Solving the Harmonic Oscillator with Complex Numbers

As a quick review, a harmonic oscillator is a system that, when displaced from equilibrium, experiences a restoring force proportional to the displacement.

Written algebraically, $$\vec{F}=-k\vec{x}$$ where $\vec{F}$ is the force vector, $\vec{x}$ is the displacement vector, and $k$ is a positive constant.

Suppose that we have a harmonic oscillator, and attached to this oscillator is a mass $m$.

Unfortunately (or perhaps fortunately), we will have to use calculus to understand the motion of the oscillator.

Using Newton's Second Law of Motion, $$\vec{F}=m\vec{a}$$ where $\vec{a}$ is the acceleration.

We know that the acceleration is the second-order derivative of position with respect to time.

Using the definition of a harmonic oscillator above, we can set up the following: $$m\frac{d^2x}{dt^2}=-kx$$ $$\frac{d^2x}{dt^2}=-\frac{k}{m}x$$

Exponential functions should come to mind! Note that in an exponential function, the rate of change is proportional to the value of the function. We will try to use this to solve our differential equation.

Suppose that we have an exponential function $f(t)=Ae^{at}$. Let us differentiate this function: $$f'(t)=Aae^{at}$$ $$f''(t)=Aa^2e^{at}$$ Sure enough, if we let $x=f(t)=Ae^{at}$, then we have: $$\frac{d^2x}{dt^2}=Aa^2e^{at}$$ $$\frac{d^2x}{dt^2}=a^2x$$ Now, we just replace our $a^2$ coefficient with $-\frac{k}{m}$, and we're done! $...$ Wait a second: $$\frac{d^2x}{dt^2}=a^2x$$ $$\frac{d^2x}{dt^2}=-\frac{k}{m}x$$ $$a^2=-\frac{k}{m}$$

The right way to go is to broaden our perspective and include complex numbers into our discussion. Now, the equation we must solve is very easy: $$a^2=-\frac{k}{m}$$ $$a=\pm i\sqrt{\frac{k}{m}}$$ $$x=Ae^{at}=Ae^{\pm i\sqrt{\frac{k}{m}}t}$$ Now, we use the famous Euler's identity, which states that $e^{ix}=\cos x+i\sin x$. Then we get the following: $$x=Ae^{\pm i\sqrt{\frac{k}{m}}t}$$ $$x=A\cos(\pm\sqrt{\frac{k}{m}}t)+Ai\sin(\pm\sqrt{\frac{k}{m}}t)$$ Now, we only care about the real part of $x$ for our oscillation, although the complex numbers did provide us with a neat mathematical maneuver: $$Re(x)=A\cos(\pm\sqrt{\frac{k}{m}}t)$$ Using the fact that the cosine function is even, we have our answer: $$Re(x)=A\cos\left(t\sqrt{\frac{k}{m}}\right)$$ Thus, we have beautifully found the relation of position and time for a harmonic oscillator.

Another Way of Solving the Simple Harmonic Oscillator

The Simple Harmonic Oscillator System can also be solved by using separation of variables, instead of complex numbers. The differential equation we are trying to solve here is $$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -\omega^2 x$$

We note that we can write $\frac{\mathrm{d}^2x}{\mathrm{d}t^2} = v \frac{\mathrm{d}v}{\mathrm{d}x}$ since $\frac{\mathrm{d}x}{\mathrm{d}t}$ and we thus obtain:

$$\int v \mathrm{d}v = \int -\omega^2 x \mathrm{d}x$$

We can later write $v = \frac{\mathrm{d}x}{\mathrm{d}t}$ and it is left as an exercise to the reader that the same answer is obtained for the equation of motion of a simple harmonic oscillator as the one obtained when complex numbers is used to solve the differential equation.