22.1 The harmonic oscillator with one degree of freedom

22.1 The harmonic oscillator with one degree of freedom

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue

An even simpler case of a particle in a potential than the Coulomb potential of chapter 21 is the case of $V(q)$ quadratic in $q.$ This is also the lowest-order approximation when one studies motion near a local minimum of an arbitrary $V(q)$ , expanding $V(q)$ in a power series around this point. We’ll write this as

$$h=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{q}^2$$

with coeficients chosen so as to make $\omega$ the angular frequency of periodic motion of the classical trajectories. These satisfy Hamilton’s equations

$$\dot{p}=-\frac{\partial V}{\partial q}=-m{\omega }^{2}q,\dot{q}=\frac{p}{m}$$

so

$$\ddot{q}=-{\omega }^{2}q$$

which will have solutions with periodic motion of angular frequency $\omega .$ . These solutions can be written as

$$q(t)=c_{+}{e}^{i\omega t}+c_{-}{e}^{-i\omega t}$$

for $c_{+},c_{-}\in \mathbf{C}$ where, since $q(t)$ must be real, we have $c_{-}={\overline{c}}_{+}$ . The space of solutions of the equation of motion is thus two real dimensional, and abstractly this can be thought of as the phase space of the system.

More conventionally, the phase space can be parametrized by initial values that determine the classical trajectories, for instance by the position $q(0)$ and momentum $p(0)$ at an initial time t(0). Since

$$p(t)=m\dot{q}=m{c}_{+}i\omega e^{i\omega t}-m{c}_{-}i\omega e^{-i\omega t}=im\omega (c_{+}{e}^{i\omega t}-{\overline{c}}_{+}{e}^{-i\omega t})$$

we have

$$q(0)=c_{+}+c_{-}=2\mathrm{Re}(c+),p(0)=im\omega (c_{+}-c_{-})=-2m\omega \mathrm{Im}(c_{+})$$

so

$$c_{+}=\frac{1}{2}q(0)-i\frac{1}{2m\omega }p(0)$$

The classical phase space trajectories are

$$q(t)=\left(\frac{1}{2}q(0)-i\frac{1}{2m\omega }p(0)\right)e^{i\omega t}+\left(\frac{1}{2}q(0)+i\frac{1}{2m\omega }p(0)\right)e^{-i\omega t}$$ $$p(t)=\left(\frac{im\omega }{2}q(0)+\frac{1}{2}p(0)\right)e^{i\omega t}+\left(\frac{-im\omega }{2}q(0)+\frac{1}{2}p(0)\right)e^{-i\omega t}$$

Instead of using two real coordinates to describe points in the phase space (and having to introduce a reality condition when using complex exponentials), one can instead use a single complex coordinate, which we will choose as

$$z(t)=\sqrt{\frac{m\omega }{2}}\left(q(t)-\frac{i}{m\omega }p(t)\right)$$

Then the equation of motion is a first-order rather than second-order diferential equation

$$\dot{z}=i\omega z$$

with solutions

$$z(t)=z(0)e^{i\omega t}\text{\hspace{1em}(22.1)}$$

The classical trajectories are then realized as complex functions of t, and parametrized by the complex number

$$z(0)=\sqrt{\frac{m\omega }{2}}\left(q(0)-\frac{i}{m\omega }p(0)\right)$$

Since the Hamiltonian is quadratic in the $p$ and $q,$ we have seen that we can construct the corresponding quantum operator uniquely using the Schr¨odinger representation. For $\mathcal{H}=L^2(\mathbf{R})$ we have a Hamiltonian operator

$$H=\frac{P^2}{2m}+\frac{1}{2}m{\omega }^2{Q}^2=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{q}^2}+\frac{1}{2}m{\omega }^2{q}^2$$

To find solutions of the Schr¨odinger equation, as with the free particle, one proceeds by first solving for eigenvectors of H with eigenvalue $E_{:}$ , which means finding solutions to

$$H{\psi }_E=\left(-\frac{{\hslash }^2}{2m}\frac{d^2}{d{q}^2}+\frac{1}{2}m{\omega }^2{q}^2\right){\psi }_E=E{\psi }_E$$

Solutions to the Schr¨odinger equation will then be linear combinations of the functions

$${\psi }_E(q)e^{-\frac{i}{\hslash }Et}$$

Standard but somewhat intricate methods for solving diferential equations like this show that one gets solutions for $E=E_n=(n+\begin{array}{l}\frac{1}{2}\end{array})\hslash \omega$ , $n$ a non-negative integer, and the normalized solution for a given n (which we’ll denote ${\psi }_n)$ will be

$${\psi }_n(q)={\left(\frac{m\omega }{\pi \hslash 2^{2n}(n!{)}^2}\right)}^{\frac{1}{4}}{H}_n\left(\sqrt{\frac{m\omega }{\hslash }}q\right)e^{-\frac{m\omega }{2\hslash }{q}^2}\text{\hspace{1em}(22.2)}$$

where $H_n$ is a family of polynomials called the Hermite polynomials. The ${\psi }_n$ provide an orthonormal basis for H (one does not need to consider nonnormalizable wavefunctions as in the free particle case), so any initial wavefunction $\psi (q,0)$ can be written in the form

$$\psi (q,0)=\sum _{n=0}^{\infty }{c}_n{\psi }_n(q)$$

with

$$c_n={\int }_{-\infty }^{+\infty }{\psi }_n(q)\psi (q,0)dq$$

(note that the ${\psi }_n$ are real-valued). At later times, the wavefunction will be

$$\psi (q,t)=\sum _{n=0}^{\infty }{c}_n{\psi }_n(q)e^{-\frac{i}{\hslash }{E}_{n}t}=\sum _{n=0}^{\infty }{c}_n{\psi }_n(q)e^{-i(n+\frac{1}{2})\omega t}$$

Chapter contents · Next section: 22.2 Creation and annihilation operators · 中文版本

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