Problem 4 - A Modern Approach to Quantum Mechanics

Question

A beam of spin- \(\frac{1}{2}\) particles with speed \(v_{0}\) passes through a series of two SGz devices. The first SGz device transmits particles with \(S_{z}=\hbar / 2\) and filters out particles with \(S_{z}=-\hbar / 2\). The second SGz device transmits particles with \(S_{z}=-\hbar / 2\) and filters out particles with \(S_{z}=\hbar / 2\). Between the two devices is a region of length \(l_{0}\) in which there is a uniform magnetic field \(B_{0}\) pointing in the \(x\) direction. Determine the smallest value of \(l_{0}\) such that exactly 25 percent of the particles transmitted by the first SGz device are transmitted by the second device. Express your result in terms of \(\omega_{0}=e g B_{0} / 2 m c\) and \(v_{0}\).

Step-by-Step Solution

Answer

The smallest length \( l_0 \) is \( l_0 = \frac{\pi v_0}{3 \omega_0} \).

Step 1: Understanding the Exercise

We have a beam of spin-\( \frac{1}{2} \) particles passing through a series of two Stern-Gerlach \( (\text{SG}_z) \) devices. The first device allows particles with \( S_z = \frac{\hbar}{2} \) and the second allows \( S_z = -\frac{\hbar}{2} \). In between, particles move through a magnetic field \( B_0 \) pointing in the \( x \)-direction. We need to calculate the smallest region length \( l_0 \) for 25% transmission through the second device.

Step 2: Spin Precession in the Magnetic Field

The key effect of a magnetic field in the \( x \)-direction on spin is causing the spins to precess around the \( x \)-axis. The frequency of this precession is given by \( \omega_0 = \frac{e g B_0}{2m c} \). The spin precesses by an angle \( \theta = \omega_0 t \) during the time \( t \) it spends in this field.

Step 3: Time Spent in the Magnetic Field

As the particles travel through a region of length \( l_0 \) at speed \( v_0 \), the time they spend in the magnetic field is \( t = \frac{l_0}{v_0} \). Substituting this into the precession angle formula gives \( \theta = \omega_0 \frac{l_0}{v_0} \).

Step 4: Probability Amplitude Calculation

To transition from \( S_z = \frac{\hbar}{2} \) to \( S_z = -\frac{\hbar}{2} \), the component of the spin must rotate to align with the \( -z \) direction. This transition probability is given by \( \sin^2 (\theta/2) = 0.25 \), meaning the precession angle must satisfy \( \sin^2 (\omega_0 l_0 / (2 v_0)) = 0.25 \).

Step 5: Solving for the Smallest Angle

Since \( \sin^2(\omega_0 l_0 / 2v_0) = 0.25 \), we have \( \sin(\omega_0 l_0 / 2v_0) = 0.5 \). This gives \( \omega_0 l_0 / 2v_0 = \frac{\pi}{6} \) because \( \sin(\pi/6) = 0.5 \). Therefore, \( \omega_0 l_0 = \frac{\pi v_0}{3} \).

Step 6: Finding the Smallest Length \( l_0 \)

Solve for \( l_0 \) in terms of the given variables: \( l_0 = \frac{\pi v_0}{3 \omega_0} \). This represents the smallest length of the region with magnetic field for the desired transmission probability.

Key Concepts

Spin-1/2 Particles

Spin-1/2 particles are fundamental components of quantum mechanics, prominently playing a role in the behavior of fermions like electrons. These particles possess an intrinsic form of angular momentum known as "spin," which can be visualized as the particle's internal rotation. However, unlike classical objects, spin-1/2 particles can only have two discrete orientations: "up" ( \(S_z = \frac{\hbar}{2}\)) or "down" ( \(S_z = -\frac{\hbar}{2}\)). This quantum spin quantization is a central concept in quantum mechanics. It leads to the probabilistic nature of measurements, highlighting the differences from classical physics.

These discrete states are integral to phenomena such as quantum entanglement and interference. Spin-1/2 particles can be manipulated using various external fields, which influence their orientation and related probabilities. These properties are crucial when analyzing experimental setups like the Stern-Gerlach experiment, where spins are separated based on their orientation.

Stern-Gerlach Experiment

The Stern-Gerlach experiment is a famous quantum mechanics experiment that demonstrated the quantized nature of angular momentum. It involves passing a beam of particles, such as silver atoms with unpaired electrons (spin-1/2 particles), through a non-uniform magnetic field. This field causes particles with different spin orientations to deflect in opposite directions.

• Magnetic Field Interaction: As particles travel through the magnetic field, those with "up" spin ( \(S_z = \frac{\hbar}{2}\)) are deflected one way, while particles with "down" spin ( \(S_z = -\frac{\hbar}{2}\)) deflect in the opposite way.
• Spin State Selection: Using this setup, experimenters can effectively "filter" particles by their spin state, allowing only certain orientations to pass through specific regions.

This experiment provided solid evidence of quantum mechanics over classical physics, as it clearly showed that angular momentum is quantized. The discrete nature of spin observed in the experiment further supports the quantum mechanical model of atoms.

Magnetic Field Effects

Magnetic fields have profound effects on spin-1/2 particles. These effects are evident when particles pass through fields aligned in particular directions, leading to changes in their spin orientation due to a phenomenon called "spin precession."

• Spin Precession: In the context of quantum mechanics, spin precession occurs when a spin-1/2 particle enters a magnetic field. The field causes the particle's spin to rotate around an axis aligned with the field direction, such as the x-axis in the exercise above.
• Precession Frequency: The rate of this rotation, or precession, is dictated by the Larmor frequency ( \(\omega_0 = \frac{e g B_0}{2mc}\)), a relation that depends on factors like the charge, magnetic characteristics of the particle, and field strength.
• Transition Probability: As spin-1/2 particles precess, the probability of them transitioning between different spin states changes. Calculating such probabilities is essential in predicting experimental outcomes and helps in understanding quantum behavior in magnetic fields.

Understanding magnetic field effects on spin is vital for interpreting experiments and technological applications that rely on quantum spin dynamics, such as magnetic resonance imaging (MRI) and quantum computing. Each of these relies on the ability to control and measure spin state transitions accurately.