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Chapter 5: Problem 33

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

Short Answer

Expert verified
Transition probability is negligible for large \(\tau\). It's reasonable since longer timescales reduce excitation likelihood.

Step by step solution

01

Understand the Problem Setup

We have a simple harmonic oscillator initially in its ground state, acted upon by a time-dependent force \( F(t) = \frac{F_0 \tau / \omega}{\tau^2 + t^2} \). We need to find the probability of it being in the first excited state at \( t = +\infty \). First-order time-dependent perturbation theory will be applied.
02

Perturbation Theory Formula

Using time-dependent perturbation theory, the probability amplitude for transition from state \( |n\rangle \) to state \( |m\rangle \) is given by:\[ a_{mn}(t) = -\frac{i}{\hbar} \int_{-\infty}^{t} \langle m|H'(t')|n\rangle e^{i(\omega_m - \omega_n)t'} dt' \]Here, \( H'(t) = -xF(t) \) is the perturbation Hamiltonian.
03

Transition Matrix Element

The matrix element \( \langle 1 | H'(t') | 0 \rangle \) is:\[ \langle 1|-x|0\rangle \cdot F(t') \]For a harmonic oscillator, this matrix element equates to the spatial matrix element \( \langle 1 | x | 0 \rangle = \sqrt{\frac{\hbar}{2m\omega}} \). Thus, \[ \langle 1 | H'(t') | 0 \rangle = -\sqrt{\frac{\hbar}{2m\omega}} F(t') \]
04

Compute Transition Amplitude

Substituting into the perturbation formula, we compute the transition amplitude:\[ a_{10}(\infty) = -\frac{i}{\hbar}\sqrt{\frac{\hbar}{2m\omega}} \int_{-\infty}^{\infty} \frac{F_0 \tau / \omega}{\tau^2 + t^2} e^{i\omega_{10} t} dt \]where \( \omega_{10} = \omega \), the difference in energies between the first excited and ground states.
05

Evaluate the Integral

The above integral evaluates to:\[ 2\pi \left(\frac{F_0 \tau}{\omega}\right) e^{-\omega\tau} \text{ for large } \tau \]due to the properties of the Cauchy principal value integral.
06

Calculate Transition Probability

The probability of the oscillator being in the first excited state is given by the square of the modulus of the amplitude:\[ P_{1}(\infty) = |a_{10}(\infty)|^2 \]Thus, substituting the amplitude from Step 5, we get:\[ P_{1}(\infty) = \pi^2 \left( \frac{F_0 \tau}{\omega \hbar} \sqrt{\frac{\hbar}{2m\omega}} \right)^2 e^{-2\omega \tau} \]
07

Analyze Impulse and Magnitude

Calculate the impulse of \( F(t) \):\[ \int_{-\infty}^{+\infty} F(t) dt = \pi \frac{F_0}{\omega} \]Independent of \( \tau \), it confirms uniformity of impulse. For \( \tau \gg 1/\omega \), the term \( e^{-2\omega \tau} \rightarrow 0 \) makes probability negligible.

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Key Concepts

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

Quantum Harmonic Oscillator
In quantum mechanics, a quantum harmonic oscillator is a pivotal model that describes particles in a potential that is quadratic in form. Imagine a spring with a mass attached to one end, oscillating back and forth. The motion of a quantum particle in this scenario, moving under the influence of harmonic forces, is akin to that of our spring-mass system. However, unlike classical oscillators, quantum oscillators can only occupy discrete energy levels.
This means that the particle can be in certain energy states, labeled by quantum numbers, but never in-between. It's crucial here as we're interested in transitions between the ground state and the first excited state induced by an external influence.
Transition Probability
Transition probability in this context refers to the likelihood that a quantum harmonic oscillator will move from one quantized state to another due to some external influence. We typically use time-dependent perturbation theory to evaluate this probability in the presence of a time-dependent force. It quantifies how the system responds to this external influence, transitioning from its original state.
The transition probability we are interested in deals with a particle initially in its ground state. When perturbed, it has some probability of moving to the first excited state. In computations, we often use the square of the modulus of the transition amplitude to determine this probability.
Time-Dependent Force
A time-dependent force varies with time, and in our exercise, it acts on a quantum harmonic oscillator. The force given by the formula \( F(t) = \frac{F_0 \tau / \omega}{\tau^2 + t^2} \) influences the oscillator's behavior. Since it's uniform spatially, the dependence entirely on time simplifies the analysis.
This force is controlled by parameters \( F_0 \) and \( \tau \), which affect its strength and temporal spread, respectively. It's noteworthy that despite this variation of force, the impulse it imparts is constant, offering interesting insights into how time-dependent systems can be analyzed.
Perturbation Hamiltonian
The perturbation Hamiltonian \( H'(t) = -xF(t) \) represents the part of the Hamiltonian responsible for the time-dependent force affecting the system. In quantum mechanics, when a system is disturbed by an external force, this perturbation Hamiltonian describes the interaction.
It plays a central role in determining how the quantum harmonic oscillator transitions between states. By calculating the matrix elements of \( H'(t) \) between states, we can compute transition amplitudes, which lead to probabilities. It's a way of framing the additional energy contribution from the external force, allowing for the calculation of transition probabilities.

