# De Broglie Waves and Complex Numbers

We used complex variables to describe the wave function. People do that in acoustics and optics too, strictly for convenience, because the real and imaginary parts are rudundant.

The wave function of quantum mechanics is "necessarily" complex, it's not just for convenience that we use complex numbers in quantum theory. Is there any physical reason for the wave function to be complex?

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blechman
I think there is more than one way to answer this question. One way to see that complex numbers are built into the theory rather than simply for convenience like your earlier examples is that Schrodinger's equation is a heat equation with imaginary dispersion coefficient. Thus the solutions are intrinsically complex.

The real and imaginary parts of these solutions would also obey the Schrodinger equation since the equation is linear, but they do not generally also match the boundary conditions, and are therefore not generally solutions to your physical systems (UNLIKE your counterexamples of acoustics and optics).

For a concrete example: a "left-moving particle" has a wavefunction $e^{ikx}$, NOT cos(kx) or sin(kx).

That might not be what you call a "physical reason", but it is a mathematical one.

I'll attempt an explanation without using equations.

The wave function represents the state of the system. Because its time and space derivatives must be proportional to the wave function itself, it takes an exponential form. If the argument in the exponent is real, the wave function will either grow without limit or decay away. If we want the system to persist, we need to make the wave function periodic, and that requires an i in the argument.

By the way, de Broglie waves don't have to be complex, but Schroedinger waves do. The Schroedinger equation contains an explicit i.

epenguin
Homework Helper
Gold Member
The wave function represents the state of the system. Because its time and space derivatives must be proportional to the wave function itself, it takes an exponential form. If the argument in the exponent is real, the wave function will either grow without limit or decay away. If we want the system to persist, we need to make the wave function periodic, and that requires an i in the argument.

By the way, de Broglie waves don't have to be complex, but Schroedinger waves do. The Schroedinger equation contains an explicit i.

Could you expand on the 'must be' please? Why must they be?

Could you expand on the 'must be' please? Why must they be?

For example, if the energy is constant, the time derivative of the wave function is proportional to the energy multiplied by the wave function (the eigenvalue equation).

If we are going to use wave functions to describe motion, then they must be complex.

olgranpappy
Homework Helper
Could you expand on the 'must be' please? Why must they be?

They don't have to be. An obvious example is the particle in a box, the solutions of which are not proportional to their odd spacial derivatives. Another example is the ground state (or any state) of a simple harmonic oscillator. Another example is *any* function other than an exponential.

So, I think that he must just be saying that the Schrodinger equation is linear. But this does not mean that the solutions are always proportional to their space and time derivatives.

Xeinstein - there is no physical reason why complex numbers should be used in QM. If it were so, it would be equivalent to saying it only worked in German, or in base 8 arithmetic.

There are perfectly good QM's that do not use complex numbers. It is merely a (great) convenience.

They don't have to be. An obvious example is the particle in a box, the solutions of which are not proportional to their odd spacial derivatives. Another example is the ground state (or any state) of a simple harmonic oscillator. Another example is *any* function other than an exponential.

So, I think that he must just be saying that the Schrodinger equation is linear. But this does not mean that the solutions are always proportional to their space and time derivatives.

You are correct, of course, wave functions do not need to be complex. I was addressing the question about why they are (when they are). But even the solutions to Schroedinger's time-independent equation that you give as examples are associated with a time-dependent factor that is complex.

Regarding the Schrödinger equation from a mathematical point of view it has just one derivative in time, whereas the Maxwell eqations have a second derivative in time. That's the big difference and requires the solution of Schrödingers equation to be complex.

Physically there is another reason: Energypreservation demands that the solution has to be invariant by time transformation. If the solution would be real and you replace the t by -t there is no invariancy. But by replacing the imaginary part it to -it accomplishes the requirement.

Another reason is the Spin description. The whole theory is only possible in the complex spere (For one Spin the Bloch sphere, minus the part for unity matrix). Spin eigen states have to be orthogonal and for one spin the vector has to be 2 dimensional (because there are two possibilites: Spin up and Spin down) . Further there have to be 3 eigen states and that is possible only if two are in real space and one is imaginary. So complex space enables more orthogonal eigen states.

And then and I guess what's most important: The only way to write a vector product as an integral is in complex space, namely in Quantum physics the Hilbert space. I can't remember exactely, but it has something to do with the bilinear form ...

There would be certainly more reasons ... but that's what's crossing my mind, wright now. (No guarantee that it is correct!)