Problem 8 - Chemistry The Central Science In Si Units

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 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

Greatest probability at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \); probability at \( x = \pi \) is 0, a node.

Step 1: Understand the Probability Density Equation

The probability density for the wave function is defined as the square of the wave function, \[ \psi^2(x) = (\sin x)^2. \]This helps us find where the electron is most likely to be found.

Step 2: Sketch the Probability Density

To sketch \( \psi^2(x) = (\sin x)^2 \) from \( x=0 \) to \( x=2\pi \), recall that \( \sin x \) oscillates between -1 and 1. Therefore, \( \sin^2 x \) oscillates between 0 and 1. The probability density will have peaks at points where \( \sin x = \pm 1 \), which occur at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

Step 3: Determine Maximum Probability Locations

The greatest probability density occurs at the peaks of \( \sin^2 x \). These are the points where \( \sin x \) is maximum, specifically\[ x = \frac{\pi}{2} \quad \text{and} \quad x = \frac{3\pi}{2}. \]

Step 4: Evaluate Probability at Specific Points

To find the probability of the electron being exactly at \( x = \pi \), calculate \( \psi^2(\pi) \):\[ \psi(\pi) = \sin(\pi) = 0, \]thus,\[ \psi^2(\pi) = 0^2 = 0. \]The probability at \( x = \pi \) is 0, indicating this is a node.

Step 5: Identify Nodes in the Wave Function

A point where the probability density is zero in a wave function is known as a node. In this wave function, \( x = \pi \) is a node due to \( \psi(x) \) being zero there.

Key Concepts

Probability Density

In quantum mechanics, probability density is a crucial concept that indicates where a particle, such as an electron, is likely to be found. It is defined as the square of the wave function, represented as \( \psi^2(x) \). This equation shows us how the likelihood of finding a particle at a certain position changes along a given path.

Since the probability density is the square of the wave function, it is always a non-negative value. For the given exercise, the wave function is \( \psi(x) = \sin x \) over the interval from \( x = 0 \) to \( x = 2\pi \). Hence, the probability density becomes \( \psi^2(x) = (\sin x)^2 \).

• The wave function, \( \sin x \), varies between -1 and 1.
• This means the probability density, \( \sin^2 x \), varies between 0 and 1.
• Where \( \sin x = \pm 1 \), \( \psi^2(x) \) reaches its maximum value of 1.

By sketching this probability density, you can visually grasp where the particle is most likely to be present along its path.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels, such as electrons and photons. This fascinating field explores how particles exhibit both wave-like and particle-like properties.

A fundamental principle of quantum mechanics is that it uses wave functions to describe the behavior of particles. These functions are vital in predicting the probability distribution of a particle's position. Wave functions can be complex, but they give us crucial insights into the probabilistic nature of particles within an atom.

In our exercise, the wave function \( \psi(x) = \sin x \) describes the state of an electron in a one-dimensional system. By squaring this wave function, quantum mechanics allows us to compute the probability density \( \psi^2(x) \). This approach provides a detailed understanding of the electron’s likely positions between \( x = 0 \) and \( x = 2\pi \).

Key points:

• Wave functions are the cornerstone of quantum mechanics.
• They provide a probabilistic framework for understanding particle positions.
• The squaring of the wave function leads to the probability density, reflecting real, observable probabilities.

Node Concept

Nodes are significant concepts when interpreting wave functions. A node is a point where the probability density is zero—it represents positions where the particle will never be found.

In the given wave function \( \psi(x) = \sin x \), a node occurs when \( \sin x = 0 \). For our interval of \( x = 0 \) to \( x = 2\pi \), a node is present at \( x = \pi \). Here, the probability of finding the electron is zero because \( \psi(\pi) = 0 \) and thus \( \psi^2(\pi) = 0 \).

Nodes play an essential role in quantum mechanics because:

• They mark regions of space that a particle cannot occupy.
• Understanding nodes helps in visualizing and analyzing wave functions.
• In molecular orbital theory, nodes can indicate different bonding patterns.

Recognizing where nodes occur in a wave function helps students understand the intricate nature of quantum particles.