Problem 8 - Chemistry The Central Science

Question

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn below, is \(\psi(x)=\sin x\) from \(x=0\) to \(x=2 \pi\) . (a) Sketch the probability density, \(\psi^{2}(x),\) from \(x=0\) to \(x=2 \pi .(\mathbf{b})\) At value or values of \(x\) will there be the greatest probability of finding the electron? (c) What is the probability that the electron will be found at \(x=\pi ?\) What is such a point in a wave function called? [Section 6.5\(]\)

Step-by-Step Solution

Answer

The probability density function for the given one-dimensional system is \(\psi^2(x) = \sin^2 x\) from \(x=0\) to \(x=2\pi\). The greatest probability of finding the electron occurs at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\). The probability of finding the electron at \(x=\pi\) is 0, and such a point is called a node in the wave function.

Step 1: We are given the wave function \(\psi(x)=\sin x\). To calculate the probability density \(\psi^2(x)\), we square the wave function: \[\psi^2(x) = (\sin x)^2 = \sin^2 x\] This is the probability density function from \(x=0\) to \(x=2\pi\). #Step 2: Sketch the probability density function#

The probability density function for the given range is \(\psi^2(x) = \sin^2 x\), which is a square of the sine function. To sketch it, plot the sine function and the resulting squared sine function between \(x=0\) and \(x=2\pi\). The squared sine function will always be positive and have peaks where the sine function reaches its maximum and minimum amplitude. #Step 3: Find the value(s) with the greatest probability of finding the electron#

Step 2: To find the value(s) with the greatest probability, analyze the probability density function, \(\psi^2(x) = \sin^2 x\). We are interested in the maximum value(s) of this function between \(x=0\) and \(x=2\pi\). The maximum values occur when the sine function is at its maximum or minimum amplitude. Since the sine function varies between -1 and 1, its maximum and minimum values are 1 and -1. Therefore, maximum probability density occurs when: \[\sin^2 x = 1\] Solving this equation, we have the following values of x within the given range: \[x = \frac{\pi}{2},\frac{3\pi}{2}\] #Step 4: Determine probability at x=pi and name the point#

To find the probability density at \(x=\pi\), we compute \(\psi^2(x)\) at that point: \[\psi^2(\pi) = \sin^2 \pi = 0\] The probability density at \(x=\pi\) is 0. Such a point, where the probability is 0, is called a node in a wave function. In conclusion, the probability density function is given by \(\psi^2(x) = \sin^2 x\) from \(x=0\) to \(x=2\pi\). The greatest probability of finding the electron occurs at \(x=\frac{\pi}{2}\) and \(x=\frac{3\pi}{2}\). The probability of finding the electron at \(x=\pi\) is 0, and such a point is called a node in the wave function.

Key Concepts

Probability Density

Understanding the probability density in quantum mechanics is crucial for predicting where an electron might be found in a given space. It's a concept that tells us the likelihood of locating a quantum particle in a specified position. When dealing with wave functions, such as \( \psi(x) = \sin x \) from \( x=0 \) to \( x=2\pi \), we can find the probability density by squaring the wave function to obtain \( \psi^2(x) \) or \( \sin^2 x \).

Here's why the squaring matters: When you square the amplitude of the wave function, which may have positive or negative values, the resulting probability density is always positive or zero. This aligns with the physical interpretation that probabilities cannot be negative. A graph of \( \psi^2(x) \) will show a pattern of peaks and troughs corresponding to the higher and lower probabilities of finding the electron along the \( x \) coordinate.

Sine Function

The sine function plays a pivotal role in generating the shape of wave functions in quantum mechanics. Represented mathematically as \( \sin x \), the sine function has a wave-like appearance, oscillating between 1 and -1. In the context of quantum wave functions, each cycle of a sine wave could represent the probability amplitude for the location of a particle.

For instance, the given exercise uses \( \sin x \) as the wave function of an electron in a one-dimensional space. The important property of the sine function, in this case, is that it crosses the axis at certain points, called nodes, where the electron’s probability density will consequently be zero. When \( \psi(x) = \sin x \) is squared to generate \( \psi^2(x) \) or \( \sin^2 x \), the negative parts of the sine wave are converted to positive values, illustrating that the probability of finding an electron is never negative.

Nodes in Wave Function

Nodes are significant points in a wave function where the probability of finding an electron is exactly zero. In our example with \( \psi(x) = \sin x \), nodes occur at values of \( x \) where the sine of \( x \) is zero. For a simple sine function, these nodes happen at integer multiples of \( \pi \).

Specifically, in our exercise, when \( \psi(x) = \sin x \) is evaluated at \( x=\pi \) for instance, it results in a node since \( \sin(\pi) = 0 \). Nodes are important because they help us understand the quantum behavior of particles. They represent areas where the electron cannot be found, and identifying these nodes enables us to visualize the confinement and potential movement of particles within a given space.

Electron Localization

Electron localization refers to the tendency of an electron to be found in certain regions of space within an atom or molecule. In quantum mechanics, the electron is described by a wave function, and the square of this function gives us the probability density, directly tied to the concept of electron localization.

In the exercise’s context, the greatest probability of finding the electron will occur where the probability density reaches its maximum, corresponding to the peaks of the \( \sin^2 x \) graph. In this sine function scenario, these peaks occur at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) — regions where the electron is most likely to be localized. Describing electron localization is an integral part of understanding the behavior of quantum systems and anticipating the outcomes of experiments and reactions in chemistry and physics.