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Chapter 5: Problem 29

Consider a one-dimensional simple harmonic oscillator whose classical angular frequency is \(\omega_{0}\). For \(t<0\) it is known to be in the ground state. For \(t>0\) there is also a time-dependent potential $$ V(t)=F_{0} x \cos \omega t $$ where \(F_{0}\) is constant in both space and time. Obtain an expression for the expectation value \(\langle x\rangle\) as a function of time using time- dependent perturbation theory to lowest nonvanishing order. Is this procedure valid for \(\omega \simeq \omega_{0} ?\)

Short Answer

Expert verified
\(\langle x \rangle(t) = \frac{F_0}{m(\omega_0^2 - \omega^2)} \sin((\omega - \omega_0) t)\) for \(\omega \neq \omega_0\)

Step by step solution

01

Understanding the System

The system is a one-dimensional simple harmonic oscillator (SHO) initially in its ground state. For times after 0, a time-dependent perturbation potential \( V(t) = F_0 x \cos(\omega t) \) is applied. The perturbation affects the dynamics of the system, specifically modifying the expectation values over time.
02

Identify Base and Perturbation Hamiltonian

The Hamiltonian for a simple harmonic oscillator is given by \( H_0 = \frac{p^2}{2m} + \frac{1}{2}m\omega_0^2 x^2 \). The perturbation Hamiltonian to be added is \( H'(t) = F_0 x \cos(\omega t) \). Thus, the total Hamiltonian is \( H(t) = H_0 + H'(t) \).
03

Use Time-Dependent Perturbation Theory

We are interested in finding the expectation value of position, \(\langle x \rangle\), due to \( H'(t) \) using first-order time-dependent perturbation theory. The first-order correction to the state is given by \[ |\psi^{(1)}(t)\rangle = -\frac{i}{\hbar} \int_{0}^{t} H'(t') |\psi_0(t')\rangle \, dt', \] where \(|\psi_0(t)\rangle\) is the unperturbed state evolution.
04

Calculate Perturbation Effects

Substituting the form of the perturbation, the matrix elements become:\[ \langle n| H'(t) |0\rangle = F_0 \langle n| x \cos(\omega t) |0\rangle. \]Only transitions \( |0\rangle \to |1\rangle \) are considered (due to selection rules, non-vanishing matrix elements for odd transitions). This simplifies to \[ \langle 1| x |0\rangle = \sqrt{\frac{\hbar}{2m\omega_0}}. \]
05

Integrate to Find Expectation Value

Calculate \(\langle x \rangle\) as:\[ \langle x \rangle(t) = \sum_{n} c_{n}(t) \langle n| x |0\rangle. \]The coefficient for \(c_1(t)\) at first order is determined as:\[ c_1(t) = -\frac{iF_0}{\hbar}\sqrt{\frac{\hbar}{2m\omega_0}} \int_0^t \cos(\omega t') e^{i\omega_0 t'}\, dt'. \]This integral evaluates to give terms with a time-dependent exponential. Upon simplifying,\[ \langle x \rangle(t) = \frac{F_0}{m(\omega_0^2 - \omega^2)} \sin(\omega t - \omega_0 t). \]
06

Validity for \(\omega \approx \omega_0\)

For this perturbation theory to be valid, \(\omega\) should not be exactly equal to \(\omega_0\), as resonant effects would complicate the calculation. If \(\omega \to \omega_0\), this expression involves division by zero and thus is not reliable in this form for resonance. Hence, the derivation is only valid as long as \(\omega\) is not too close to \(\omega_0\).

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Key Concepts

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

Time-Dependent Perturbation Theory
In quantum mechanics, time-dependent perturbation theory is used when a system is subject to a time-varying external influence. This theory helps us to calculate how a system originally in an eigenstate is influenced by perturbations over time, especially when those perturbations are not strong. For our simple harmonic oscillator, a perturbative potential \( V(t) = F_0 x \cos(\omega t) \) is added.
This potential represents an external force affecting the particle's motion, which is not considered in the unperturbed state. In first-order theory, we calculate the new state by integrating the effect of this time-dependent interaction over time. This allows us to derive the perturbative effects on the system's wavefunction, often expressed as a small correction to the original state. Here, it's crucial to look at how the unperturbed system evolves and how the perturbation modifies it slightly to gain insights into observables like the position expectation value. This small correction is then captured and expressed mathematically to reflect the time-dependent changes in the system.
Simple Harmonic Oscillator
The simple harmonic oscillator (SHO) is a fundamental model in physics representing a system where the restoring force is proportional to the displacement from equilibrium. For a quantum mechanical perspective, the system is characterized by specific energy eigenstates and energy spacings. These eigenstates are "quantized," meaning they have specific discrete energy levels.
The classic example is a mass attached to a spring which oscillates with frequency \( \omega_0 \). However, in quantum mechanics, it is abstractly represented with a Hamiltonian \( H_0 \) given as:\[ H_0 = \frac{p^2}{2m} + \frac{1}{2}m\omega_0^2 x^2. \]The ground state or lowest energy state is of particular interest and is often used as the starting point for perturbative calculations. In the context of the problem, the SHO is initially in its ground state, which provides a stable basis for analyzing the time-dependent effects introduced with the additional potentialF鈧� x cos(蠅t). When addressing the simple harmonic oscillator, key attention is on how these discrete states interact with external perturbations.
Expectation Value in Quantum Mechanics
The expectation value in quantum mechanics refers to the average value of a quantity that can be expected from multiple measurements on a system identically prepared. This is particularly useful when determining position, momentum, or energy distributions. Mathematically, the expectation value of a position \( x \) is calculated as \( \langle x \rangle = \langle \psi | x | \psi \rangle \), where \( | \psi \rangle \) describes the system's quantum state.
In the exercise, we compute how \( \langle x \rangle \) evolves over time due to the applied perturbation. This involves using the modified state's coefficients resulting from the perturbation theory calculations. For the first order, the expectation value requires integration over time, reflecting how the system's behavior harmonically oscillates according to the parameters of the disturbance.
  • The perturbative force introduces oscillations in \( \langle x \rangle \) with frequencies dependent on both \( \omega \) and \( \omega_0 \).
  • Accurate calculation of these terms aids in understanding how the system evolves and provides insights into the overall quantum dynamics when subjected to time-dependent forces.
Though initially straightforward, this process demonstrates how quantum systems handle real-world perturbations, showcasing the intricate interplay between theory and observable phenomena.

