Question

Show that unless \(\ell=s . L\) and \(S\) cannot be exactly opposite: that is, show that at its minimum possible value, for which \(j=\ell-s\), the magnitude \(J\) of the total angular momenturn is strictly greater than the difference \(L-S \mid\) berween the magnitudes of the orbital and intrinsic angular momentum vectors.

Short answer

Upon applying the triangle inequality, it is verified that the total angular momentum is always greater than the difference between the orbital and the intrinsic angular momenta, hence confirming the inquiry from the exercise. The only exception occurs when the two momenta are diametrically opposed.

Step-by-step solution

Understand the Terms

Firstly, it is key to understand that angular momentum is a measure of the amount of rotation an object has, taking into account its mass, shape, and speed. There are two forms here: intrinsic angular momentum \(S\), inherent to an object, and orbital angular momentum \(L\), which relates to the object's motion about a point outside itself.

Showing that \(J\) is always greater than \(|L - S|\)

Recall that \(J = |\ell - s|\) is the magnitude of the total angular momentum, and that \(\ell = s . L\) is given in the problem. Substituting the latter equation into the former, we get \(J = |\ell - s| = |s.L - s| = |s (L - 1)|\). Now, there is a rule that for any vectors \(A\) and \(B\), \(|A - B| \geq |A| - |B|\), known as the triangle inequality. Applying this to \(s (L - 1)\), we get \(|s (L - 1)| \geq |s| - |L - 1|\). Since \(s\) and \((L - 1)\) are both positive, we can remove the absolute value signs, giving \(s (L - 1) \geq s - (L - 1) = J\), re-arranging for \(J\), we obtain \(s (L - 1) + (L - 1) \geq J\), thus proving that at a minimum, \(J\) is always greater than \(|L - S|\).

Interpretation

In essence, this result confirms the original premise: regardless of their orientations, the magnitude of the total angular momentum will always be greater than the difference in magnitude between the orbital and intrinsic angular momenta, barring the case where they are exactly opposite. This represents a key governing principle in the dynamics of angularly rotating objects.

Key concepts

Orbital Angular Momentum

Orbital angular momentum is a fundamental concept in physics, encapsulating the idea of rotational motion for objects in orbit. In classical mechanics, it represents an object's momentum attributed to its rotation around a fixed center of mass, much akin to how a planet revolves around a star.

For an object revolving in a two-dimensional plane, the magnitude of its orbital angular momentum, denoted by the symbol \(L\), can be expressed as the product of the object's mass \(m\), its velocity \(v\), and the perpendicular distance from the center of rotation to the line of motion, known as the moment arm \(r\). The formula is simply \( L = mvr \).

This physical quantity becomes even more intriguing when we travel into the realm of quantum mechanics. Here, \(L\) is quantized, meaning it can only take specific discrete values, and it plays a pivotal role in determining the energy levels of electrons in an atom. An understanding of orbital angular momentum is crucial as it helps explain the structure of the atom, the nature of chemical bonds, and the spectral lines of elements.

Intrinsic Angular Momentum

Intrinsic angular momentum, more famously known as spin, represents the internal rotation of a particle on its own axis. Unlike orbital angular momentum, intrinsic angular momentum does not involve the motion of particles around an external point. Instead, it is a fundamental property of particles just like mass or charge, and every elementary particle has a specific 'spin' value.

In quantum mechanics, the spin \(S\) is characterized by its own set of quantum numbers, and it greatly influences a particle’s quantum behavior, particularly in magnetic environments. An electron, for instance, can have a spin of \(+1/2\) or \(–1/2\). The concept of spin is necessary to understand phenomena such as the Pauli exclusion principle, which explains the arrangement of electrons in an atom and is the foundation for the entire field of chemistry and solid-state physics.

Vector Magnitudes

In physics, a vector represents a quantity that possesses both magnitude and direction. Common examples include force, velocity, and acceleration. The magnitude of a vector is its 'length' in a spatial sense, and it is always a non-negative number.

To calculate the magnitude of a vector \( \vec{A} \), which has components \(A_x, A_y\), and \(A_z\) in three-dimensional space, one would use the formula \( |\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} \). Understanding vector magnitudes is crucial because they tell us about the 'size' of the vector quantity, irrespective of the direction in which it's pointing. This measure is essential when combining multiple vectors to determine the net effect, such as the total displacement or force acting on an object.

Triangle Inequality

The triangle inequality theorem is a principle that states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the remaining side. This concept is also applied in the context of vector algebra in physics.

When dealing with vector magnitudes, the triangle inequality implies that the magnitude of the sum of two vectors will never be less than the magnitude of the difference between them. In mathematical terms, for vectors \( \vec{A} \) and \( \vec{B} \), the following always holds true: \( |\vec{A} + \vec{B}| \geq ||\vec{A}| - |\vec{B}|| \).

Application in Angular Momentum

In the realm of angular momentum, the triangle inequality is used to understand the relationship between various types of angular momentum. For instance, when we discuss the total angular momentum \(J\) of a system, this value can never be less than the difference in magnitudes of the orbital \(L\) and intrinsic \(S\) angular momenta, reaffirming the notion that individual contributions to a system's total momentum are intricately linked.