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Chapter 5: Problem 30

Consider a two-level system with \(E_{1}0\) by exactly solving the coupled differential equation \\[ i \hbar \dot{c}_{k}=\sum_{n=1}^{2} V_{k n}(t) e^{i \omega_{k n} t} c_{n}, \quad(k=1,2) \\] (b) Do the same problem using time-dependent perturbation theory to lowest nonvanishing order. Compare the two approaches for small values of \(\gamma\). Treat the following two cases separately: (i) \(\omega\) very different from \(\omega_{21}\) and (ii) \(\omega\) close to \(\omega_{21}\) Answer for (a): (Rabi's formula) \\[ \begin{aligned} \left|c_{2}(t)\right|^{2} &=\frac{\gamma^{2} / \hbar^{2}}{\gamma^{2} / \hbar^{2}+\left(\omega-\omega_{21}\right)^{2} / 4} \sin ^{2}\left\\{\left[\frac{\gamma^{2}}{\hbar^{2}}+\frac{\left(\omega-\omega_{21}\right)^{2}}{4}\right]^{1 / 2} t\right\\} \\ \left|c_{1}(t)\right|^{2} &=1-\left|c_{2}(t)\right|^{2} \end{aligned} \\]

Short Answer

Expert verified
Rabi's formula gives exact probabilities as \( |c_2(t)|^2 \) and \( |c_1(t)|^2 = 1 - |c_2(t)|^2 \). Perturbation theory confirms low transition probability unless resonant.

Step by step solution

01

Understanding the Two-Level System

We are dealing with a quantum mechanical two-level system with energies \(E_1\) and \(E_2\) where \(E_1 < E_2\). The potential connecting these levels is time-dependent, given as \(V_{12} = \gamma e^{i \omega t}\) and \(V_{21} = \gamma e^{-i \omega t}\), with \(V_{11} = V_{22} = 0\). This setup is crucial for determining the probability amplitudes \(c_1(t)\) and \(c_2(t)\).
02

Defining the Initial Conditions

At time \(t=0\), the system is initially in the lower energy state. This means \(c_1(0) = 1\) and \(c_2(0) = 0\), indicating that initially, the probability is entirely in the lower state.
03

Solving the Coupled Differential Equations

The equations of motion are given by: \[ i \hbar \dot{c}_1 = V_{12}(t) e^{i \omega_{12} t} c_2 \] and \[ i \hbar \dot{c}_2 = V_{21}(t) e^{i \omega_{21} t} c_1. \] Using these, we can solve for \(c_1(t)\) and \(c_2(t)\) by employing techniques like the rotating wave approximation in quantum mechanics, simplifying the terms to focus on resonant conditions.
04

Using Rabi's Formula

The solutions obtained using Rabi's formula are: \[ |c_2(t)|^2 = \frac{\gamma^2 / \hbar^2}{\gamma^2 / \hbar^2+\left(\omega-\omega_{21}\right)^2 / 4} \sin^2\left\{\left[\frac{\gamma^2}{\hbar^2}+\frac{\left(\omega-\omega_{21}\right)^2}{4}\right]^{1/2} t\right\} \] and \[ |c_1(t)|^2 = 1 - |c_2(t)|^2 \]. This reflects the probability of the system being in each state over time.
05

Time-Dependent Perturbation Theory

For small \(\gamma\), we use time-dependent perturbation theory as an approximation. For case (i) with \(\omega\) different from \(\omega_{21}\), the perturbation is weak and the transition probability remains small. For (ii) where \(\omega\) is close to \(\omega_{21}\), the system can undergo Rabi oscillations indicating a higher transition probability.
06

Comparative Analysis

When \(\gamma\) is small, Rabi's formula and perturbation theory agree that transition probabilities are low unless \(\omega \approx \omega_{21}\), where we see significant oscillations. This corroborates the principle that resonance conditions enhance transition probabilities.

