# On the nonsense of complex springs

#### kiwiheretic

I've just been attending lectures on differential equations at the university of YouTube

Noticed something interesting...

For a simple spring system F= -kx so mx'' = -kx or x''+kx/m = 0. This is a homogeneous 2nd order linear DE so the solutions are:
x = e^{+/- i*sqrt(k/m)} where i = sqrt(-1). Now convention has it that there only the real solution is taken because we don't assume that the spring is also producing harmonic motion in higher dimensions.

My question is why does quantum mechanics assert that quantum complex valued wavefunctions are ontologically "real" when only real valued observations are observed in experiments? And why in contrast to this simple spring in which we make no such claims about complex valued mechanical springs?

#### topsquark

Forum Staff
I've just been attending lectures on differential equations at the university of YouTube

Noticed something interesting...

For a simple spring system F= -kx so mx'' = -kx or x''+kx/m = 0. This is a homogeneous 2nd order linear DE so the solutions are:
x = e^{+/- i*sqrt(k/m)} where i = sqrt(-1). Now convention has it that there only the real solution is taken because we don't assume that the spring is also producing harmonic motion in higher dimensions.

My question is why does quantum mechanics assert that quantum complex valued wavefunctions are ontologically "real" when only real valued observations are observed in experiments? And why in contrast to this simple spring in which we make no such claims about complex valued mechanical springs?
First, the general solution of the Classical SHO equation is a linear combination of both exponentials: $$\displaystyle A~e^{it \sqrt{k/m}} + B~e^{-it \sqrt{k/m}}$$. With a little work we can see this is the same as $$\displaystyle C~cos( t \sqrt{k/m}) + D~sin( t \sqrt{k/m} )$$. So the solutions are real despite the apparent complex behavior in the solutions.

Second, the Quantum SHO is different. It is the solution of the Schrodinger equation with a potential $$\displaystyle V = \frac{m \omega ^2 x^2}{2}$$. The wavefunctions are the Hermite polynomials. We usually use a phase convention that makes the wavefunctions real but this is not really necessary. Remember it is the observables that must be real not the wavefunctions themselves.

-Dan

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#### kiwiheretic

Doesn't this require your constants C and D to be complex in order to maintain the same generality? I guess I am saying that with the mechanical string problem they simply strip out the imaginary terms and deal with the real terms only. However they dont do that with QM. However I understand the mechanical spring problem can be solved in other ways yielding only real solutions. (Perhaps integration by parts?) How do we know this can't be done with QM?

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#### HallsofIvy

So, basically, what you are saying is that because you have never learn "complex number" or "complex analysis", no one else should be allowed to use it.

#### kiwiheretic

So, basically, what you are saying is that because you have never learn "complex number" or "complex analysis", no one else should be allowed to use it.
That's hardly an accurate summary of this thread. It was to do with why do we discard the imaginary component when dealing with mechanical systems but we don't with quantum systems and why do we insist that we're not using complex numbers simply to "encode" a notion of phase in simpler mathematical terms.

#### Fox333

@kiwiheretic, do you still remember the link to the lecture? If yes, could you share it, please?

#### kiwiheretic

@kiwiheretic, do you still remember the link to the lecture? If yes, could you share it, please?
It was awhile ago but I believe it was this MIT OCW series: https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/ Those lectures weren't about physics as such but simply differential equations with some practical examples.

This history of this post came from a bit of a controversy I started some time back with http://physicshelpforum.com/philosophy-physics/11298-complex-valued-psi-really-ontological.html and my complaints about imaginary numbers (square root of -1) being used as the description of the physical quantum world. Electrical engineering uses imaginary numbers for calculations but all the outputs, (ie volts, amps, impedance, inductance etc predictions) are real quantities. Quantum Mechanics breaks this rule by insisting that in its world view imaginary numbers and are ontological in its outputs. A claim I struggle with mainly on philosophical grounds but I've made some progress in being able to describe physical quantum outputs without resorting to complex math.

#### Fox333

I see. Btw, there are some alternative researchers who believe that even negative numbers are "fake", not only complex numbers. From my point of view, any numbers aren't ontological, they are just a math abstraction in order to describe real things.

I've made some progress in being able to describe physical quantum outputs without resorting to complex math.
This might be interesting. How have you managed to do?

My interest was caught with that:
we don't assume that the spring is also producing harmonic motion in higher dimensions.
So, my first thought was: "what if it really does?"

#### studiot

The imaginary component is needed in the Konig-Penny equation for metallic and semiconductor bonding, and appears in many other places in electromagnetic dynamics and also in quantum mechanics.