Question

The subatomic omega particle has spin \(s=\frac{3}{2}\). What angles might its intrinsic angular inomentum vector make with the \(z\) -axis?

Short answer

The four possible angles that the intrinsic angular momentum vector can make with the z-axis corresponds to the four \(m_s\) values: \(-\frac{3}{2}\), \(-\frac{1}{2}\), \(\frac{1}{2}\), and \(\frac{3}{2}\). Determine the corresponding angles by calculating \(θ\) for each \(m_s\) value using the relation \(cos θ = \frac{L_z}{|L|}\).

Step-by-step solution

Understanding the Spin

The spin quantum number, \(s\), is a property of subatomic particles. For the omega particle in this exercise, the given spin value is \(s = \frac{3}{2}\). This means that the particle may have \(2s+1 = 2(\frac{3}{2}) + 1 = 4\) possible orientations in space relative to an external magnetic field.

Calculating the \(m_s\) Values

The z-component of the spin, \(m_s\), can have 2s+1 integer values ranging from \(-s\) to \(+s\). For the omega particle with \(s = \frac{3}{2}\), the possible \(m_s\) values are \(-\frac{3}{2}\), \(-\frac{1}{2}\), \(\frac{1}{2}\), and \(\frac{3}{2}\). These values represent the four different orientations that the intrinsic angular momentum vector can take with respect to the z-axis.

Translating \(m_s\) into Angles

The \(m_s\) values give the quantized component of the spin along the z-axis. To find the angles these components represent, recall that the magnitude of the angular momentum is \(|L| = \sqrt{s(s+1) \hbar}\). So, if \(L_z = m_s \hbar\), the angle \(θ\) relative to the z-axis can be found by \(cos θ = \frac{L_z}{|L|}\). Substituting and calculating for each \(m_s\) value will give the corresponding angles.

Key concepts

Spin Quantum Number

The spin quantum number, denoted as \(s\), is a fundamental property of subatomic particles, much like charge or mass. It's a bit of a misnomer because it doesn't involve the particle physically spinning around an axis like a planet. Instead, spin is a form of intrinsic angular momentum carried by particles, which is a quantum property with no classical counterpart.

The spin quantum number can take on either whole or half-integer values such as 0, 1/2, 1, 3/2, etc. This value determines the number of different orientations a particle's spin can have in the presence of an external magnetic field. The formula to calculate the possible orientations is \(2s + 1\). For instance, an omega particle with a spin quantum number of 3/2 can exist in one of four distinct orientations. These different orientations are critical for understanding magnetic properties and the behavior of particles in quantum systems.

Intrinsic Angular Momentum

Intrinsic angular momentum is essential to the concept of spin as it represents the internal moment of momentum that a particle possesses. Unlike orbital angular momentum which arises from a particle's motion through space, intrinsic angular momentum is an inherent characteristic of the particle itself.

It is quantized, meaning it can only take on certain discrete values. This quantization reflects the fundamentally granular nature of the quantum world, distinguishing it sharply from classical physics where quantities such as angular momentum can vary continuously. Each particle has a fixed value of intrinsic angular momentum determined by its spin quantum number \(s\), and this value remains constant unless acted upon by an external force, such as an electromagnetic field.

Z-component of Spin

The z-component of spin, denoted as \(m_s\), describes the orientation of a particle's spin along the z-axis, which is typically defined by the direction of an external magnetic field. Mathematically, the z-component of spin is a projection of the intrinsic angular momentum vector along the z-axis and it provides a measure of how much of the spin is aligned with this axis.

For a given spin quantum number \(s\), the z-component can take on specific values ranging from \(-s\) to \(+s\) in integer steps. This means for an omega particle with \(s = \frac{3}{2}\), the z-component can have four possible values: \(-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \text{ and } \frac{3}{2}\). These values correspond to the quantized orientations of the particle's spin vector with respect to the z-axis, crucial for understanding phenomena like the Stern-Gerlach experiment which demonstrated the discrete nature of quantum states.

Quantization of Angular Momentum

Quantization of angular momentum is a cornerstone of quantum mechanics. It implies that angular momentum cannot vary arbitrarily but must change in discrete amounts, called quanta. This principle applies to both orbital and intrinsic (spin) angular momenta.

In fact, the concept of quantized angular momentum comes from the early days of quantum theory, with the Bohr-Sommerfeld model of the atom, where electrons only occupy certain orbits with quantized angular momentum. The generalization to intrinsic angular momentum or spin was one of the profound steps in the development of quantum mechanics. Quantization ensures that any measurement of angular momentum along any axis, such as the z-component, will result in certain allowed values, explained by the associated quantum numbers. For example, the quantization of an omega particle's spin, which reveals itself as four potential angular positions relative to the z-axis, reflects this fundamental quantum property.