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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

Expert verified
For (a) values: \\(L = \sqrt{l(l+1)}\hslash\\), \\(L_z = m_l\hslash\\); For (b) angles: calculate using \\(\cos \theta = \frac{L_z}{L}\\); For (c), angles vary from 0° to 180°.

Step by step solution

01

Finding Possible Values of L

In quantum mechanics, the angular momentum quantum number \(l\) is related to the principal quantum number \(n\) by \(l = 0, 1, 2, ..., n-1\). For \(n = 5\), the possible values of \(l\) are \([0, 1, 2, 3, 4]\). The magnitude of angular momentum \(L\) is given by \(L = \sqrt{l(l+1)}\hslash\).
02

Determine Possible Values of Lz

The magnetic quantum number \(m_l\) can range from \(-l\) to \(l\), giving possible values \(m_l = -l, -(l-1), ..., 0, ..., (l-1), l\). The z-component of angular momentum is given by \(L_z = m_l \hslash\). Each value of \(l\) will have \(2l+1\) corresponding \(m_l\) values.
03

Calculating L for Each l Value

For each possible \(l\):- \(l = 0\): \(L = 0\)- \(l = 1\): \(L = \sqrt{2}\hslash\)- \(l = 2\): \(L = \sqrt{6}\hslash\)- \(l = 3\): \(L = \sqrt{12}\hslash\)- \(l = 4\): \(L = \sqrt{20}\hslash\)
04

Calculating Lz for Each l and ml Value

For each \(l\), calculate possible \(L_z\):- \(l = 0\): \(L_z = 0\)- \(l = 1\): \(L_z = \{-1\hslash, 0, 1\hslash\}\)- \(l = 2\): \(L_z = \{-2\hslash, -1\hslash, 0, 1\hslash, 2\hslash\}\)- \(l = 3\): \(L_z = \{-3\hslash, -2\hslash, -1\hslash, 0, 1\hslash, 2\hslash, 3\hslash\}\)- \(l = 4\): \(L_z = \{-4\hslash, -3\hslash, -2\hslash, -1\hslash, 0, 1\hslash, 2\hslash, 3\hslash, 4\hslash\}\)
05

Finding Possible Angles between L and z-axis

The angle \(\theta\) between \(\vec{L}\) and the z-axis is given by \(\cos \theta = \frac{L_z}{L}\). Compute \(\theta\) for each combination of \(L\) and \(L_z\).For example, \(l = 1\), \(L = \sqrt{2}\hslash\):\(-\hslash/\sqrt{2}\hslash, 0/\sqrt{2}\hslash, \hslash/\sqrt{2}\hslash\) corresponding to angles \(135^\circ, 90^\circ, \text{and } 45^\circ\). Repeat for other values of \(l\).
06

Maximum and Minimum Angles Calculation

The maximum angle corresponds to the minimum value of \(\cos \theta\) and the minimum angle to the maximum value of \(\cos \theta\).- For maximum angle: The smallest \(|m_l|/L\)- For minimum angle: The largest \(|m_l|/L\) corresponds to angles close to \(0^\circ\) and \(180^\circ\). Compute each angle from this equation for all combinations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hydrogen Atom
The hydrogen atom is the simplest atom in the universe, consisting of just one proton and one electron. In quantum mechanics, the behavior of the electron in a hydrogen atom is described by the Schrödinger equation. This equation helps us understand the probabilistic nature of particles at the quantum level.
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
Angular momentum in quantum mechanics is a fundamental concept used to describe rotational motion. It is similar but not identical to classical angular momentum. In quantum systems, angular momentum is quantized, meaning it can only take on discrete values.
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
Quantum numbers are essential to defining the quantum state of an electron in an atom. They provide a set of values used to solve the Schrödinger equation, representing different physical properties. The four primary quantum numbers are:
  • 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.
These numbers work together to describe completely the quantum state of an electron, much like a unique address in a city. Each combination of quantum values provides insight into the electron's energy, position, and magnetic characteristics.
Magnetic Quantum Number
The magnetic quantum number \(m_l\) is crucial in understanding how an electron's orbital aligns within an atom. For a given angular momentum quantum number \(l\), \(m_l\) can take any integer value from -l to l. These integers depict the number of possible orientations an orbital can have.In a hydrogen atom, the magnetic quantum number allows us to study the distribution of electrons in a magnetic field. It ties directly to the z-component of the angular momentum, represented as \(L_z = m_l \hslash\). These discrete \(L_z\) values dictate how angular momentum is aligned relative to the z-axis, which becomes important in contexts such as spectral emission and electron spin resonance.Understanding the magnetic quantum number helps illustrate why electrons behave in specific ways under external magnetic influences, supporting the foundation of quantum mechanics in atomic theory.

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Most popular questions from this chapter

A photon is emitted when an electron in a threedimensional cubical box of side length 8.00 \(\times\) 10\(^{-11}\) m makes a transition from the n\(_X\) = 2, n\(_Y\) = 2, n\(_Z\) = 1 state to the n\(_X\) = 1, n\(_Y\) = 1, n\(_Z\) = 1 state. What is the wavelength of this photon?

(a) Write out the ground-state electron configuration (\(1s^2, 2s^2,\dots\)) for the beryllium atom. (b) What element of nextlarger \(Z\) has chemical properties similar to those of beryllium? Give the ground-state electron configuration of this element. (c) Use the procedure of part (b) to predict what element of nextlarger \(Z\) than in (b) will have chemical properties similar to those of the element you found in part (b), and give its ground-state electron configuration.

\(\textbf{Classical Electron Spin}\). (a) If you treat an electron as a classical spherical object with a radius of 1.0 \(\times\) 10\(^{-17}\) m, what angular speed is necessary to produce a spin angular momentum of magnitude \(\sqrt{3\over4}\hslash\) ? (b) Use \(v = r\omega\) and the result of part (a) to calculate the speed \(v\) of a point at the electron's equator. What does your result suggest about the validity of this model?

In studying electron screening in multielectron atoms, you begin with the alkali metals. You look up experimental data and find the results given in the table. The ionization energy is the minimum energy required to remove the least-bound electron from a ground-state atom. (a) The units kJ/mol given in the table are the minimum energy in kJ required to ionize 1 mol of atoms. Convert the given values for ionization energy to the energy in eV required to ionize one atom. (b) What is the value of the nuclear charge \(Z\) for each element in the table? What is the n quantum number for the least-bound electron in the ground state? (c) Calculate \(Z$$_{eff}\) for this electron in each alkali-metal atom. (d) The ionization energies decrease as \(Z\) increases. Does \(Z$$_{eff}\) increase or decrease as \(Z\) increases? Why does \(Z$$_{eff}\) have this behavior?

A hydrogen atom initially in an \(n\) = \(3,\) \(l\) = 1 state makes a transition to the \(n\) = \(2\), \(l\) = \(0\), \(j\) = \\(\frac{1}{2}\\) state. Find the difference in wavelength between the following two photons: one emitted in a transition that starts in the \(n\) = \(3\), \(l\) = \(1\), \(j\) = \\(\frac{3}{2}\\) state and one that starts instead in the \(n\) = \(3\), \(l\) = \(1\), \(j\) = \\(\frac{1}{2}\\) state. Which photon has the longer wavelength?

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