Question

Find the probability of finding an electron trapped in a one-dimensional infinite well of width \(2.00 \mathrm{nm}\) in the \(n=2\) state between 0.800 and \(0.900 \mathrm{nm}\) (assume that the left edge of the well is at \(x=0\) and the right edge is at \(x=2.00 \mathrm{nm}\) ).

Short answer

Answer: The probability of finding an electron in the n=2 state between 0.800 nm and 0.900 nm is approximately 0.049874.

Step-by-step solution

Write down the wave function for an electron in an infinite well

The wave function for an electron trapped in a one-dimensional infinite well of width L in the nth state is given by: $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$$ Here, L=2.00 nm is the width of the well, and n=2 is the energy state.

Square the wave function to get the probability density

To calculate the probability of finding an electron in a certain region, we need to find the probability density (the square of the wave function): $$\rho_n(x) = |\psi_n(x)|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right)$$

Write the probability density function for n=2 and L=2.00 nm

Substitute n=2 and L=2.00 nm into the probability density function: $$\rho_2(x) = \frac{2}{2.00 \mathrm{nm}}\sin^2\left(\frac{2\pi x}{2.00 \mathrm{nm}}\right)$$

Integrate the probability density function over the specified range

To find the probability of finding the electron between 0.800 nm and 0.900 nm, integrate the probability density function over this range: $$P = \int_{0.800 \mathrm{nm}}^{0.900 \mathrm{nm}} \rho_2(x) dx$$

Simplify and evaluate the integral

Use the antiderivative identity: $$\int \sin^2(ax) dx = \frac{1}{4a}\left(x - \frac{1}{2a}\sin(2ax)\right) + C$$ Substitute a=2π/2 and the limits of integration: $$P = \left[\frac{1}{4(2\pi/2)}\left(x - \frac{1}{2(2\pi/2)}\sin\left(2\pi x\right)\right)\right]_{0.800 \mathrm{nm}}^{0.900 \mathrm{nm}}$$ Evaluate the expression: $$P \approx 0.049874$$ The probability of finding an electron in the n=2 state between 0.800 nm and 0.900 nm is approximately 0.049874.

Key concepts

Probability in Quantum Systems

In the world of quantum mechanics, probability plays a crucial role in predicting the behavior of particles at the microscopic level. Unlike classical mechanics, where objects follow predictable paths, quantum systems are governed by probability.
To determine the probability of locating a particle, such as an electron, in a specified region, we rely on the concept of probability density derived from the wave function.
A wave function, often denoted by \( \psi(x) \), is a mathematical description of the quantum state of a particle.

• The square of the wave function, \( |\psi(x)|^2 \), gives us the probability density, which we use to find the probability of locating the particle in a given region.
• This probability density function helps us understand the likelihood of a particle's presence in a specific area within the constraints of a potential well or other boundary conditions.
• The integral of the probability density over the desired range provides the actual probability of the particle being found there.

These probabilistic interpretations are unique to quantum systems and highlight the inherently uncertain and fundamentally discrete nature of these systems.

Infinite Potential Well

The infinite potential well is a fundamental concept in quantum mechanics, serving as a model to study particles confined within a region. Imagine it as a box with infinitely high walls that a particle cannot escape from.

• In a one-dimensional infinite potential well, an electron, for example, is trapped between two impenetrable barriers at well-defined positions. For our exercise, these barriers are located at \( x = 0 \) and \( x = 2.00 \, \text{nm} \).
• The electron cannot leave this region, and its potential energy outside the walls is infinite, hence it must remain within the well's boundaries.

This confinement means the electron's wave function must satisfy boundary conditions, being zero at the walls. Consequently, only certain discrete energy levels, known as quantum states, are allowed for the electron.
The quantization of energy in the well signifies that the electron can exist only in specific "harmonic modes" of vibration within the box.

Wave Function Analysis

Wave functions are central to understanding quantum systems, as they contain all the information needed to describe a particle's state.

• For a particle in an infinite potential well, the wave function \( \psi_n(x) \) in the \( n \)-th quantum state is given by the formula \( \psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right) \), where \( L \) is the well's width and \( n \) is the state number.
• This mathematical function exhibits sine waves that signify the standing wave nature of the particle in the well, with nodes (points of zero amplitude) at the boundaries.
• The higher the quantum state \( n \), the greater the number of wavelengths fitting within the well, which corresponds to higher energy levels.

Wave function analysis involves finding these solutions and understanding their implications for the particle's behavior and probabilities within the confinement of an infinite potential well.

Electron Probability Density

The probability density function in quantum mechanics gives us insights into the likelihood of locating an electron at different positions within a potential well.

• For an electron in a one-dimensional infinite potential well, the probability density \( \rho_n(x) \) can be calculated by squaring the wave function: \( \rho_n(x) = |\psi_n(x)|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right) \).
• This function indicates that certain regions within the well have higher probabilities for finding the electron based on the squared sine function's oscillations.
• To find the probability of locating the electron in a specific range, say between \(0.800 \, \text{nm}\) and \(0.900 \, \text{nm}\), one must integrate the probability density over that interval.

This approach allows physicists to predict electron behavior in constrained systems and provides foundational understanding for atomic and molecular models.