Question

Figure 8.3 shows the Stern-Gerlach apparans. It reveals that spin- \(\frac{1}{2}\) particles have just two possible spin states. Assume that when these rwo beams are separated inside the channel (though still near its centerline). we can choose to block one or the other for study. Now a second such apparatus is added after the first. Their channels are aligned. but the second one is rotated about the \(x\) -axis by an angle \(\phi\) from the first. Suppose we block the spin-down beam in the first apparatus, allowing only the spin-up beam into the second. There is no wave function for spin. but we can still talk of a probability amplitude, which we square to give a probability. After the first apparatus' spin-up beam passes chrough the second apparatus, the probability amplitude is \(\cos (\phi / 2) T_{2 n d}+\sin (\phi / 2) b_{2 n d}\). where the arrows indicate the two possible findings for spin in the second apparatus. (a) What is the probability of finding the particle spin up in the second apparatus? Of finding it spin down? Argue thatthese probabilities make sense individually for representative values of \(\phi\) and that their sum is also sensible. (b) By contrasting this spin prohability anplitude with a sporial probability amplitude, such as \(\psi(x)=A e^{-b x^{2}}\), argue that although the surhitrariness of \(\phi\) gives the spin case an infinite number of values. it is still justified to refer to it as 8 "two-state system." while the spatial case is an infinite.state system.

Short answer

The probabilities for the spin to be measured up and down are respectively \(cos^2(\phi / 2)\) and \(sin^2(\phi / 2)\). These probabilities are sensible as their sum is always 1. Despite the infinite degree of freedom introduced by the apparatus rotation, the spinner remains a 'two-state system' contrasting the spatial probability amplitude that spans an infinite number of states.

Step-by-step solution

Calculate Probability of spin-up

The probability of a quantum state is given by the square of the magnitude of its corresponding amplitude. For the spin-up state in the second apparatus, this is given by \(|cos(\phi / 2)T_{2 n d}|^2 = cos^2(\phi / 2)\)

Calculate Probability of spin-down

Similarly, for the spin-down state, the probability is \(|sin(\phi / 2)B_{2 n d}|^2 = sin^2(\phi / 2)\)

Validate Probabilities

The sum of the probabilities for all possible states must equal to 1. We check this by adding the above calculated probabilities. \(cos^2(\phi / 2) + sin^2(\phi / 2) = 1\). This is always true for all real values of \(phi\), and it confirms that these probabilities are logical.

Compare with a Spatial Probability Amplitude

The spatial state is characterized by an infinite number of possibilities, as dictated by the function \(\psi(x)=A e^{-b x^{2}}\). Its range is continuous, allowing an infinite number of states. On the other hand, even though the apparatus rotation introduces an infinite angular range, the spinner can only assume two states i.e. a 'two-state system'

Key concepts

Understanding Quantum Spin

Quantum spin is a fundamental property of particles, similar to charge or mass, but with unique quantum characteristics. Unlike classical spinning, quantum spin doesn't mean the particle is physically rotating. Instead, it's an intrinsic form of angular momentum. This property is intrinsic to particles such as electrons, making them behave like tiny magnets.

In quantum mechanics, spin is measured in units of the reduced Planck's constant. For certain particles, like electrons, this value is \(rac{1}{2}\). Thus, they are called spin-\(rac{1}{2}\) particles. The Stern-Gerlach experiment beautifully demonstrates this property, showing how particles split into discrete spin states when passed through a magnetic field.

These spin states—spin-up and spin-down—are the only possibilities for a spin-\(rac{1}{2}\) particle, confirming their quantum two-state behavior. Understanding this makes it easier to grasp how these states influence behaviors in various quantum systems.

Exploring Probability Amplitude

In quantum mechanics, probability amplitude is a crucial concept used to determine the likelihood of a particle's state. It is not a probability by itself but rather a complex number whose magnitude squared gives the probability.

For spin-\(rac{1}{2}\) particles, the probability amplitude can describe the likelihood of finding a particle in a specific spin state. If we consider the Stern-Gerlach experiment with apparatus aligned and rotated at an angle \(\phi\), the probability amplitude for spin-up is given by \(\cos(\phi / 2)\), and for spin-down by \(\sin(\phi / 2)\).

These amplitudes are critical in quantum mechanics because they allow predictions of probabilities after passing through the apparatus. For example, the probability of finding the particle in the spin-up state is \(\cos^2(\phi / 2)\). The use of probability amplitudes highlights the wave-like nature of quantum entities.

Definition of a Two-State System

A two-state system in quantum mechanics refers to a system where only two possible states exist. This simplicity allows for profound insights into quantum behavior. Spin-\(rac{1}{2}\) particles, like those in the Stern-Gerlach experiment, are quintessential examples.

When a particle is subjected to a magnetic field, it reveals only two outcomes: spin-up or spin-down. Even though the angle \(\phi\) of the apparatus can vary infinitely, it doesn't generate new states. Instead, it adjusts the likelihood of each outcome within the two available states.

This system contrasts with spatial probability distributions like \(\psi(x) = A e^{-bx^2}\), where states are continuous and infinite. In spatial cases, a particle can be in an unlimited number of positions. The two-state system, by being limited, provides a clear framework for understanding fundamental quantum processes.

Characteristics of Spin-1/2 Particles

Spin-\(\frac{1}{2}\) particles include particles like electrons and are fundamental to the study of quantum mechanics. Their spin value of \(\frac{1}{2}\) means that only two spin states are possible, making this an ideal candidate for studying quantum phenomena.

These particles have distinct behaviors because of their intrinsic spin:

• They exhibit magnet-like properties, contributing to magnetic moment.
• They split into two distinct paths in a magnetic field, as shown in the Stern-Gerlach experiment.
• They comply with the Pauli exclusion principle, preventing identical fermions from occupying the same quantum state.

Understanding spin-\(\frac{1}{2}\) particles is essential in fields like quantum computing and particle physics, revealing the intricacies of the quantum world.