Question

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 $\overrightarrow{L}$ 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

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

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 $L=\sqrt{l(l+1)}\hslash$.

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.

Calculating L for Each l Value

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

Calculating Lz for Each l and ml Value

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

Finding Possible Angles between L and z-axis

The angle $\theta$ between $\overrightarrow{L}$ 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 $l$.

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.

Key concepts

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.