How to interpret complex solutions to simple harmonic oscillator?

In summary, the equation of motion for a simple harmonic oscillator is ##m\ddot {x}(t)=-kx(t).## The solutions are ##x(t)=Ae^{i\omega t}+Be^{-i\omega t},## where ##\omega=\sqrt{\frac{k}{m}}## and constants ##A## and ##B##. A complex solution means that both the real and imaginary parts are solutions to the differential equation. However, for physical applications, the solution must be matched to boundary conditions, which constrain the actual result. In order to satisfy real initial conditions, the constants ##A## and ##B## can be complex, but ##B=A^*## to ensure a real solution.
  • #1
schniefen
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Consider the equation of motion for a simple harmonic oscillator:
##m\ddot {x}(t)=-kx(t).##​
The solutions are
##x(t)=Ae^{i\omega t}+Be^{-i\omega t},##​
where ##\omega=\sqrt{\frac{k}{m}}##, and constants ##A## and ##B##. Physically, what does it mean for a solution to be complex? Is it only the real part that is of interest or also the imaginary part? Of course, using Euler's formula, the solution can be rewritten as
##(A+B)\cos{(\omega t)}+i(A-B)\sin{(\omega t)},##​
and one can introduce new, complex constants. However, there is still the issue of a real and imaginary part with that representation.
 
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  • #2
Both the real and imaginary parts are solutions to the differential equation by construction. If you put in real initial values, your determined constants ##A## and ##B## will result in a real solution (i.e., their sum will be real and their difference imaginary - in other words ##B = A^*##).
 
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  • #3
You can form linear combinations of your basis solutions to get real values solutions in terms of sine and cosine. That can form your new basis if you wish.

Edit: and your constants A and B can be complex in order to make this happen.
 
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  • #4
Of course at some point your solution to a physical problem must be matched to boundary conditions in space and time. That will constrain the actual result .
 
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  • #5
Another way to think about it is to say that you only look for solutions with ##x(t) \in \mathbb{R}##. That constrains the "allowed values" for the complex coefficients of you solution to ##B=A^*##. You get still the complete solutions for the real differential equation, because the complex coefficient ##A=A_r + \mathrm{i} A_i## consists of the two real numbers ##A_r## and ##A_i##, which can be used to satisfy the real (!) initial conditions ##x(0)=x_0 \in \mathbb{R} ##, ##\dot{x}(0)=v_0 \in \mathbb{R}##.
 
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1. What is a simple harmonic oscillator?

A simple harmonic oscillator is a system that exhibits oscillatory motion, where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction of the displacement.

2. How are complex solutions interpreted in the context of a simple harmonic oscillator?

Complex solutions in the context of a simple harmonic oscillator represent the amplitude and phase of the oscillatory motion. The real part of the solution represents the amplitude, while the imaginary part represents the phase.

3. What does it mean when a solution to a simple harmonic oscillator is purely real or purely imaginary?

A purely real solution means that the motion is purely oscillatory, with no phase shift. A purely imaginary solution means that the motion is purely exponential, with no oscillation.

4. How do complex solutions affect the period and frequency of a simple harmonic oscillator?

The period and frequency of a simple harmonic oscillator are determined by the real part of the solution. The imaginary part does not affect the period or frequency, but it does affect the phase of the oscillation.

5. Can complex solutions be physically observed in a simple harmonic oscillator?

No, complex solutions cannot be physically observed in a simple harmonic oscillator. The physical motion is always described by the real part of the solution, while the imaginary part is a mathematical construct used to represent the phase of the oscillation.

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