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Chapter 5: Problem 26

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
The probability of excitation is negligible for large \( \tau \). This is reasonable since the force's effect diminishes as it spreads over time.

Step by step solution

01

Understand the Given Problem

We are given a simple harmonic oscillator with angular frequency \( \omega \) and a time-dependent, uniform force \( F(t) \). We need to calculate the probability that the oscillator transitions from the ground state \( \left| 0 \right> \) to the first excited state \( \left| 1 \right> \) under the influence of \( F(t) \). This involves using first-order time-dependent perturbation theory.
02

Set Up the Perturbation Theory

In time-dependent perturbation theory, the transition probability from state \( \left| n \right> \) to state \( \left| n' \right> \) is given by:\[P_{n \to n'} = \left| \frac{-i}{\hbar} \int_{-\infty}^{+\infty} F(t) \left< n' \right| x \left| n \right> e^{i \omega_{n'n} t} dt \right|^2\]where \( \omega_{n'n} = \omega(n'-n) \) and \( \left< n' \right| x \left| n \right> \) is the matrix element.
03

Compute the Matrix Element

For our transition from \( \left| 0 \right> \) to \( \left| 1 \right> \), the relevant matrix element is:\[\left< 1 \right| x \left| 0 \right> = \sqrt{\frac{\hbar}{2m\omega}} \sqrt{1+1} = \sqrt{\frac{\hbar}{m\omega}}\]
04

Evaluate the Integral

The transition probability formula becomes:\[P_{0 \to 1} = \left| \frac{-i}{\hbar} \int_{-\infty}^{+\infty} \frac{F_0 \tau / \omega}{\tau^2 + t^2} \sqrt{\frac{\hbar}{m\omega}} e^{i \omega t} dt \right|^2\]Use the Fourier transform property to evaluate the integral, resulting in:\[\int \frac{1}{\tau^2 + t^2} e^{i\omega t} dt = \frac{\pi}{\tau}e^{-\omega \tau}\]Thus, the transition probability is:\[P_{0 \to 1} = \frac{\pi^2 F_0^2 \hbar}{m \omega^3 \tau^2} e^{-2\omega \tau}\]
05

Analyze the Result

Observe that as \( \tau \to \infty\), the term \( e^{-2\omega\tau} \) becomes very small, making \( P_{0 \to 1} \) negligible, even though \( F(t) \) has a constant impulse. This is reasonable because, with larger \( \tau \), the force becomes spread out over a longer duration, diminishing its effect on excitation.

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Key Concepts

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

Time-Dependent Perturbation Theory
Time-dependent perturbation theory is a powerful tool used in quantum mechanics. It helps to understand how quantum systems evolve when subjected to time-dependent external influences.
The theory extends the concepts of time-independent perturbation theory to scenarios where the perturbation鈥攁 small external force or potential鈥攃hanges with time. In the context of the exercise, we consider a harmonic oscillator acted upon by a time-dependent force. The primary goal is to calculate the probability of transition between quantum states under this influence. By assuming that the perturbation is small, we consider the system's evolution from a known initial state, like the ground state of the harmonic oscillator, to a possible final state, like the first excited state.
This approach uses a series expansion, where only the first-order term is typically considered for small perturbations, making calculations more manageable.
Probability of Transition
The probability of transition refers to the chance that a quantum system in an initial state will move to a different state due to an external perturbation.
In this exercise, we are interested in the system's transition from the ground state to the first excited state under the influence of the time-dependent force. Using first-order time-dependent perturbation theory, we can express the probability of such a transition, denoted as \( P_{0 \to 1} \).To calculate this, we integrate over time the product of the perturbing force and the initial and final state matrix elements. This integral captures how the system's state changes over time due to the external influence.
For the given exercise, the probability is influenced by the characteristics of the applied force \( F(t) \). As the variable \( \tau \) (related to the force's duration) increases, the probability of the transition becomes negligible. This reflects the fact that longer duration translations lower perturbation strength, demonstrating the delicate balance between external forces and quantum state transitions.
Matrix Elements
Matrix elements are crucial in quantum mechanics to calculate probabilities of state transitions. They relate to how two quantum states overlap in the presence of some observable or interaction.
In this scenario, the matrix element \( \left< 1 \right| x \left| 0 \right> \) is needed for calculating the transition probability of our harmonic oscillator system.This matrix element describes how the position operator \( x \) connects the ground state \( \left| 0 \right> \) and the first excited state \( \left| 1 \right> \). It is derived from the properties of the harmonic oscillator and expressed as \( \sqrt{\frac{\hbar}{m\omega}} \) for this specific transition.
Understanding matrix elements provides insight into how different states in a quantum system may be affected by interactions or forces, bridging the gap between theoretical calculations and observable phenomena in quantum physics.

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Most popular questions from this chapter

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} \\]

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.

Work out the quadratic Zeeman effect for the ground-state hydrogen atom \([\langle\mathbf{x} | 0\rangle=\) \(\left.(1 / \sqrt{\pi a_{0}^{3}}) e^{-r / a_{0}}\right]\) due to the usually neglected \(e^{2} \mathbf{A}^{2} / 2 m_{e} c^{2}\) -term in the Hamiltonian taken to first order. Write the energy shift as \\[ \Delta=-\frac{1}{2} \chi \mathbf{B}^{2} \\] and obtain an expression for diamagnetic susceptibility, \(\chi .\) The following definite integral may be useful: \\[ \int_{0}^{\infty} e^{-\alpha r} r^{n} d r=\frac{n !}{\alpha^{n+1}} \\]

Estimate the lowest eigenvalue \((\lambda)\) of the differential equation \\[ \frac{d^{2} \psi}{d x^{2}}+(\lambda-|x|) \psi=0, \quad \psi \rightarrow 0 \quad \text { for }|x| \rightarrow \infty \\] using the variational method with \(\psi=\left\\{\begin{array}{ll}c(\alpha-|x|), & \text { for }|x|<\alpha \\ 0, & \text { for }|x|>\alpha\end{array} \quad(\alpha \text { to be varied })\right.\) as a trial function. (Caution: \(d \psi / d x\) is discontinuous at \(x=0 .\) ) Numerical data that may be useful for this problem are \\[ 3^{1 / 3}=1.442, \quad 5^{1 / 3}=1.710, \quad 3^{2 / 3}=2.080, \quad \pi^{2 / 3}=2.145 \\] The exact value of the lowest eigenvalue can be shown to be 1.019

Consider a particle bound to a fixed center by a spherically symmetrical potential \(V(r)\) (a) Prove \\[ |\psi(0)|^{2}=\left(\frac{m}{2 \pi \hbar^{2}}\right)\left\langle\frac{d V}{d r}\right) \\] for all \(s\) -states, ground and excited. (b) Check this relation for the ground state of a three-dimensional isotropic oscillator, the hydrogen atom, and so on. (Note: This relation has actually been found to be useful in guessing the form of the potential between a quark and an antiquark.)
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