Problem 16 - College Physics

Question

Calculate, in units of \(\hbar ,\) the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of \(2,20 ,\) and 200 . Compare each with the value of \(n \hbar\) postulated in the Bohr model. What trend do you see?

Step-by-Step Solution

Answer

The quantum mechanical angular momentum is always less than \( n \hbar \) but approaches it as \( n \) increases.

Step 1: Understanding the Maximum Orbital Angular Momentum

The orbital angular momentum of an electron in a quantum mechanical model is given by the expression: \( L = \sqrt{l(l+1)} \hbar \), where \( l \) is the azimuthal quantum number. The maximum value of \( l \) is given by \( n - 1 \). Thus, the maximum orbital angular momentum for a given principal quantum number \( n \) is found by using \( l = n - 1 \).

Step 2: Calculating Maximum Orbital Angular Momentum for n = 2

For \( n = 2 \), the maximum \( l \) is \( 2 - 1 = 1 \). Hence, the maximum orbital angular momentum is calculated as:\[ L_{max} = \sqrt{1(1+1)} \hbar = \sqrt{2} \hbar. \]

Step 3: Calculating Maximum Orbital Angular Momentum for n = 20

For \( n = 20 \), the maximum \( l \) is \( 20 - 1 = 19 \). The maximum orbital angular momentum is:\[ L_{max} = \sqrt{19(19+1)} \hbar = \sqrt{19 \times 20} \hbar = \sqrt{380} \hbar. \]

Step 4: Calculating Maximum Orbital Angular Momentum for n = 200

For \( n = 200 \), the maximum \( l \) is \( 200 - 1 = 199 \). The maximum orbital angular momentum is:\[ L_{max} = \sqrt{199(199+1)} \hbar = \sqrt{199 \times 200} \hbar = \sqrt{39800} \hbar. \]

Step 5: Comparing with Bohr Model Prediction

In the Bohr model, the angular momentum is quantized as \( n \hbar \). Comparing the calculated maximum orbital angular momenta:- For \( n = 2 \): \( \sqrt{2} \hbar \approx 1.41 \hbar < 2 \hbar \)- For \( n = 20 \): \( \sqrt{380} \hbar \approx 19.49 \hbar < 20 \hbar \)- For \( n = 200 \): \( \sqrt{39800} \hbar \approx 199.5 \hbar < 200 \hbar \)All calculated values are less than \( n \hbar \).

Step 6: Identifying the Trend

As the principal quantum number \( n \) increases, the value of \( \sqrt{l(l+1)} \) for \( l = n-1 \) approaches \( n \), especially for large \( n \). Thus, the quantum mechanical angular momentum values become closer to the Bohr model prediction \( n \hbar \) with increasing \( n \).

Key Concepts

Quantum Mechanics: Understanding Orbital Angular Momentum

Quantum mechanics is a fundamental theory in physics that provides a framework for understanding the behavior of particles at the smallest scales, such as electrons in an atom. One of the key concepts within this field is orbital angular momentum. This refers to the motion of an electron as it orbits around the nucleus of an atom. In quantum mechanics, the orbital angular momentum is not simply a direct measure but instead, it quantifies the extent of the electron's rotational movement. It is calculated using the formula:

• \[ L = \sqrt{l(l+1)} \hbar \]

where \( l \) is the azimuthal quantum number and \( \hbar \) is the reduced Planck's constant.

• The azimuthal quantum number, \( l \), can take integer values from 0 up to \( n-1 \), where \( n \) is the principal quantum number.

This model describes how the movement and positioning of electrons is not definite but rather probabilistic, a core aspect of quantum mechanics. Understanding this probabilistic nature allows scientists to predict electron behavior, which in turn, helps in delineating atomic structures and reactions.
The orbital angular momentum's variability marks a significant departure from classical systems, where angular momentum is often considered as a straightforward multiplication of mass, velocity, and distance from the rotation axis. In quantum mechanics, the assessment is more complex due to the directive of quantum numbers and the inherent uncertainty present in the electron's positioning.

Bohr Model: A Simplified View of Atomic Structure

The Bohr model is a pivotal concept in the study of atomic structure, particularly in the history of quantum mechanics. Proposed by Niels Bohr in 1913, the model presents an early effort to explain the discrete energy levels in atoms and the emission spectrum of hydrogen. Unlike prior models where electrons orbited in simple circular paths, the Bohr model suggests certain quantized conditions:

• Electrons travel in fixed orbits around the nucleus.
• These orbits correspond to specific energy levels or states.

In these quantized orbits, an electron's angular momentum is given by \(n \hbar\), where \( n \) is an integer termed as the principal quantum number.

• This was a groundbreaking idea because it introduced the concept of quantization to physics, implying discrete rather than continuous energy values.

Although the Bohr model simplifies many complex quantum mechanical phenomena, it falls short in explaining multi-electron systems and does not account for electron-electron interactions. However, it laid the groundwork for future quantum theories and remains a useful conceptual tool for understanding atomic behavior at a glance.
In your studies, remember that while the Bohr model provides a visually intuitive framework, it is limited and should be complemented by modern quantum mechanical interpretations for a complete picture of atomic interactions.

Principal Quantum Number: Key to Electron Energy Levels

The principal quantum number, usually denoted as \( n \), is a fundamental concept in quantum mechanics and the Bohr model. It is essential in defining the energy level of an electron within an atom.

• The principal quantum number determines the electron's shell and is integral to understanding both the structure and behavior of the atom.

As \( n \) increases, the electron occupies higher energy orbits, which are further from the nucleus. This relationship is crucial in calculating the energy and angular momentum of the electron.

• In terms of energy, the higher the principal quantum number, the greater the energy level and further distance of the electron from the nucleus.
• The principal quantum number also influences the number of sub-shells available to an electron.

These sub-shells, in turn, designate the possible values of the azimuthal quantum number \( l \), as each \( n \) level contains \( n \) sub-shells.Thus, the principal quantum number plays a critical role not just in determining the electron's position but also in the corresponding energy state within an atom.
This notion is underscored when comparing the quantum mechanical and classical Bohr model predictions for the orbital angular momentum, demonstrating an evolution in understanding as \( n \) increases.