Probability Calculation For Two Non-Identical Spin 1/2 Fermions In State |s=1, M=0>
Introduction
In the fascinating realm of quantum mechanics, understanding the behavior of systems composed of multiple particles is crucial. This article delves into the intricacies of a system comprising two non-identical spin 1/2 fermions. Specifically, we will explore the probability of finding this system in a particular state, given that one fermion is in a state with s₁ₓ = ħ/2 and the other in a state s₂y = -ħ/2. This exploration requires a solid grasp of concepts such as spin angular momentum, quantum states, and probability calculations within the quantum mechanical framework. Understanding the spin states of multiple particles is not just an academic exercise; it has profound implications in various fields, including condensed matter physics, quantum computing, and particle physics. The behavior of these fermions, governed by the principles of quantum mechanics, dictates the properties of matter and energy at the most fundamental level. The concept of spin, an intrinsic form of angular momentum possessed by particles, is central to this discussion. Unlike classical angular momentum, spin is quantized, meaning it can only take on discrete values. For spin 1/2 particles like electrons, the spin quantum number can be either +ħ/2 or -ħ/2 along a given axis. When we consider systems with multiple particles, the individual spins combine to form a total spin, which also has quantized values. In this context, fermions, particles that obey Fermi-Dirac statistics, play a key role. Fermions, such as electrons, protons, and neutrons, have half-integer spins (e.g., 1/2, 3/2, etc.) and are subject to the Pauli exclusion principle, which dictates that no two identical fermions can occupy the same quantum state simultaneously. This principle has profound consequences for the structure of atoms and the behavior of materials. When we have a system of two spin 1/2 fermions, the total spin can be either 0 (a singlet state) or 1 (a triplet state). The singlet state is characterized by the spins of the two particles being anti-aligned, while the triplet state corresponds to the spins being aligned in various ways. The problem at hand asks us to calculate the probability of finding the system in a specific state, namely the |s=1, m=0> state, given the initial spin states of the individual fermions. This calculation involves expressing the given initial state as a superposition of the total spin eigenstates and then determining the squared magnitude of the coefficient corresponding to the desired state. This article aims to provide a comprehensive and accessible explanation of this process, shedding light on the quantum mechanical principles at play.
Problem Statement: Two Non-Identical Spin 1/2 Fermions
The core of our investigation lies in a system comprising two distinct spin 1/2 fermions. These fermions, while sharing the same intrinsic spin magnitude, are distinguishable due to other properties, such as their spatial location or internal quantum numbers. Our initial condition specifies that one fermion is in a state where its spin projection along the x-axis (s₁ₓ) is +ħ/2, while the other fermion is in a state where its spin projection along the y-axis (s₂y) is -ħ/2. This seemingly simple setup opens the door to a rich exploration of quantum mechanical principles. To fully grasp the problem, we must first define the spin states of individual fermions. A spin 1/2 particle has two possible spin states along any given axis, typically represented as |+> and |->. These states correspond to the spin being aligned either parallel or anti-parallel to the chosen axis. In our case, we have one fermion with a definite spin projection along the x-axis and another along the y-axis. This requires us to express these states in a common basis, usually the z-axis basis, to facilitate calculations. The spin states along different axes are related by unitary transformations. For instance, the eigenstates of the Sx operator (spin along the x-axis) can be written as superpositions of the eigenstates of the Sz operator (spin along the z-axis), and vice versa. Similarly, the eigenstates of the Sy operator (spin along the y-axis) can be expressed in terms of the Sz eigenstates. The key question we aim to answer is: What is the probability of finding this system in the state |s=1, m=0>? Here, |s=1> represents the triplet state with total spin quantum number s=1, and |m=0> specifies the projection of the total spin along the z-axis, which is zero. To find this probability, we need to express the initial state of the system as a linear combination of the eigenstates of the total spin operator. The coefficients in this linear combination will then allow us to calculate the probability amplitude for finding the system in the |s=1, m=0> state. Squaring the magnitude of this probability amplitude will give us the desired probability. The process involves several steps, including expressing the individual spin states in a common basis, constructing the total spin states, and calculating the overlap between the initial state and the target state. The fact that the fermions are non-identical simplifies the analysis somewhat, as we don't need to worry about the exchange symmetry requirements that arise when dealing with identical fermions. However, the fundamental quantum mechanical principles remain the same, and the problem provides a valuable illustration of how spin states combine in multi-particle systems.
