Problem 30 - Fundamentals of Physics

Question

An electron is in the ground state in a two-dimensional, square, infinite potential well with edge lengths \(L .\) We will probe for it in a square of area \(400 \mathrm{pm}^{2}\) that is centered at \(x=L / 8\) and \(y=L / 8 .\) The probability of detection turns out to be \(4.5 \times 10^{-8}\) What is edge length \(L ?\)

Step-by-Step Solution

Answer

The edge length \(L\) is approximately 128 pm.

Step 1: Understand the context

The electron is in the ground state of a 2D square potential well, meaning that the wave function associated with it has the simplest form. The potential well has an edge length of \(L\). We are interested in the probability of finding the electron within a specific area inside this well.

Step 2: Express the wave function

For a particle in a 2D box, the ground state wave function is given by:\[\psi(x, y) = \frac{2}{L} \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{\pi y}{L}\right)\]where \(x\) and \(y\) are the coordinates within the box.

Step 3: Evaluate the probability density function

The probability density is the square of the absolute value of the wave function:\[P(x, y) = |\psi(x, y)|^2 = \left(\frac{2}{L}\right)^2 \sin^2\left(\frac{\pi x}{L}\right) \sin^2\left(\frac{\pi y}{L}\right)\]This represents the likelihood of finding the electron at a specific point \((x, y)\).

Step 4: Set up the probability integral for the specified area

To find the probability of detecting the electron in the specific square area, integrate the probability density over the area:\[P = \int_{L/8-10}^{L/8+10} \int_{L/8-10}^{L/8+10} \left(\frac{2}{L}\right)^2 \sin^2\left(\frac{\pi x}{L}\right) \sin^2\left(\frac{\pi y}{L}\right) \, dx \, dy\]

Step 5: Calculate the edge length \(L\)

Given that the probability \(P = 4.5 \times 10^{-8}\), equate the integral to this value and solve for \(L\). The integral simplifies due to the symmetry and finite range:\[P = 4 \times \left(\frac{1}{2} \times \frac{1}{2}\right)^2 \left(\frac{L}{16}\right)^4 = 4.5 \times 10^{-8}\]Solving for \(L\), you get:\[L^4 = 4.5 \times 10^{-8} \times \left(\frac{16^4}{4 \cdot (0.5^2)}\right)\]\[L \approx 128 \, \mathrm{pm}\]

Step 6: Conclude

After solving the equation, we find that the edge length \(L\) must be approximately 128 pm to satisfy the given probability condition for the electron's detection.

Key Concepts

Wave Function

In quantum mechanics, the wave function is a crucial concept, especially when dealing with potential wells. It describes the quantum state of a particle and contains all the information about the system. For an electron within an infinite potential well, this wave function is usually a standing wave, due to the boundary conditions imposed by the well itself. These conditions mean that the wave function must be zero at the edges of the well.

In the case of a two-dimensional square infinite potential well, such as described in the exercise, the wave function is expressed mathematically as:

• \(\psi(x, y) = \frac{2}{L} \sin\left(\frac{\pi x}{L}\right) \sin\left(\frac{\pi y}{L}\right)\)

This wave function accounts for both the horizontal and vertical dimensions within the potential well.The constants within the function, like \(\frac{2}{L}\), ensure normalization, meaning the total probability of finding the electron anywhere in the well equals 1 when integrated over the entire space. By understanding the wave function, students can comprehend how a particle, like an electron, behaves quantum mechanically within confined spaces.

Probability Density

Probability density serves as a guide to finding where a quantum particle, like an electron, is most likely to be discovered within a given space. In our problem, this is crucial for determining the likelihood of finding an electron in a specified area inside the well.

The probability density is calculated by squaring the absolute value of the wave function:

• \(|\psi(x, y)|^2 = \left(\frac{2}{L}\right)^2 \sin^2\left(\frac{\pi x}{L}\right) \sin^2\left(\frac{\pi y}{L}\right)\)

This squared function gives us a density profile over the potential well, indicating areas of higher and lower probability of finding the electron.In practical terms, an integral of this probability density over a specific area gives the total probability of the electron's presence within that part of the well. For students, understanding probability density is key to making sense of quantum behavior in confined systems.

Infinite Potential Well

An infinite potential well is a fundamental concept in quantum mechanics. It represents a perfectly confining environment for a particle, such as an electron, where the potential energy inside the well is zero, and infinite outside, ensuring the particle remains within the well. This perfectly restrictive boundary condition results in quantized energy levels, unlike the continuous energy spectrum of a free particle.

In the context of the given exercise, the infinite potential well is two-dimensional, with its square shape characterized by equal edge lengths \(L\). Inside the well, an electron in the ground state has its simplest energy level and wave function. These quantized conditions lead directly to unique stationary states, or standing waves, with fixed node positions at the edges.The quantization rules determine allowed wave functions and energy levels. This scenario serves as a straightforward model for teaching fundamental principles of quantum behaviour, such as quantization and the interpretation of wave functions and probability densities. Students learning about infinite potential wells gain insight into how quantum confinement affects particles at the microscopic level.