91Ó°ÊÓ

Consider a composite system made up of two spin \(\frac{1}{2}\) objects. For \(t<0\), the Hamiltonian does not depend on spin and can be taken to be zero by suitably adjusting the energy scale. For \(t>0\), the Hamiltonian is given by $$ H=\left(\frac{4 \Delta}{\hbar^{2}}\right) \mathbf{S}_{1} \cdot \mathbf{S}_{2} $$ Suppose the system is in \(|+-\rangle\) for \(t \leq 0\). Find, as a function of time, the probability for being found in each of the following states \(|++\rangle,|+-\rangle,|-+\rangle\), and \(|--\rangle\). a. By solving the problem exactly. b. By solving the problem assuming the validity of first-order time-dependent perturbation theory with \(H\) as a perturbation switched on at \(t=0\). Under what condition does (b) give the correct results?

Step by step solution

01

Understanding the Initial State

Initially, the system is in the state \(|+ -\rangle\). For \(t \leq 0\), the Hamiltonian is zero, meaning the system does not evolve in time. The basis states are \(|+ +\rangle, |+ -\rangle, |- +\rangle, |--\rangle\).

02

Writing the Hamiltonian for t > 0

For \(t > 0\), the Hamiltonian is given by \(H=\left(\frac{4 \Delta}{\hbar^{2}}\right) \mathbf{S}_{1} \cdot \mathbf{S}_{2}\). The dot product can be expressed using the identity \(\mathbf{S}_{1} \cdot \mathbf{S}_{2} = \frac{1}{2}(S^2 - S_1^2 - S_2^2)\). For two spin-\(\frac{1}{2}\) particles: \((S^2) |s, m\rangle = s(s+1)\hbar^2 |s, m\rangle\), where \(s\) is the total spin quantum number.

03

Calculate Using the Total Spin Eigenstates

The possible total spin quantum numbers are \(s=1\) and \(s=0\). Change the basis to total spin states: triplet \(\{ |1,1\rangle=|++\rangle, |1,0\rangle=\frac{1}{\sqrt{2}}(|+-\rangle + |-+\rangle), |1,-1\rangle=|--\rangle \}\) and singlet \(\{ |0,0\rangle=\frac{1}{\sqrt{2}}(|+-\rangle - |-+\rangle) \}\).

04

Solving for Time Evolution

- At \(t=0\), \(|+-\rangle = \frac{1}{\sqrt{2}}(|1,0\rangle + |0,0\rangle)\).- Total spin basis: \(H|1, m\rangle = \tfrac{\Delta}{\hbar^2}|1, m\rangle\) and \(H|0,0\rangle = -\tfrac{3 \Delta}{\hbar^2}|0,0\rangle\).- The states evolve as: \[ |1, m\rangle(t) = e^{-i(\frac{\Delta t}{\hbar})}|1, m\rangle \] \[ |0,0\rangle(t) = e^{i(\frac{3 \Delta t}{\hbar})}|0,0\rangle \]

05

Return to Original Basis

Combine the time-evolved states: \[ |+-\rangle(t) = \frac{e^{-i(\frac{\Delta t}{\hbar})}}{\sqrt{2}}|1,0\rangle + \frac{e^{i(\frac{3 \Delta t}{\hbar})}}{\sqrt{2}}|0,0\rangle \]Convert back: \[ |+-\rangle(t) = \frac{1}{2}(e^{-i(\frac{\Delta t}{\hbar})} + e^{i(\frac{3 \Delta t}{\hbar})})|+-\rangle + \frac{1}{2}(e^{-i(\frac{\Delta t}{\hbar})} - e^{i(\frac{3 \Delta t}{\hbar})})|-+\rangle \]

06

Calculate the Probabilities

- The probability for \( |+-\rangle \) is \( P_{|+-\rangle}(t) = |\frac{1}{2}(e^{-i(\frac{\Delta t}{\hbar})} + e^{i(\frac{3 \Delta t}{\hbar})})|^2 = \cos^2(\frac{2\Delta t}{\hbar})\).- The probability for \( |-+\rangle \) is \( P_{|-+\rangle}(t) = |\frac{1}{2}(e^{-i(\frac{\Delta t}{\hbar})} - e^{i(\frac{3 \Delta t}{\hbar})})|^2 = \sin^2(\frac{2\Delta t}{\hbar})\).- The probability for \( |++\rangle \) and \( |--\rangle \) is \( P_{|++\rangle}(t) = P_{|--\rangle}(t) = 0\).

