Problem 7 - Chemistry The Central Science

Question

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn at the top of the next page, is \(\psi(x)=\sin x\) from \(x=0\) to \(x=2 \pi\). Sketch the probability density, \(\psi^{2}(x),\) from \(x=0\) to \(x=2 \pi\). (b) At what 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 is given by \(\psi^2(x) = (\sin x)^2\). The maximum probability of finding the electron occurs at \(x = \pi/2\) and \(x = 3\pi/2\). The probability of finding the electron at \(x=\pi\) is 0, and such a point is called a node.

Step 1: Calculate the probability density, \(\psi^2(x)\)

To calculate the probability density, square the wave function: \[\psi^2(x) = (\sin x)^2\]

Step 2: Sketch the probability density function

To sketch the probability density function, observe that the function \((\sin x)^2\) has a similar shape as the sine function, but lies entirely in the positive quadrant. The function starts at 0, increases to 1 at \(x=\pi/2\), decreases to 0 at \(x=\pi\), increases to 1 at \(x=3\pi/2\), and finally decreases to 0 at \(x=2\pi\). Thus, the shape of the graph is a series of peaks and valleys.

Step 3: Determine the maximum points of the probability density function

To find the maximum points of the probability density function, \(\psi^2(x) = (\sin x)^2\), we differentiate the function with respect to x and set the result equal to 0. \[\frac{d (\sin^2 x)}{dx} = 0\] We find that: \[\frac{d (\sin^2 x)}{dx} = 2 \sin x \cos x\] Setting this equal to zero, we find that: \[2 \sin x \cos x = 0\] This implies that either \(\sin x = 0\) or \(\cos x = 0\). For \(\sin x = 0\), the possible solutions within the interval \(0 \le x \le 2\pi\) are \(x = 0, \pi, 2\pi\). For \(\cos x = 0\), the possible solutions within the interval \(0 \le x \le 2\pi\) are \(x = \pi/2, 3\pi/2\). The maximum probability of finding the electron occurs at \(x = \pi/2\) and \(x = 3\pi/2\).

Step 4: Calculate the probability of finding the electron at \(x=\pi\)

To find the probability of finding the electron at \(x = \pi\), we evaluate the probability density function, \(\psi^2(x)\), at \(x = \pi\): \[\psi^2(\pi) = (\sin\pi)^2 = 0\] The probability of finding the electron at \(x=\pi\) is 0. Such a point in a wave function is called a node.

Key Concepts

Wave Function

In quantum mechanics, the wave function, denoted as \( \psi(x) \) for a one-dimensional case, is a mathematical representation of the quantum state of a system. It is a core concept that encapsulates how the quantum state of a particle, like an electron in our exercise, varies with respect to position. The wave function can be complex, but for this fictitious one-dimensional system, it is \( \psi(x)=\sin x \) within the interval from \( x=0 \) to \( x=2\pi \).

This wave function is crucial as it allows us to calculate important physical properties, such as the probability of finding a particle at a certain position, which is given by squaring the wave function to obtain the probability density.

Probability Density

Probability density \( \psi^2(x) \) is a key concept that expresses how the probability that a particle is present varies along different points in space. For our exercise, we calculate it by squaring the original wave function: \( \psi^2(x) = (\sin x)^2 \). Sketched, it forms a series of peaks and valleys, with zeros at \( x=0, \pi, \) and \( 2\pi \) indicating the electron is least likely to be found at these positions.

Understanding the graph of the probability density allows students to visualize where an electron is most and least likely to be found in a given space, leading to a better grasp of physical implications of the quantum state.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties at the atomic and subatomic levels. It differs from classical mechanics as it incorporates principles such as quantization of energy, wave-particle duality, and the uncertainty principle.

In the context of our exercise, quantum mechanics governs how wave functions are used to describe the behavior of electrons. The squared wave function, in particular, embodies one of the quantum mechanical predictions about how we can calculate the likelihood of finding an electron at a given point in space.

Nodes in Wave Functions

Nodes in wave functions represent specific points where the probability of finding a particle is exactly zero. They are important in understanding the quantum mechanical behavior of particles. In our exercise, \( x=\pi \) is a node since the probability density \( \psi^2(x) \) at that point is zero.

Nodes can occur at points where the wave function crosses zero and are indicative of the particle's wave-like nature since they are analogous to points of destructive interference in wave phenomena.