Theoretical Framework: Spin Angular Momentum and Quantum States
To solve this problem, a solid foundation in the theoretical underpinnings of spin angular momentum and quantum states is essential. Spin, an intrinsic property of particles, is a form of angular momentum that arises even when the particle is at rest. Unlike classical angular momentum, which is associated with the physical rotation of an object, spin is an inherent quantum mechanical property. For spin 1/2 particles, the spin angular momentum is quantized, meaning it can only take on specific discrete values. The magnitude of the spin angular momentum is given by √(s(s+1))ħ, where s is the spin quantum number (s=1/2 for spin 1/2 particles) and ħ is the reduced Planck constant. The projection of the spin angular momentum along any chosen axis is also quantized, taking on values mħ, where m is the spin projection quantum number. For spin 1/2 particles, m can be either +1/2 or -1/2. These two spin states are often referred to as spin-up and spin-down, respectively. In quantum mechanics, the state of a system is described by a wave function, which is a mathematical function that contains all the information about the system. For a spin 1/2 particle, the spin state can be represented as a linear combination of the spin-up and spin-down states along a particular axis. For example, the spin state along the z-axis can be written as |ψ> = a|+> + b|->, where a and b are complex coefficients and |+> and |-> represent the spin-up and spin-down states, respectively. The squares of the magnitudes of these coefficients, |a|² and |b|², give the probabilities of measuring the particle to be in the spin-up and spin-down states, respectively. When dealing with systems of multiple particles, the total spin angular momentum is obtained by combining the individual spin angular momenta. For two spin 1/2 particles, the total spin quantum number can be either s=0 or s=1. The s=0 state is called the singlet state, and it corresponds to the spins of the two particles being anti-aligned. The s=1 state is called the triplet state, and it corresponds to the spins being aligned in various ways. The triplet state has three possible projections along the z-axis: m=-1, m=0, and m=+1. These states are often denoted as |1,-1>, |1,0>, and |1,+1>, respectively. The singlet state has only one possible projection, m=0, and is denoted as |0,0>. To calculate the probability of finding the system in a particular state, we need to express the initial state of the system as a linear combination of the eigenstates of the total spin operator. This involves using Clebsch-Gordan coefficients, which are mathematical factors that arise when combining angular momenta in quantum mechanics. The probability of finding the system in a specific eigenstate is then given by the square of the magnitude of the corresponding coefficient in the linear combination. Understanding these theoretical concepts is crucial for tackling the problem of determining the probability of finding the two-fermion system in the |s=1, m=0> state. By applying the principles of spin angular momentum and quantum state superposition, we can unravel the quantum mechanical behavior of this system.
Methodology: Calculating the Probability
The central challenge lies in calculating the probability of finding the system in the |s=1, m=0> state. This involves a systematic approach that combines the principles of quantum mechanics with careful mathematical manipulation. The initial step is to express the given state, where one fermion has s₁ₓ = ħ/2 and the other has s₂y = -ħ/2, in terms of the standard basis states, namely the eigenstates of the Sz operator (spin along the z-axis). We denote these eigenstates as |+>z and |->z, representing spin-up and spin-down along the z-axis, respectively. Similarly, we can define eigenstates |+>x and |->x for spin along the x-axis, and |+>y and |->y for spin along the y-axis. The key relationships between these eigenstates are:
- |+>x = (1/√2)(|+>z + |->z)
- |->x = (1/√2)(|+>z - |->z)
- |+>y = (1/√2)(|+>z + i|->z)
- |->y = (1/√2)(|+>z - i|->z)
Given that the first fermion is in the state |+>x and the second is in the state |->y, the combined initial state can be written as:
|ψinitial> = |+>x ⊗ |->y
Substituting the expressions for |+>x and |->y in terms of the z-basis states, we get:
|ψinitial> = (1/√2)(|+>z + |->z) ⊗ (1/√2)(|+>z - i|->z)
Expanding this tensor product, we obtain:
|ψinitial> = (1/2)(|+>z|+>z - i|+>z|->z + |->z|+>z - i|->z|->z)
This expression represents the initial state as a superposition of four basis states in the z-basis. The next step is to express the target state, |s=1, m=0>, in terms of the same basis states. The |s=1, m=0> state, which is one of the triplet states, can be written as:
|1,0> = (1/√2)(|+>z|->z + |->z|+>z)
This state represents a situation where the total spin is 1, but the projection of the spin along the z-axis is zero. This means that the spins of the two fermions are anti-aligned along the z-axis, but in a symmetric way. To find the probability of finding the system in the |1,0> state, we need to calculate the probability amplitude, which is the inner product of the initial state and the target state:
Amplitude = <1,0|ψinitial>
Substituting the expressions for |ψinitial> and |1,0>, we get:
Amplitude = <(1/√2)(|+>z|->z + |->z|+>z)|(1/2)(|+>z|+>z - i|+>z|->z + |->z|+>z - i|->z|->z)>
Using the orthonormality of the basis states, we can simplify this expression. The inner product of two different basis states is zero, and the inner product of a basis state with itself is one. This allows us to eliminate several terms in the expression and focus on the terms that contribute to the overlap. After performing the inner product and simplifying, we obtain:
Amplitude = (1/2√2)(-i + 1)
The probability of finding the system in the |1,0> state is the square of the magnitude of this amplitude:
Probability = |Amplitude|² = |(1/2√2)(-i + 1)|²
Calculating the magnitude and squaring it, we find:
Probability = (1/8)(1 + 1) = 1/4
Therefore, the probability of finding the system in the |s=1, m=0> state is 1/4 or 25%. This result highlights the probabilistic nature of quantum mechanics and demonstrates how spin states combine in multi-particle systems.
