Question

Assume that the researchers place an atom in a state with \(n\) = 100, \(l\) = 2. What is the magnitude of the orbital angular momentum \(L\) associated with this state? (a) \(\sqrt{2} \space\hslash \); (b) \(\sqrt{6} \space\hslash\); (c) \(\sqrt{200}\space \hslash\); (d) \(\sqrt{10,100}\space \hslash \).

Short answer

The magnitude of the orbital angular momentum is \( \sqrt{6} \hbar \), option (b).

Step-by-step solution

Understanding the Orbital Angular Momentum Formula

The orbital angular momentum for a given state in quantum mechanics is given by the formula \( L = \sqrt{l(l+1)}\hbar \), where \( \hbar \) is the reduced Planck constant and \( l \) is the azimuthal (or orbital) quantum number. In this problem, \( l = 2 \).

Applying the Formula

Substitute \( l = 2 \) into the formula to find the magnitude of \( L \). Thus, we compute \( L = \sqrt{2(2+1)}\hbar = \sqrt{2 \cdot 3}\hbar = \sqrt{6}\hbar \).

Identifying the Correct Option

Looking at the given choices, we see that option (b), \( \sqrt{6} \hbar \), matches our calculated result. Therefore, the magnitude of the orbital angular momentum for this state is \( \sqrt{6} \hbar \).

Key concepts

Orbital Angular Momentum

Orbital angular momentum is a crucial concept in quantum mechanics related to the motion of particles in an atom. It is analogous to the angular momentum in classical physics. However, in the quantum realm, it takes on discrete values rather than a continuous range.
In quantum mechanics, the orbital angular momentum of an electron in an atom is quantized and determined by the azimuthal quantum number, denoted as \( l \). The quantum number \( l \) can take on integer values from 0 up to \( n-1 \), where \( n \) is the principal quantum number.

• The formula for orbital angular momentum is \( L = \sqrt{l(l+1)}\hbar \), where \( \hbar \) is the reduced Planck constant.
• This formula shows that the magnitude of the orbital angular momentum depends on \( l \), and is directly proportional to \( l(l+1) \).
• Each value of \( l \) corresponds to a different shape of the electron's orbital, which affects how it moves around the nucleus.

By plugging in the quantum number \( l = 2 \), we find that the orbital angular momentum is \( \sqrt{6}\hbar \), as indicated in the problem's solution.

Quantum Numbers

Quantum numbers are sets of numerical values that provide important information about the quantum state of a particle. They are like the particle's "addresses" in the quantum world and are essential for understanding the arrangement of electrons in atoms.
Every electron in an atom is described by a unique set of quantum numbers:

• Principal Quantum Number (\( n \)): This number indicates the energy level of the electron and can be any positive integer. In the exercise, \( n = 100 \).
• Azimuthal (or Orbital) Quantum Number (\( l \)): It specifies the shape of the orbital and can range from 0 to \( n-1 \). In our scenario, \( l = 2 \).
• Magnetic Quantum Number (\( m_l \)): This indicates the orientation of the orbital around the nucleus, varying from \( -l \) to \( +l \).
• Spin Quantum Number (\( m_s \)): It describes the intrinsic angular momentum (or spin) of the electron, typically \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

These quantum numbers collectively help predict and describe the behavior and interaction of electrons within an atom.

Reduced Planck Constant

The reduced Planck constant, denoted by \( \hbar \), is a fundamental constant in quantum mechanics that appears frequently in equations dealing with quantized quantities. It is derived from the Planck constant \( h \), divided by \( 2\pi \):\[ \hbar = \frac{h}{2\pi} \]
This smaller version of the Planck constant is particularly useful in the context of angular momentum and its quantization in systems such as atoms.

• \( \hbar \) quantifies the discreteness of quantum states, serving as the "scale" for action in quantum mechanical systems.
• It allows the representation of quantities like \( L = \sqrt{l(l+1)}\hbar \) succinctly and conveniently.
• The reduced Planck constant is critical in describing phenomena where wave-like properties are fundamental, such as the wave-particle duality of electrons.

Overall, \( \hbar \) helps in measuring changes in quantum systems, linking the quantum world with the classical understanding through the concept of quantization.