07

Use Time-Dependent Perturbation Theory

For time-dependent perturbation theory, assume \(H(t)=\theta(t)\hat{V}\) is a perturbation. The first-order probability amplitude is \( c_{f}^{(1)}(t) = -\frac{i}{\hbar}\int_{0}^{t}\langle f | \hat{V} | i \rangle e^{i\omega_{fi}t'} dt'\), mainly non-zero for first-order transitions. This simplifies to valid results when \( \Delta t / \hbar << 1\).

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Key Concepts

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

Spin-1/2 Systems

In quantum mechanics, a **spin-1/2 system** is one of the simplest types of quantum systems, which often involves particles like electrons. Spin, a fundamental property of particles, can be visualized as a type of intrinsic angular momentum. For spin-1/2 particles, this angular momentum can exist in one of two possible states: **up** \((|+\rangle)\) or **down** \((|-\rangle)\). This binary nature is akin to a classical bit and forms the foundation of quantum computing as a **qubit**.

• These particles obey the Pauli exclusion principle, restricting them to unique quantum states in a system.
• The mathematics of spin involves Pauli matrices, which describe rotations and other transformations.
• In a composite system of two spin-1/2 particles, like in the exercise, the combined system has four possible states: \(|++\rangle, |+-\rangle, |-+\rangle,\) and \(|--\rangle\).

This simplicity makes spin-1/2 systems an essential model for learning about more complex quantum systems. Notably, its discrete state space allows clear demonstrations of quantum entanglement and superposition, which are key phenomena in quantum mechanics.

Hamiltonian Dynamics

The **Hamiltonian** is a critical concept in both classical and quantum mechanics, essentially acting as the generator of time evolution in a system. In our context, the Hamiltonian determines how the quantum state of the system evolves over time. For a system of spin-1/2 particles, the Hamiltonian may involve interactions terms in the form of spin operators, which capture the coupling between different particles.
In the original exercise:

• The Hamiltonian for \(t > 0\) is given by \(H=\left(\frac{4 \Delta}{\hbar^{2}}\right) \mathbf{S}_{1} \cdot \mathbf{S}_{2}\).
• This indicates a coupling between the spins of the two particles, where \(\mathbf{S}_{1}\) and \(\mathbf{S}_{2}\) are spin operators for particles 1 and 2, respectively.
• Using the identity \(\mathbf{S}_{1} \cdot \mathbf{S}_{2} = \frac{1}{2}(S^2 - S_1^2 - S_2^2)\), it's clear that the system's energy depends on the total configuration of spins.

The eigenvalues of the Hamiltonian correspond to possible energy levels of the system. Understanding these dynamics allows physicists to predict the behavior of quantum systems under various forces, interactions, and transformations.

Quantum State Evolution

**Quantum state evolution** concerns how quantum states change over time, driven by the system's Hamiltonian. Initially, for \(t \leq 0\), there is no evolution because the Hamiltonian is zero. However, when \(t > 0\), the system described evolves under this Hamiltonian influence.
Key aspects in quantum state evolution:

• The time-dependent Schrödinger equation governs the evolution of quantum states: \(i\hbar \frac{d}{dt} |\psi(t)\rangle = H|\psi(t)\rangle\).
• For the given problem, the total spin states \(|1, m\rangle\) and the singlet state \(|0,0\rangle\) evolve as \(|1, m\rangle(t) = e^{-i(\frac{\Delta t}{\hbar})}|1, m\rangle\) and \(|0,0\rangle(t) = e^{i(\frac{3 \Delta t}{\hbar})}|0,0\rangle\).
• These transformations show how initial quantum states change into superpositions of base states due to interactions defined in the Hamiltonian.

This time dependence is crucial for determining the probability of finding the system in any given state at a later time. These probabilities, calculated by taking the absolute square of the state's amplitude, help predict physical phenomena like saturation and oscillation effects in quantum systems.