A particle of mass m executes simple harmonic motion at...

A particle of mass $m$ executes simple harmonic motion at angular frequency $\omega$. Initially it is in its ground state but from $t=0$ its motion is disturbed by a steady force $F .$ Show that at time $t>0$ and to first order in $F$ the state is $$ |\psi, t\rangle=\mathrm{e}^{-\mathrm{i} E_{0} t / \hbar}|0\rangle+a_{1} \mathrm{e}^{-\mathrm{i} E_{1} t / \hbar}|1\rangle $$ where $$ a_{1}=\frac{\mathrm{i}}{\sqrt{2 m \hbar \omega}} \int_{0}^{t} \mathrm{~d} t^{\prime} F\left(t^{\prime}\right) \mathrm{e}^{\mathrm{i} \omega t^{\prime}} $$ Calculate $\langle x\rangle(t)$ and show that your expression coincides with the classical solution $$ x(t)=\int_{0}^{t} \mathrm{~d} t^{\prime} F\left(t^{\prime}\right) G\left(t-t^{\prime}\right) $$ where the Green's function is $G\left(t-t^{\prime}\right)=\sin \left[\omega\left(t-t^{\prime}\right)\right] / m \omega .$ Show that a suitable displacement of the point to which the oscillator's spring is anchored could give rise to the perturbation.

A particle of mass $m$ executes simple harmonic motion at angular frequency $\omega$. Initially it is in its ground state but from $t=0$ its motion is disturbed by a steady force $F .$ Show that at time $t>0$ and to first order in $F$ the state is
$$
|\psi, t\rangle=\mathrm{e}^{-\mathrm{i} E_{0} t / \hbar}|0\rangle+a_{1} \mathrm{e}^{-\mathrm{i} E_{1} t / \hbar}|1\rangle
$$
where
$$
a_{1}=\frac{\mathrm{i}}{\sqrt{2 m \hbar \omega}} \int_{0}^{t} \mathrm{~d} t^{\prime} F\left(t^{\prime}\right) \mathrm{e}^{\mathrm{i} \omega t^{\prime}}
$$
Calculate $\langle x\rangle(t)$ and show that your expression coincides with the classical solution
$$
x(t)=\int_{0}^{t} \mathrm{~d} t^{\prime} F\left(t^{\prime}\right) G\left(t-t^{\prime}\right)
$$
where the Green's function is $G\left(t-t^{\prime}\right)=\sin \left[\omega\left(t-t^{\prime}\right)\right] / m \omega .$ Show that a suitable displacement of the point to which the oscillator's spring is anchored could give rise to the perturbation.

Key Concepts

Displaced Equilibrium Position

A displacement of the oscillator’s anchoring point, or the point at which the spring is fixed, effectively introduces a force that shifts the equilibrium position. This physical interpretation demonstrates that the perturbation can be envisioned as a slight shift in the potential’s minimum, thereby altering the oscillator’s dynamics in a way analogous to applying an external force.

Correspondence Principle

The correspondence principle states that the predictions of quantum mechanics must converge to those of classical mechanics in the appropriate limit. In this scenario, showing that the expectation value ?x? matches the classical solution illustrates how quantum mechanics recovers classical behavior when the system is under a weak disturbance.

Green's Function

Green's functions are mathematical tools used to solve inhomogeneous differential equations subject to certain boundary conditions. In the classical analysis of a forced harmonic oscillator, the Green's function encapsulates the oscillator’s impulse response and is used to construct the solution for the displacement by integrating over the history of the applied force.

Perturbation Theory

Perturbation theory is used to analyze how a quantum system responds to a small external influence or disturbance. In this context, the steady force F is treated as a perturbation to the unperturbed harmonic oscillator, and the state is expanded to first order in F, which allows calculation of corrections to the wave function and related observables.

Simple Harmonic Motion

This concept describes the dynamics of a system oscillating around an equilibrium point with a specific angular frequency, ?. In quantum mechanics, the simple harmonic oscillator provides a fundamental model with evenly spaced energy levels and eigenstates that are widely used due to its solvability and its role as an approximation for many physical systems.

Quantum Expectation Value

The quantum expectation value, such as ?x?, is the average value of an observable obtained by integrating the product of the wave function, the observable, and the complex conjugate of the wave function over all space. In the problem, computing ?x? reveals how the quantum state’s evolution under the influence of a force reproduces the behavior expected from classical mechanics.

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