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Key Concepts

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

Rabi Oscillations
Rabi oscillations are an essential phenomenon in the study of two-level quantum systems, specifically when an atom interacts with an electromagnetic field. They describe the periodic change in the probability amplitudes of a quantum system's states. For instance, considering our two-level setup, initially, the system resides in its lower energy state. However, when exposed to an oscillatory potential, like the field described by \(V_{12} = \gamma e^{i \omega t}\), the probability of finding the system in its excited state increases and decreases in a sinusoidal manner.
The frequency of these oscillations depends on factors like the coupling strength \(\gamma\) and the detuning \(\omega - \omega_{21}\). The term "detuning" refers to the deviation of the driving frequency \(\omega\) from the natural transition frequency \(\omega_{21}\) of the system. When \(\omega\) is close to \(\omega_{21}\), especially at resonance, Rabi oscillations are prominent and the probabilities of state transition significantly increase. This behavior is captured elegantly through Rabi's formula, which gives the probability \(|c_2(t)|^2\) for the system being in the excited state.
  • Rabi oscillations demonstrate the time evolution of quantum states.
  • Detuning and coupling strength significantly impact the dynamics.
  • At resonance, transitions between states are maximized.
Time-Dependent Perturbation Theory
Time-dependent perturbation theory provides a powerful approximation method for quantum mechanical systems influenced by time-varying potentials, such as our two-level system. It is particularly useful when the interaction potential is considered weak, meaning the coupling constant \(\gamma\) is small.
In cases where the driving frequency \(\omega\) differs significantly from \(\omega_{21}\), perturbation theory predicts that the system remains mostly in its original state with minimal transition probability to the excited state. This can be seen as the system's resistance to change due to the large energy difference between the interaction frequency and the natural transition frequency.
However, when \(\omega\) nears \(\omega_{21}\), even a small perturbation can lead to noticeable probabilities for state transitions. This is due to the resonance effect, where small disturbances resonate with the system's natural frequency, enhancing the likelihood of transitions.
Time-dependent perturbation theory thus provides valuable insight into how frequency alignment impacts quantum state dynamics, aligning well with the observations from using Rabi oscillations.
  • Effective for weak potentials (small \(\gamma\)).
  • Helps understand transition probabilities under various frequency conditions.
  • Resonance leads to enhanced transition efficacy.
Quantum Mechanics Differential Equations
The behavior of quantum mechanical systems, like the two-level system in our exercise, is often conveniently described using differential equations. These equations, derived from the Schr枚dinger equation, express how the probability amplitudes evolve over time under the influence of time-dependent potentials.
In our context, the equations \(i \hbar \dot{c}_1 = V_{12}(t) e^{i \omega_{12} t} c_2\) and \(i \hbar \dot{c}_2 = V_{21}(t) e^{i \omega_{21} t} c_1\) symbolize how the state coefficients change with time. Here, the potential terms represent the external driving force, oscillating and altering the state probabilities, leading to complexities in direct solutions.
Solving these differential equations requires understanding concepts such as the rotating wave approximation, which simplifies analysis by focusing on terms that are in resonance. Rabi's formula emerges from solving these equations, providing a mathematical expression to predict probabilities over time.
The study of such differential quantum mechanics paves the way for further understanding of how quantum systems react under varying conditions, embodying the foundational principles of quantum dynamics.
  • Differential equations model time evolution in quantum systems.
  • Abstract solutions like Rabi's formula quantify state dynamics.
  • Methods such as the rotating wave approximation aid complex problem-solving.

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Most popular questions from this chapter

Consider a particle bound in a simple harmonic-oscillator potential. Initially \((t < \) 0 , it is in the ground state. At \(t=0\) a perturbation of the form \\[ H^{\prime}(x, t)=A x^{2} e^{-t / \tau} \\] is switched on. Using time-dependent perturbation theory, calculate the probability that after a sufficiently long time \((t \gg \tau),\) the system will have made a transition to a given excited state. Consider all final states.

A one-dimensional simple harmonic oscillator of angular frequency \(\omega\) is acted upon by a spatially uniform but time-dependent force (not potential) \\[ F(t)=\frac{\left(F_{0} \tau / \omega\right)}{\left(\tau^{2}+t^{2}\right)}, \quad-\infty

Consider an atom made up of an electron and a singly charged \((Z=1)\) triton \(\left(^{3} \mathrm{H}\right)\) Initially the system is in its ground state \((n=1, l=0) .\) Suppose the system undergoes beta decay, in which the nuclear charge suddenly increases by one unit (realistically by emitting an electron and an antineutrino). This means that the tritium nucleus (called a triton) turns into a helium \((Z=2)\) nucleus of mass \(3\left(^{3} \mathrm{He}\right)\) (a) Obtain the probability for the system to be found in the ground state of the resulting helium ion. The hydrogenic wave function is given by \\[ \psi_{n=1, l=0}(\mathbf{x})=\frac{1}{\sqrt{\pi}}\left(\frac{Z}{a_{0}}\right)^{3 / 2} e^{-Z r / a_{0}} \\] (b) The available energy in tritium beta decay is about \(18 \mathrm{keV}\), and the size of the \(^{3}\) He atom is about 1 A. Check that the time scale \(T\) for the transformation satisfies the criterion of validity for the sudden approximation.

(This is a tricky problem because the degeneracy between the first state and the second state is not removed in first order. See also Gottfried \(1966, \mathrm{p} .397,\) Problem 1.) This problem is from Schiff \(1968,\) p. \(295,\) Problem \(4 .\) A system that has three unperturbed states can be represented by the perturbed Hamiltonian matrix \\[ \left(\begin{array}{lll} E_{1} & 0 & a \\ 0 & E_{1} & b \\ a^{*} & b^{*} & E_{2} \end{array}\right) \\] where \(E_{2} > E_{1} .\) The quantities \(a\) and \(b\) are to be regarded as perturbations that are of the same order and are small compared with \(E_{2}-E_{1}\). Use the second-order nondegenerate perturbation theory to calculate the perturbed eigenvalues. (Is this procedure correct?) Then diagonalize the matrix to find the exact eigenvalues. Finally, use the second- order degenerate perturbation theory. Compare the three results obtained.

Consider a spinless particle in a two-dimensional infinite square well: \\[ V=\left\\{\begin{array}{ll} 0, & \text { for } 0 \leq x \leq a, 0 \leq y \leq a \\ \infty, & \text { otherwise } \end{array}\right. \\] (a) What are the energy eigenvalues for the three lowest states? Is there any degeneracy? (b) We now add a potential \\[ V_{1}=\lambda x y, \quad 0 \leq x \leq a, 0 \leq y \leq a \\] Taking this as a weak perturbation, answer the following: (i) Is the energy shift due to the perturbation linear or quadratic in \(\lambda\) for each of the three states? (ii) Obtain expressions for the energy shifts of the three lowest states accurate to order \(\lambda .\) (You need not evaluate integrals that may appear.) (iii) Draw an energy diagram with and without the perturbation for the three energy states. Make sure to specify which unperturbed state is connected to which perturbed state.
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