Results and Discussion: Probability of Finding the System in |s=1, m=0>
Our detailed calculations reveal that the probability of finding the two-fermion system in the |s=1, m=0> state is precisely 1/4 or 25%. This result provides valuable insights into the quantum mechanical behavior of the system and underscores the probabilistic nature of quantum measurements. The fact that the probability is not 100% indicates that the initial state is not a pure |s=1, m=0> state but rather a superposition of different total spin states. The initial state, defined by one fermion having s₁ₓ = ħ/2 and the other having s₂y = -ħ/2, is a specific configuration of individual spin projections. However, when we consider the total spin of the system, this initial state can be decomposed into a linear combination of eigenstates of the total spin operator. These eigenstates include the singlet state (s=0) and the triplet states (s=1, m=-1, 0, +1). Our calculation effectively quantifies the contribution of the |s=1, m=0> triplet state to this superposition. The probability of 25% signifies that this particular triplet state component is present in the initial state with a non-negligible amplitude. This has important implications for understanding the system's response to external fields and its interactions with other particles. For instance, if we were to apply a magnetic field along the z-axis, the different total spin states would experience different energy shifts due to the Zeeman effect. The |s=1, m=0> state would have a specific energy shift, and the probability of finding the system in this state would influence the overall magnetic properties of the system. Furthermore, the result highlights the importance of quantum superposition in multi-particle systems. The initial state is not simply a classical combination of the individual spin states; it is a quantum superposition where the system exists in multiple states simultaneously until a measurement is made. The measurement process forces the system to collapse into one of the eigenstates, with the probability of collapsing into a particular state determined by the square of the amplitude of that state in the superposition. In this context, our calculation demonstrates how the initial spin configuration, defined in terms of individual spin projections along different axes, translates into probabilities for the total spin states. This connection between individual particle properties and collective system behavior is a hallmark of quantum mechanics and is crucial for understanding the properties of matter at the atomic and subatomic levels. The result also underscores the role of Clebsch-Gordan coefficients in combining angular momenta in quantum mechanics. These coefficients, which implicitly appear in our calculation when we express the total spin states in terms of the individual spin states, dictate the allowed combinations of angular momenta and their corresponding probabilities. The specific value of 1/4 arises from the interplay of these coefficients and the initial spin configuration. In summary, the probability of 1/4 for finding the system in the |s=1, m=0> state is a direct consequence of the quantum mechanical principles of spin angular momentum, superposition, and probability. This result provides a quantitative measure of the contribution of this particular triplet state to the overall state of the system and has implications for understanding its physical properties and behavior.
Conclusion
In conclusion, our analysis of the system composed of two non-identical spin 1/2 fermions, with one fermion in a state with s₁ₓ = ħ/2 and the other in a state s₂y = -ħ/2, reveals a fascinating aspect of quantum mechanics. We have determined that the probability of finding this system in the |s=1, m=0> state is 1/4 or 25%. This result, obtained through a rigorous application of quantum mechanical principles, highlights the probabilistic nature of quantum measurements and the importance of quantum superposition in multi-particle systems. The initial spin configuration, defined by the individual spin projections along different axes, does not uniquely determine the total spin state of the system. Instead, it leads to a superposition of different total spin states, with the |s=1, m=0> state being one of the components in this superposition. The calculated probability quantifies the contribution of this particular triplet state to the overall state of the system. This finding has several important implications. It underscores the fact that in quantum mechanics, systems can exist in multiple states simultaneously, a concept known as superposition. The act of measurement forces the system to collapse into one of these states, with the probability of collapsing into a particular state determined by the square of the amplitude of that state in the superposition. In our case, the 25% probability indicates that the |s=1, m=0> state is a significant component of the initial state, but it is not the only one. The result also highlights the role of Clebsch-Gordan coefficients in combining angular momenta in quantum mechanics. These coefficients, which implicitly appear in our calculation, dictate the allowed combinations of angular momenta and their corresponding probabilities. The specific value of 1/4 arises from the interplay of these coefficients and the initial spin configuration. Furthermore, the analysis provides a concrete example of how individual particle properties (in this case, the spin projections of the two fermions) translate into collective system behavior (the total spin state). This connection between microscopic and macroscopic properties is a central theme in quantum mechanics and is crucial for understanding the behavior of matter at the atomic and subatomic levels. The methods and concepts used in this analysis have broader applications in various fields of physics, including condensed matter physics, quantum computing, and particle physics. Understanding the spin states of multiple particles is essential for describing the magnetic properties of materials, the behavior of quantum gates in quantum computers, and the interactions of elementary particles. In summary, our investigation of the two-fermion system provides a valuable illustration of the power and subtlety of quantum mechanics. The result of 1/4 for the probability of finding the system in the |s=1, m=0> state is a testament to the probabilistic nature of quantum measurements, the importance of quantum superposition, and the intricate interplay of angular momenta in multi-particle systems.
