Problem 2 - College Physics

Question

Consider states with \(l = 3 .\) (a) In units of \(\hbar ,\) what is the largest possible value of \(L _ { z } ?\) (b) In units of \(\hbar ,\) what is the value of \(L ?\) Which is larger, \(L\) or the maximum possible \(L _ { z } ?\) (c) Assume a model in which \(\vec { \boldsymbol { L } }\) is described as a classical vector. For each allowed value of \(L _ { z } ,\) what angle does the vector \(\vec { L }\) make with the \(+ z\) axis?

Step-by-Step Solution

Answer

(a) Largest \( L_z = 3\hbar \), (b) \( L = 2\sqrt{3}\hbar \), \( L \) is larger, (c) \( \theta \) varies with \( m \).

Step 1: Understanding Quantum Numbers

The quantum number \( l \) corresponds to the azimuthal or orbital angular momentum quantum number and defines the shape of the electron's orbital and thus its angular momentum. For a given \( l \), the possible values of the magnetic quantum number \( m \) range from \( -l, -l+1, ..., l-1, l \).

Step 2: Calculating Largest Possible \( L_z \)

The largest possible value of the magnetic quantum number \( m \) is \( +l \). Therefore, the largest value of \( L_z \) can be calculated as \( m \hbar = l\hbar = 3\hbar \).

Step 3: Calculating Total Angular Momentum \( L \)

The total angular momentum \( L \) is given by \( L = \sqrt{l(l+1)} \hbar \). Substituting \( l = 3 \), we have \( L = \sqrt{3(3+1)} \hbar = \sqrt{12} \hbar = 2\sqrt{3} \hbar \).

Step 4: Comparing \( L \) and Maximum \( L_z \)

Now, we compare \( L = 2\sqrt{3} \hbar \) with the maximum \( L_z = 3\hbar \). Numerically, \( 2\sqrt{3} \approx 3.464 \), which means that \( L = 2\sqrt{3} \hbar \) is larger than the maximum possible \( L_z = 3\hbar \).

Step 5: Calculating Angle with +z Axis

Assuming \( \vec{L} \) is a classical vector, the angle \( \theta \) between \( \vec{L} \) and the \(+z\) axis can be calculated using the formula \( \cos \theta = \frac{L_z}{L} \). For each allowed value of \( m \), calculate \( \theta \) using the corresponding \( L_z = m\hbar \).

Key Concepts

Angular Momentum

Angular momentum in quantum mechanics is a measure of the amount of rotational motion an object possesses. Unlike classical physics, where angular momentum is straightforward, in quantum mechanics, it follows certain quantized rules. This means that the angular momentum cannot take just any value; it is quantized. In any quantum system, particularly electrons within atoms, the angular momentum is characterized by quantum numbers. These numbers stem from solutions to the Schrödinger equation. Now, let's explore these quantum numbers and see how they help us understand quantum systems better.

Quantum Numbers

Quantum numbers are a set of numerical values that describe the unique quantum state of an electron in an atom. These include:

• Principal Quantum Number ( n ): It signifies the shell or energy level an electron is in. Higher values denote electrons further from the nucleus, and hence, at higher energies.
• Azimuthal Quantum Number ( l ): It also refers to as angular momentum quantum number, which indicates the shape of the electron's orbital.
• Magnetic Quantum Number ( m ): This number relates to the orientation of the orbital's angular momentum component along a specified axis, typically the z-axis.
• Spin Quantum Number ( s ): Although not covered here in detail, s refers to the intrinsic spin of the electron, which can be either +1/2 or -1/2.

Think of these like an address system, specifying not only the electron's location but also its orientation and potential energy states.

Magnetic Quantum Number

The magnetic quantum number, denoted as m , arises from the azimuthal quantum number l . It defines the orientation of an electron's orbital angular momentum relative to an applied magnetic field, usually along the z-axis.

For a given azimuthal quantum number l , the magnetic quantum number can take integer values ranging from -l to +l , including zero. This means for l = 3 , m can be -3, -2, -1, 0, +1, +2, +3. Each m determines how tilted the orbitals are with respect to, say, an external magnetic field or a chosen axis.

This quantum number is crucial in understanding the Zeeman effect where atomic energy levels split in the presence of a magnetic field. On a broader level, it can affect how atoms absorb light and participate in chemical bonding.

Orbital Angular Momentum

Orbital angular momentum is a key idea in the quantization of electron motion within an atom. It indicates the electron's wave pattern around the nucleus and is symbolized by the quantum number l .

The magnitude of orbital angular momentum L is derived from quantum mechanics using the formula:\[L = \sqrt{l(l+1)} \hbar\]This highlights how dramatically different quantum mechanics is from classical mechanics, with quantized values rather than continuous ones. For example, with l = 3 , L computes to 2\sqrt{3} \hbar , which is the inherent rotation measure an electron possesses in its orbital.

Understanding this helps explain why not all electrons in quantum orbitals have the same energy, even if they share the same shell or principal quantum number. It's also why discrete energy levels arise and electrons distribute themselves in defined orbitals.