Question

A linear harmonic oscillator is subjected to a spatially uniform external force $F(t)=C \eta(t) e^{-\lambda t}$ where $\lambda$ is a positive constant and $\eta(t)$ the Heaviside step function (A.23). If the oscillator is in the ground state at $t<0$, calculate the probability of finding it at time $t$ in an oscillator eigenstate with quantum number $n$. Assuming $C=\left(\hbar m \lambda^3\right)^{1 / 2}$, examine the variation of the transition probabilities with $n$ and with the ratio $\lambda / \omega, \omega$ being the natural frequency of the harmonic oscillator.

   A linear harmonic oscillator is subjected to a spatially uniform external force $F(t)=C \eta(t) e^{-\lambda t}$ where $\lambda$ is a positive constant and $\eta(t)$ the Heaviside step function (A.23). If the oscillator is in the ground state at $t<0$, calculate the probability of finding it at time $t$ in an oscillator eigenstate with quantum number $n$. Assuming $C=\left(\hbar m \lambda^3\right)^{1 / 2}$, examine the variation of the transition probabilities with $n$ and with the ratio $\lambda / \omega, \omega$ being the natural frequency of the harmonic oscillator.

Quantum mechanics

Eugen Merzbacher 3rd Edition

Chapter 14, Problem 2

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Step 1

The harmonic oscillator is initially in the ground state, denoted by $|0\rangle$, at $t < 0$. The external force applied is $F(t) = C \eta(t) e^{-\lambda t}$, where $\eta(t)$ is the Heaviside step function, meaning the force is applied at $t \geq 0$. Show more…

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A linear harmonic oscillator is subjected to a spatially uniform external force $F(t)=C \eta(t) e^{-\lambda t}$ where $\lambda$ is a positive constant and $\eta(t)$ the Heaviside step function (A.23). If the oscillator is in the ground state at $t<0$, calculate the probability of finding it at time $t$ in an oscillator eigenstate with quantum number $n$. Assuming $C=\left(\hbar m \lambda^3\right)^{1 / 2}$, examine the variation of the transition probabilities with $n$ and with the ratio $\lambda / \omega, \omega$ being the natural frequency of the harmonic oscillator.

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