of detecting the electron.

Electron Wavefunction

The electron wavefunction is a cornerstone of quantum mechanics, describing the state of the electron in the potential well. It is a solution to the Schrödinger equation for the system.

In this problem, the wavefunction for an electron in a one-dimensional infinite well is \(\psi_n(x) = \sqrt{\frac{2}{L}} \sin \left(\frac{n\pi x}{L}\right)\). For the electron in the third excited state (fourth energy level), \(n = 4\).

• The wavefunction provides amplitude information, influencing where the electron might be.
• It incorporates quantization, highlighting that only specific wave patterns (energy levels) are permissible within the well.

Analyzing the wavefunction helps us comprehend how electrons behave in confined spaces.

Energy Levels

Energy levels in an infinite potential well arise from constraining an electron's movement and defining discrete states or quantization. These levels correspond to different electron states, with more nodes (zero points) appearing as the energy level increases.

Each energy level is denoted by a quantum number "n," with the ground state being \(n = 1\). In the current exercise, the electron is said to be in the fourth energy level, hence, \(n = 4\).

Key characteristics of energy levels include:

• Energy levels are quantized, meaning the electron can only occupy certain states with fixed energy.
• As "n" increases, the electron resides higher in energy and has more nodes in its wavefunction.
• The spacing between energy levels in the well grows tighter, which reflects the increasing complexity of electron behavior in higher states.

Understanding energy levels allows us to predict electron locations and interaction based on quantum mechanics principles.