Problem 7 - A Modern Approach to Quantum Mechanics

Question

Determine the four states with \(s=\frac{3}{2}\) that can be formed by three spin- \(\frac{1}{2}\) particles. Suggestion: Start with the state \(\left|\frac{3}{2}, \frac{3}{2}\right\rangle\) and apply the lowering operator as in (5.36).

Step-by-Step Solution

Answer

There are four states: \( \left| \frac{3}{2}, \frac{3}{2} \right\rangle, \left| \frac{3}{2}, \frac{1}{2} \right\rangle, \left| \frac{3}{2}, -\frac{1}{2} \right\rangle, \left| \frac{3}{2}, -\frac{3}{2} \right\rangle. \)

Step 1: Understanding the System

We need to determine the states for three particles, each with spin \( \frac{1}{2} \), combined to form a total spin \( s = \frac{3}{2} \). The total spin state is characterized by \( \left| s, m_s \right\rangle \), where \( m_s \) ranges from \(-s\) to \(s\) in integer steps.

Step 2: Starting with Maximum Spin State

The highest spin state \( \left| \frac{3}{2}, \frac{3}{2} \right\rangle \) can be formed when all three spins are aligned. This corresponds to \( \left| \uparrow \uparrow \uparrow \right\rangle \) with each spin being \( +\frac{1}{2} \).

Step 3: Applying the Lowering Operator

To find the next state \( \left| \frac{3}{2}, \frac{1}{2} \right\rangle \), we apply the lowering operator \( S_- = S_{1-} + S_{2-} + S_{3-} \) to \( \left| \frac{3}{2}, \frac{3}{2} \right\rangle \). This operator changes one spin from \( +\frac{1}{2} \) to \( -\frac{1}{2} \).

Step 4: Calculate Lowered State

The result of applying \( S_- \) to \( \left| \frac{3}{2}, \frac{3}{2} \right\rangle \) is a superposition of states with two spins up and one spin down: \[ \left| \frac{3}{2}, \frac{1}{2} \right\rangle = \frac{1}{\sqrt{3}} ( \left| \downarrow \uparrow \uparrow \right\rangle + \left| \uparrow \downarrow \uparrow \right\rangle + \left| \uparrow \uparrow \downarrow \right\rangle ). \]

Step 5: Apply Lowering Operator Twice

Apply the lowering operator again to find \( \left| \frac{3}{2}, -\frac{1}{2} \right\rangle \), resulting in:\[ \left| \frac{3}{2}, -\frac{1}{2} \right\rangle = \frac{1}{\sqrt{3}} ( \left| \downarrow \downarrow \uparrow \right\rangle + \left| \uparrow \downarrow \downarrow \right\rangle + \left| \downarrow \uparrow \downarrow \right\rangle ). \]

Step 6: Final Lowering for Lowest State

Lastly, applying the lowering operator one more time gives us \( \left| \frac{3}{2}, -\frac{3}{2} \right\rangle \), corresponding to all spins down: \[ \left| \frac{3}{2}, -\frac{3}{2} \right\rangle = \left| \downarrow \downarrow \downarrow \right\rangle. \]

Step 7: Verify Spin States

We've confirmed that the states are consistent with the allowed \( m_s \) values for \( s = \frac{3}{2} \), which are \( +\frac{3}{2}, +\frac{1}{2}, -\frac{1}{2}, -\frac{3}{2} \).

Key Concepts

Quantum Spin

Quantum spin is a fundamental property of particles similar to angular momentum but intrinsic to the particles themselves. Just like classical objects have spin or rotation, quantum particles such as electrons have what we refer to as "spin." However, unlike classical rotation, spin is an inherent characteristic, not dependent on any motion through space. It's specified by a quantum number, typically denoted by "s." For electrons, which are fermions, this spin quantum number is \(s = \frac{1}{2}\), meaning they have two possible states: "spin-up" and "spin-down." In our exercise, we explore systems where we have multiple particles, each with its own spin, and how these combine to create larger spin states like the spin-3/2 states from three separate spin-1/2 particles.

Lowering Operator

The lowering operator, often denoted as \(S_-\), is a mathematical tool used in quantum mechanics to decrease the magnetic quantum number \(m_s\) of a state without changing its total spin \(s\). When dealing with multiple spins, the total lowering operator can be written as \(S_- = S_{1-} + S_{2-} + S_{3-}\) in our system of three spin-1/2 particles. Applying this operator to a state effectively 'flips' one of the spins from up to down. This operation is critical when constructing states with different figures of \(m_s\), such as turning \(\left| \frac{3}{2}, \frac{3}{2} \right\rangle\) into \(\left| \frac{3}{2}, \frac{1}{2} \right\rangle\). It allows us to systematically explore all possible states by working our way from the highest \(m_s\) down to the lowest.

Superposition of States

In quantum mechanics, superposition is the principle that a quantum system can exist in multiple states simultaneously. Instead of thinking of a state as being in one distinct possibility, it exists in a combination or 'superposition' of several possible outcomes. For example, when determining states such as \(\left| \frac{3}{2}, \frac{1}{2} \right\rangle\) from the original \(\left| \frac{3}{2}, \frac{3}{2} \right\rangle\) state, the result is not just one simple configuration, but rather a mix of all permutations where two spins remain up, and one goes down. The concept of superposition is pivotal in quantum mechanics and is what grants particles their rich and often counterintuitive behavior. It also underlies technologies such as quantum computing.

Total Angular Momentum

Total angular momentum in a quantum system is a measure of the amount of rotation a particle or system of particles possesses. It's composed of both orbital angular momentum, related to a particle’s motion around an axis, and intrinsic spin, which is the inherent property of the particle discussed in quantum spin. For our collection of three spin-1/2 particles, the combined system can exhibit total angular momentum characterized by the quantum number \(s = \frac{3}{2}\). This means the system behaves as if it were a single particle with a higher spin quantum number. Analyzing the total angular momentum of such systems allows us to understand how particles interact and combine, providing deeper insights into the behavior of complex quantum systems.