Question

Find the probability of finding an electron trapped in a onedimensional 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 the electron in the n=2 state between 0.800 and 0.900 nm is approximately 0.049 or 4.9%.

Step-by-step solution

Write down wavefunction for n=2 state

First, we need to find the wavefunction for the electron in the n=2 state within the given well. In general, the wavefunction \(\Psi_n(x)\) of a particle in a 1D well is given by: \(\Psi_n(x) = \sqrt{\frac{2}{L}} \sin{(\frac{n \pi x}{L})}\) For n=2, \(\Psi_2(x) = \sqrt{\frac{2}{L}} \sin{(\frac{2 \pi x}{L})}\)

Calculate the probability density function

To calculate the probability density function \(|\Psi_2(x)|^2\), we will square the modulus of \(\Psi_2(x)\): \(|\Psi_2(x)|^2 = (\sqrt{\frac{2}{L}} \sin{(\frac{2 \pi x}{L})})^2\)

Integrate the probability density function over the given interval

We will now integrate \(|\Psi_2(x)|^2\) over the given interval, from x=0.800 to x=0.900 nm. \(\int_{0.800}^{0.900} |\Psi_2(x)|^2 dx = \int_{0.800}^{0.900} (\sqrt{\frac{2}{L}} \sin{(\frac{2 \pi x}{L})})^2 dx\)

Perform integration and find the final result

Now, we can input the given width of the well, L=2 nm, and perform the integration: \(\int_{0.800}^{0.900} (\sqrt{\frac{2}{2}} \sin{(\frac{2 \pi x}{2})})^2 dx = \int_{0.800}^{0.900} (\sin{(\pi x)})^2 dx\) Using integration by parts and simplifying, we get: \(P(0.800 \le x \le 0.900) = \int_{0.800}^{0.900} (\sin{(\pi x)})^2 dx = \frac{1}{2} (x - \frac{\sin{(2\pi x)}}{2\pi})\Big|_{0.800}^{0.900}\) Now, we can evaluate the integral: \(P(0.800 \le x \le 0.900) = \frac{1}{2} \left(\left(0.900 - \frac{\sin{(2\pi\cdot 0.900)}}{2\pi}\right) - \left(0.800 - \frac{\sin{(2\pi\cdot 0.800)}}{2\pi}\right)\right) \approx 0.049\) So, the probability of finding the electron in the n=2 state between 0.800 and 0.900 nm is approximately 0.049 or 4.9%.

Key concepts

wavefunction

The wavefunction is a fundamental concept in quantum mechanics that describes the quantum state of a particle or system. It is typically represented by the symbol \(\Psi(x)\) or \(\phi(x)\), and it contains all the information about a particle's position, momentum, energy, and other physical properties. The wavefunction is a complex function, meaning it can have both a magnitude and a phase.

For a particle in a one-dimensional infinite potential well, the wavefunction takes a specific form, given by a sine function that depends on the quantum number \(n\), the particle's position \(x\), and the length of the well \(L\). The wavefunction for the \(n=2\) state, which was demonstrated in the exercise, is \(\Psi_2(x) = \sqrt{{2}{L}} \sin{(\frac{2 \pi x}{L})}\), representing the state of an electron in a well with a quantized energy level associated with \(n=2\).

probability density function

The probability density function (PDF) in quantum mechanics is a tool used to calculate the likelihood of finding a particle in a given position within the space it can occupy. It is obtained by taking the square of the modulus of the wavefunction, denoted as \(|\Psi(x)|^2\). The PDF reflects the probability per unit length at any point \(x\) within the potential well.

In the context of the exercise, the probability density function for the \(n=2\) state is \(|\Psi_2(x)|^2 = (\sqrt{{2}{L}} \sin{(\frac{2 \pi x}{L})})^2\). This expression helps determine the likelihood of finding an electron at specific points within the confines of an infinite potential well. The given problem involves calculating this likelihood over a particular interval, which is an application of the probability density function in quantum mechanics.

quantum mechanics integration

Quantum mechanics integration is a crucial process used for determining probabilities associated with the wavefunction. In the given exercise, integrating the probability density function over a region gives us the probability of finding the electron within that region. When you integrate the PDF \(|\Psi_2(x)|^2\) from \(x=0.800\) nm to \(x=0.900\) nm, as shown in the step-by-step solution, you are effectively summing up all the infinitesimal probabilities within this range to find the overall probability.

Mathematically, this is depicted as \(\int_{0.800}^{0.900} |\Psi_2(x)|^2 dx\), which requires knowledge of integral calculus to solve. In quantum mechanics, these integrals often involve complex functions, but they are vital for predicting experimental outcomes and understanding the behavior of particles at the quantum level.

infinite potential well

An infinite potential well is an idealized model where a particle, such as an electron, is confined within a region of space where the potential energy is zero. Outside this region, the potential energy is considered to be infinitely large, making it impossible for the particle to be found there. The boundaries of the well are perfectly rigid and impenetrable.

This model helps to understand quantization of energy levels, as seen in the exercise, where the electron can only occupy discrete energy states. The width of the well \(L\) is a crucial parameter as it determines the spatial extent over which the wavefunction can exist and affects the quantization of energy levels. In the exercise, the width of the well is \(2.00\ nm, and calculations are done under the assumption that the electron cannot exist outside this range, hence the name 'infinite' potential well.