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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 an isotropic harmonic oscillator in two dimensions. The Hamiltonian is $$ H_{0}=\frac{p_{x}^{2}}{2 m}+\frac{p_{y}^{2}}{2 m}+\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}\right) $$ a. What are the energies of the three lowest-lying states? Is there any degeneracy? b. We now apply a perturbation $$ V=\delta m \omega^{2} x y $$ where \(\delta\) is a dimensionless real number much smaller than unity. Find the zerothorder energy eigenket and the corresponding energy to first order [that is, the unperturbed energy obtained in (a) plus the first-order energy shift] for each of the three lowest-lying states. c. Solve the \(H_{0}+V\) problem exactly. Compare with the perturbation results obtained in (b).

A particle of mass \(m\) moves in a potential well \(V(x)=m \omega^{2} x^{2} / 2\). Treating relativistic effects to order \(\beta^{2}=(p / m c)^{2}\), find the ground-state energy shift.

This problem highlights anomalies in the "exponential" decay of a state. It is inspired by Winter, Phys. Rev., 123 (1961) 1503 , but modern computer applications make it straightforward to directly evaluate the integrals numerically. Consider a particle of mass \(m\) that is initially inside a "well" bounded by an infinite wall to the left and a \(\delta\)-function potential on the right: The infinite wall is located at \(x=-a\), and the potential at \(x=0\) is \(U \delta(x)\) where \(U\) is a positive constant. The figure also shows a plausible "ground state" initial wave function \(\Psi(x, t=0)=(2 / a)^{1 / 2} \sin (n \pi x / a)\) with \(n=1\). a. Show that the wave function at all times can be written as $$ \Psi(x, t)=2 n\left(\frac{2}{a}\right)^{1 / 2} \int_{0}^{\infty} d q \frac{e^{-i T q^{2}} q \sin q[q \sin (l+1) q+f]}{\left(q^{2}-n^{2} \pi^{2}\right)\left(q^{2}+G q \sin 2 q+G^{2} \sin ^{2} q\right)} $$ where \(q \equiv\left[a(2 m E)^{1 / 2}\right] / \hbar\) for a particle with energy \(E, T \equiv \hbar t / 2 m a^{2}, l \equiv x / a\), and \(G \equiv 2 m a U / \hbar^{2}\) are all dimensionless quantities, and \(f=0\) for \(-a \leq x \leq 0\) and \(f=G \sin q \sin l q\) for \(x \geq 0\). This is most easily done by expanding the wave function in energy eigenstates \(|E\rangle\), as $$ \Psi(x, t)=\int_{0}^{\infty} d E \phi_{E}(x) e^{-i E t / \hbar} $$ where \(\phi_{E}(x)\) is an energy eigenfunction and \(\left\langle E \mid E^{\prime}\right\rangle=\delta\left(E-E^{\prime}\right)\). b. Write a computer program to (numerically) integrate the probability of finding the particle inside the well. Carry out the integration for a series of values of \(T\) between zero and 12 , using the same parameters as Winter, namely \(n=1\) and \(G=6\). Plotting these probabilities as a function of \(T\) should resemble Figure 2 of Winter's paper. Fit the points for \(2 \leq T \leq 8\) to an exponential, and compare the decay time to Winter's value of \(0.644\). c. Examine the behavior for \(T \geq 8\), and compare to the behavior Winter found for the current at \(x=0\). This suggests an experimental measurement. See Norman et al., Phys. Rev. Lett., \(\mathbf{6 0}\) (1988) \(2246 .\)

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 \(\pm \hbar \omega\) ?
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