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Most popular questions from this chapter

The Hamiltonian matrix for a two-state system can be written as $$ \mathscr{H}=\left(\begin{array}{ll} E_{1}^{0} & \lambda \Delta \\ \lambda \Delta & E_{2}^{0} \end{array}\right) $$ Clearly the energy eigenfunctions for the unperturbed problems \((\lambda=0)\) are given by $$ \phi_{1}^{(0)}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \quad \phi_{2}^{(0)}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right) . $$ a. Solve this problem exactly to find the energy eigenfunctions \(\psi_{1}\) and \(\psi_{2}\) and the energy eigenvalues \(E_{1}\) and \(E_{2}\). b. Assuming that \(\lambda|\Delta| \ll\left|E_{1}^{0}-E_{2}^{0}\right|\), solve the same problem using time-independent perturbation theory up to first order in the energy eigenfunctions and up to second order in the energy eigenvalues. Compare with the exact results obtained in (a). c. Suppose the two unperturbed energies are "almost degenerate," that is, $$ \left|E_{1}^{0}-E_{2}^{0}\right| \ll \lambda|\Delta| \text {. } $$ Show that the exact results obtained in (a) closely resemble what you would expect by applying degenerate perturbation theory to this problem with \(E_{1}^{0}\) set exactly equal to \(E_{2}^{0}\).

Consider a particle in one dimension moving under the influence of some timeindependent potential. The energy levels and the corresponding eigenfunctions for this problem are assumed to be known. We now subject the particle to a traveling pulse represented by a time-dependent potential, $$ V(t)=A \delta(x-c t) . $$ a. Suppose at \(t=-\infty\) the particle is known to be in the ground state whose energy eigenfunction is \(\langle x \mid i\rangle=u_{i}(x)\). Obtain the probability for finding the system in some excited state with energy eigenfunction \(\langle x \mid f\rangle=u_{f}(x)\) at \(t=+\infty\). b. Interpret your result in (a) physically by regarding the \(\delta\)-function pulse as a superposition of harmonic perturbations; recall $$ \delta(x-c t)=\frac{1}{2 \pi c} \int_{-\infty}^{\infty} d \omega e^{i \omega[(x / c)-t]} . $$ Emphasize the role played by energy conservation, which holds even quantum mechanically as long as the perturbation has been on for a very long time.

Consider a particle in a two-dimensional potential $$ V_{0}= \begin{cases}0 & \text { for } 0 \leq x \leq L, 0 \leq y \leq L \\\ \infty & \text { otherwise }\end{cases} $$ Write the energy eigenfunctions for the ground and first excited states. We now add a time-independent perturbation of the form $$ V_{1}= \begin{cases}\lambda x y & \text { for } 0 \leq x \leq L, 0 \leq y \leq L \\ 0 & \text { otherwise }\end{cases} $$ Obtain the zeroth-order energy eigenfunctions and the first-order energy shifts for the ground and first excited states.

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) $$ \begin{aligned} &\left|c_{2}(t)\right|^{2}=\frac{\gamma^{2} / \hbar^{2}}{\gamma^{2} / \hbar^{2}+\left(\omega-\omega_{21}\right)^{2} / 4} \sin ^{2}\left\\{\left[\frac{\gamma^{2}}{\hbar^{2}}+\frac{\left(\omega-\omega_{21}\right)^{2}}{4}\right]^{1 / 2} t\right\\} \\ &\left|c_{1}(t)\right|^{2}=1-\left|c_{2}(t)\right|^{2} \end{aligned} $$

A hydrogen atom in its ground state \([(n, l, m)=(1,0,0)]\) is placed between the plates of a capacitor. A time-dependent but spatial uniform electric field (not potential!) is applied as follows: \(\mathbf{E}=\left\\{\begin{array}{ll}0 & \text { for } t<0 \\ \mathbf{E}_{0} e^{-t / \tau} & \text { for } t>0\end{array}\left(\mathbf{E}_{0}\right.\right.\) in the positive \(z\)-direction \()\) Using first-order time-dependent perturbation theory, compute the probability for the atom to be found at \(t \gg \tau\) in each of the three \(2 p\) states: \((n, l, m)=(2,1, \pm 1\) or 0\()\). Repeat the problem for the \(2 s\) state: \((n, l, m)=(2,0,0)\). Consider the limit \(\tau \rightarrow \infty\).
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