91Ó°ÊÓ

The unperturbed Hamiltonian of a two-state system is represented by $$ H_{0}=\left(\begin{array}{cc} E_{1}^{0} & 0 \\ 0 & E_{2}^{0} \end{array}\right) \text {. } $$ There is, in addition, a time-dependent perturbation $$ V(t)=\left(\begin{array}{cc} 0 & \lambda \cos \omega t \\ \lambda \cos \omega t & 0 \end{array}\right) \quad(\lambda \mathrm{real}) $$ a. At \(t=0\) the system is known to be in the first state, represented by $$ \left(\begin{array}{l} 1 \\ 0 \end{array}\right) $$ Using time-dependent perturbation theory and assuming that \(E_{1}^{0}-E_{2}^{0}\) is not close to \(\pm \hbar \omega\), derive an expression for the probability that the system be found in the second state represented by $$ \left(\begin{array}{l} 0 \\ 1 \end{array}\right) $$ as a function of \(t(t>0)\). b. Why is this procedure not valid when \(E_{1}^{0}-E_{2}^{0}\) is close to \(\pm \hbar \omega\) ?

Step by step solution

01

Understanding the Problem

We need to find the probability that a system initially in state \( \left(\begin{array}{l} 1 \ 0 \end{array}\right) \) transitions to state \( \left(\begin{array}{l} 0 \ 1 \end{array}\right) \) due to a time-dependent perturbation \( V(t) \), using time-dependent perturbation theory.

02

Determining the Perturbation Theory Approach

Using time-dependent perturbation theory, the first-order probability amplitude for transitioning from state \( |1\rangle \) (initial) to state \( |2\rangle \) (final) is given by:\[c_{2}^{(1)}(t) = -\frac{i}{\hbar} \int_{0}^{t} e^{i \omega_{21} t'} V_{21}(t') dt'\]where \( \omega_{21} = \frac{E_2^0 - E_1^0}{\hbar} \) and \( V_{21}(t) = \lambda \cos(\omega t) \).

03

Calculating the Integral

Substitute \( V_{21}(t) = \lambda \cos(\omega t) \) into the integral.\[c_{2}^{(1)}(t) = -\frac{i \lambda}{\hbar} \int_{0}^{t} e^{i \omega_{21} t'} \cos(\omega t') dt'\]Utilize the identity \( \cos(\omega t') = \frac{1}{2}(e^{i \omega t'} + e^{-i \omega t'}) \) to split this into two separate integrals.

04

Solve the Integrals Separately

The integral becomes the sum of two exponential integrals:\[c_{2}^{(1)}(t) = -\frac{i \lambda}{2 \hbar} \left( \int_{0}^{t} e^{i(\omega_{21} + \omega)t'} dt' + \int_{0}^{t} e^{i(\omega_{21} - \omega)t'} dt' \right)\]Compute each integral separately:\[\int e^{i\alpha t'} dt' = \frac{e^{i\alpha t'}}{i\alpha}\]

05

Evaluate the Integrals at Boundaries

For each integral,\[\int_{0}^{t} e^{i(\omega_{21} + \omega)t'} dt' = \frac{1}{i(\omega_{21} + \omega)} \left( e^{i(\omega_{21} + \omega)t} - 1 \right)\]\[\int_{0}^{t} e^{i(\omega_{21} - \omega)t'} dt' = \frac{1}{i(\omega_{21} - \omega)} \left( e^{i(\omega_{21} - \omega)t} - 1 \right)\]

06

Combine Results into Transition Amplitude

Substitute these results back into the expression for \( c_{2}^{(1)}(t) \) and simplify:\[c_{2}^{(1)}(t) = \frac{\lambda}{2\hbar} \left( \frac{ e^{i(\omega_{21} + \omega)t} - 1 }{ \omega_{21} + \omega } + \frac{ e^{i(\omega_{21} - \omega)t} - 1 }{ \omega_{21} - \omega } \right)\]

07

Calculate the Transition Probability

The probability of finding the system in state \( |2\rangle \) is the modulus squared of the transition amplitude:\[P_{1\rightarrow 2}(t) = \left| c_{2}^{(1)}(t) \right|^2\]

08

Addressing Energy Level Conditions

This procedure becomes invalid when \( E_{1}^{0} - E_{2}^{0} \approx \pm \hbar \omega \) due to the resonance condition, causing denominators in \( c_{2}^{(1)}(t) \) to approach zero and making the perturbative approach insufficient.

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Key Concepts

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

Two-State System

A two-state system is a simple quantum mechanical model that describes a system with exactly two possible states or configuration. This model is often used because it's simple yet meaningful, providing insight into more complex quantum phenomena.
In the context of this exercise, the two-state system is represented as a 2x2 matrix with

• two energy eigenstates, each associated with specific energy levels \( E_1^0 \) and \( E_2^0 \).
• The initial state of the system is the first energy level, depicted as \( \left( \begin{array}{c} 1 \ 0 \end{array} \right) \), which implies full occupancy of the first state and none in the second.

Understanding this simple model helps in analyzing how quantum mechanical systems evolve over time, focusing on how a system transitions between these states.

Quantum Mechanics

Quantum mechanics is the branch of physics dealing with the behavior of systems on very small scales, such as atoms and subatomic particles. It is a foundational theory that describes nature at the smallest scales of energy levels.

• It works with probabilities instead of certainties, giving rise to phenomena like superposition and entanglement.
• In a quantum system, particles do not have definite positions or velocities until they are observed.

In this exercise, quantum mechanics enables the investigation into how a quantum system, specifically a two-state system, behaves under a time-dependent perturbation, and how the likelihood of different states changes with time. The probabilistic nature of quantum mechanics requires the use of probability amplitudes and integrals, as in the provided solution, to determine the outcome of quantum transitions.

Hamiltonian Operator

The Hamiltonian operator in quantum mechanics represents the total energy of the system, including kinetic and potential energy. It plays a crucial role in determining the evolution of a quantum system, as it is central to the Schrödinger equation.
In our two-state system, the Hamiltonian is represented by

• an initial, unperturbed Hamiltonian \( H_0 \) with diagonal elements \( E_1^0 \) and \( E_2^0 \), indicating energies of the states without perturbation.
• a time-dependent perturbation \( V(t) \) that introduces a coupling between the states, modulated by \( \lambda \cos \omega t \).

This perturbation essentially shifts the energies of the states over time, leading to the possibility of transitions between states. The Hamiltonian operator is thus the tool used to determine these transitions as well as the overall dynamics of the quantum system.

Transition Probability

In quantum mechanics, transition probability refers to the probability that a system will transition from one state to another over time when subjected to a perturbation. It is a critical concept for understanding how quantum states evolve, especially under influence of external forces or fields.
In this context:

• We seek the probability of the system transitioning from state \( |1\rangle \) to state \( |2\rangle \).
• This involves calculating the transition amplitude, which is a complex number, and then squaring its magnitude to obtain the transition probability.
• The result shows how likely the system is found in the second state after time \( t \).

The exercise uses first-order time-dependent perturbation theory to derive this probability, but notes that the theory becomes problematic in cases of near-resonance, where the energy difference \( E_1^0 - E_2^0 \) approaches \( \pm \hbar \omega \). Here, the approximation may lead to inaccuracies due to resonance effects amplifying the transition.