91Ó°ÊÓ

(a) At time \(t=0\) a large ensemble of spin-1/2 particles is prepared, all of them in the spin-up state (with respect to the \(z\) axis). 74 They are not subject to any forces or torques. At time \(t_{1}>0\) each spin is measured some along the \(z\) direction and others along the \(x\) direction (but we aren't some along the \(z\) direction and others along the \(x\) direction (but we aren't told the results). At time \(t_{2}>t_{1}\) their spin is measured again, this time along the \(x\) direction, and those with spin up (along \(x\) ) are saved as a subensemble (those with spin down are discarded). Question: Of those remaining (the subensemble), what fraction had spin up (along \(z\) or \(x\), depending on which was measured) in the first measurement? (b) Part (a) was easy-trivial, really, once you see it. Here's a more pithy generalization: At time \(t=0\) an ensemble of spin-1/2 particles is prepared, all in the spin-up state along direction a. At time \(t_{1}>0\) their spins are measured along direction b (but we are not told the results), and at time \(t_{2}>t_{1}\) their spins are measured along direction c. Those with spin up (along \(\mathrm{c}\) ) are saved as a subensemble. Of the particles in this subensemble, what fraction had spin up (along b) in the first measurement? Hint: Use Equation 4.155 to show that the probability of getting spin up (along b) in the first measurement is \(P_{+}=\cos ^{2}\left(\theta_{a b} / 2\right)\), and (by extension) the probability of getting spin up in both measurements is \(P_{++}=\cos ^{2}\left(\theta_{a b} / 2\right) \cos ^{2}\left(\theta_{b c} / 2\right)\). Find the other three probabilities \(\left(P_{+\rightarrow}, P_{-+}, \text {and } P_{--}\right) .\) Beware: If the outcome of the first measurement was spin down, the relevant angle is now the supplement of \(\theta_{b c}\).

Step by step solution

01

Examine Initial Conditions

Initially, at time \( t = 0 \), all particles are spin up along the \( z \)-axis. Hence, we can represent the state of each particle as \( |+z\rangle \). Spin measurements are made along \( z \) at \( t_{1} \) and then along \( x \) at \( t_{2} \). We need to calculate the fraction of particles that have spin up along \( z \) or \( x \) in the first measurement and are spin up along \( x \) in the second measurement.

02

Spin State Evolution

After the first measurement along the \( z \) or \( x \) direction, the outcome doesn't affect the spin evolution until the second measurement. Between measurements, since no forces act on the particles, their state doesn't change. Consequently, the initial state transitions according to measurement probability, but not in physical state development.

03

Measurement Probabilities

When measuring spin along \( z \), initially prepared as \( |+z\rangle \), the probability of measuring spin up along \( z \) is 1. If measured along \( x \), use \( |+z\rangle = \frac{1}{\sqrt{2}}(|+x\rangle + |-x\rangle) \). Hence, the probability for spin up along \( x \) is \( \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2} \).

04

Subensemble Preparation at t2

At \( t_{2} \), spin is measured along \( x \) and those with spin up are isolated. From Step 3, particles measured from \( |+z\rangle \) at \( t_1 \) have a 50% chance of spin up along \( x \) at \( t_2 \), and those initially \( |+x\rangle \) have a 100% chance to remain \( |+x\rangle \) at \( t_2 \). Thus, the subensemble community is a 50-50 mix from prior \( z \) and potential \( x \) measurements.

05

Fraction Calculation

The particles initially measured spin down on first measurement are excluded from subensemble as they do not meet second measurement \(|+x\rangle\) criteria. Fraction of those initially measuring spin up include those measurements at time \( t_1 \). If all particles measured on \( x \) at \( t_1 \), subensemble retains \( 1/2 \) from \( |+x\rangle\). Hence, description holds: the final fraction who were spin-up in the first MAJORITY case is usually \(1/2\).

06

General Case Analysis for Part B

In general, starting along direction \( a \), the spin up probability along \( b \) direction is \( P_+ = \cos^2(\theta_{ab}/2) \). Given it's spin up along \( c \) direction finally, probability of spinning up twice is \( P_{++} = \cos^2(\theta_{ab}/2) \cos^2(\theta_{bc}/2) \) by cascading rotational consistency. Since particle keeps same trajectory, trajectory config. supports precise bold calculations.

07

Remaining Probability Computations

When spin measurement in first is down (spin measures against expectation relative to chosen frame), complement angle applies: \( P_{-+} = \sin^2(\theta_{ab}/2) \cos^2(\pi-\theta_{bc}/2) \). Additionally, alternative parallel computation \( P_{+ o} = \cos^2(\theta_{bc}/2) \), and \( P_{--} = \sin^2(\theta_{ab}/2)\sin^2(\theta_{bc}/2)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spin-1/2 Particles

Spin-1/2 particles are a fundamental concept in quantum mechanics and are often used as a model system for understanding quantum spin. These particles have a spin quantum number of 1/2, which is a way of describing their intrinsic angular momentum. The term "1/2" does not imply any real or classical rotation, but rather indicates the quantized nature of this property.

For spin-1/2 particles, there are only two possible measurable states along any axis. These states are often referred to as "spin up" or "spin down." For example, if a spin-1/2 particle is measured along the z-axis, the possible outcomes are either "spin up" (denoted as \( |+z\rangle \)) or "spin down" (denoted as \( |-z\rangle \)).

• Spin-up and spin-down states are vector states in a quantum Hilbert space.
• The quantization of spin is a core feature of quantum mechanics and has no direct classical analogy.

Spin-1/2 systems serve as the simplest model for studying more complex quantum systems and illustrate fundamental principles such as superposition and entanglement.

Spin Measurement Probabilities

In quantum mechanics, measurement plays a crucial role and can affect the state of a quantum system. When we measure a spin-1/2 particle, the process changes the spin state depending on the axis along which the measurement is performed. The probability of obtaining a particular spin state after the measurement is determined by the initial state of the particle and the axis of measurement.

For example, if we start with a spin-1/2 particle prepared in the state \( |+z\rangle \) and measure its spin along the x-axis, the probability of finding it in the state \( |+x\rangle \) is \( \left| \frac{1}{\sqrt{2}} \right|^2 = \frac{1}{2} \).

• The outcome probability is computed using the inner product of the initial and final states.
• This probability reflects the concept of quantum superposition, where multiple states contribute to the final observed state.

Understanding these probabilities helps us to predict the behavior of quantum systems as they undergo measurements and demonstrates the intrinsic randomness of quantum events.

Quantum State Evolution

Quantum state evolution concerns how quantum states change over time in the absence of measurement influences or external forces. For spin-1/2 particles, if no external fields are acting on the particles, their quantum states remain constant over time unless measured.

In the described problem, spin-1/2 particles start in a defined state at time \( t = 0 \), for example, \( |+z\rangle \), and remain in this state until a measurement occurs. When a measurement does take place (e.g., along the x-axis), it effectively "collapses" the quantum state into one of the possible outcome states, altering the initial setup.

• This evolution without external influence means the system's physical description remains stable between measurements.
• The measurement process itself introduces change shown in probability distributions, rather than the state evolving.

By understanding quantum state evolution, we grasp how systems persist over time and how measurements impact the observed state of quantum particles.