Chapter 41: Problem 8
An electron is in the hydrogen atom with \(n\) = 5. (a) Find the possible values of \(L\) and \(L$$_z\) for this electron, in units of \(\hslash\). (b) For each value of \(L\), find all the possible angles between \(\vec{L}\) and the z-axis. (c) What are the maximum and minimum values of the magnitude of the angle between \(L\) S and the z-axis?
Short Answer
Step by step solution
Finding Possible Values of L
Determine Possible Values of Lz
Calculating L for Each l Value
Calculating Lz for Each l and ml Value
Finding Possible Angles between L and z-axis
Maximum and Minimum Angles Calculation
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hydrogen Atom
Atoms are defined by their quantum states, which encompass specific energy levels. Each state is characterized by several quantum numbers. For hydrogen, these quantum states tell us about the electron's position and energy surrounding the nucleus. Understanding these states provides a framework to study more complex atoms beyond hydrogen.
Angular Momentum
The magnitude of angular momentum, denoted as \(L\), is determined by the formula \(L = \sqrt{l(l+1)}\hslash\). This formula highlights that angular momentum is dependent on the quantum number \(l\). It reflects the intrinsic spin and orbital motion of particles, making it crucial for understanding atomic structure and behavior. Angular momentum is also tied to stability and energy conservation in atoms.
Quantum Numbers
- Principal quantum number \(n\): indicates the energy level or shell.
- Angular momentum quantum number \(l\): informs about the shape of the electron's orbital.
- Magnetic quantum number \(m_l\): describes the orientation of the orbital in space.
- Spin quantum number \(m_s\): accounts for the intrinsic spin of the electron.