Problem 15 - College Physics

Question

If you are twice as likely to find an electron at a distance of \(0.0400 \mathrm{nm}\) than \(0.0500 \mathrm{nm}\) from the nucleus, what is the ratio of the absolute value of the wave function at \(0.0400 \mathrm{nm}\) to that at \(0.0500 \mathrm{nm} ?\)

Step-by-Step Solution

Answer

The ratio of the wave function at 0.0400 nm to 0.0500 nm is \( \sqrt{2} \).

Step 1: Understanding the Relationship

The probability of finding an electron is related to the square of the absolute value of its wave function. Let's denote the wave function at distance 0.0400 nm as \( \psi(0.0400) \) and at 0.0500 nm as \( \psi(0.0500) \). The probability ratio is given, so \( \frac{|\psi(0.0400)|^2}{|\psi(0.0500)|^2} = 2 \).

Step 2: Expressing the Ratio of Wave Functions

Since the probability is the wave function's squared value, the relationship \( \frac{|\psi(0.0400)|^2}{|\psi(0.0500)|^2} = 2 \) can be simplified to \( \left(\frac{|\psi(0.0400)|}{|\psi(0.0500)|}\right)^2 = 2 \).

Step 3: Solving for the Wave Function Ratio

To find the ratio of the wave functions, take the square root of both sides: \( \frac{|\psi(0.0400)|}{|\psi(0.0500)|} = \sqrt{2} \).

Step 4: Substitute Constants to Check

Since the relationships are purely algebraic here, checking is about ensuring the algebraic manipulations are properly tracked, that \( (\sqrt{2})^2 = 2 \).

Key Concepts

Probability

The concept of probability is fundamental to quantum mechanics, particularly when discussing wave functions and the behavior of electrons. In classical mechanics, we can often precisely predict where an object will be at any given time. However, in quantum mechanics, particles like electrons do not have definite positions. Instead, there is a probability associated with finding an electron at a specific position.

• The wave function, often represented as \( \psi \), helps in determining this probability.
• Probability is given by the square of the wave function's absolute value, \( |\psi(x)|^2 \).
• This squared value indicates the likelihood of locating the electron at a particular distance from the nucleus.

When it comes to the exercise, we are given that the probability of finding the electron at one distance is twice that at another distance. From this, a ratio is established between the squares of their wave function values, leading us to significant insights into the wave functions at those points.

Electron

Electrons are subatomic particles that play a critical role in the makeup of atoms and the principles of quantum mechanics. Unlike other classical particles, electrons exhibit both wave-like and particle-like properties, a duality that is captured mathematically by the wave function.

• Electrons are considered basic units of matter, lacking any known substructure.
• In an atom, they exist in regions of probability (often referred to as orbitals) around the nucleus.
• Understanding an electron's position requires probability-based predictions rather than direct measurements.

In the exercise problem, we are examining where an electron can likely be found relative to a nucleus. This reflects the quantum mechanical behavior of electrons, where probability distributions given by wave functions tell us the chances of locating an electron at different distances.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of very small particles like electrons, where classical physics fails to accurately describe their behavior. This field introduces new and often non-intuitive principles like wave-particle duality, uncertainty, and superposition.

• Wave function is a fundamental concept that describes the quantum state of a system.
• According to quantum mechanics, it’s impossible to know everything about a particle's position and momentum simultaneously (Heisenberg's Uncertainty Principle).
• Probability and wave functions replace the certainty of classical physics with statistical descriptions.

The exercise relates to quantum mechanics as it demonstrates the probabilistic nature of electron positions via the wave function. The problem highlights how such wave functions provide critical insights into the spatial distribution of electrons, forming the basis of much of quantum theory.