0
$\begingroup$

I understand that the wavefunction in Quantum Mechanics is usually treated as a complex vector with one real and one imaginary component. Does it make an actual difference in terms of the answers we get, if we treat the wavefunction as having two real components instead of one real and one imaginary component, or is it just a matter of convention to treat the wavefunction as having one real and one imaginary component instead of two real components?

$\endgroup$
4
  • 1
    $\begingroup$ A complex number can always be thought of as two real numbers along with a funky rule for multiplication, so there's no different. It would just be less convenient. $\endgroup$
    – knzhou
    Aug 24, 2020 at 2:32
  • $\begingroup$ It's kind of like asking if $\mathbf{F} = m \mathbf{a}$ can be treated as having three components or being one vector equation. $\endgroup$
    – knzhou
    Aug 24, 2020 at 2:33
  • $\begingroup$ Complex numbers usually appear in systems where oscillations, waves, periodic motion etc., is involved. It turns out that complex functions are ideal for describing this and therefore quantum systems. However, I am not sure exactly what you mean by "the wavefunction as having two real components instead of one real and one imaginary component". Perhaps you could see this article hear about the wave function: physics.stackexchange.com/q/129496 $\endgroup$
    – joseph h
    Aug 24, 2020 at 3:05
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/32422/2451 and links therein. $\endgroup$
    – Qmechanic
    Aug 24, 2020 at 6:14

1 Answer 1

0
$\begingroup$

There exists the general wavefunction , a solution of a wave differential equation which is a mathematical expression that allows solutions in real numbers and in imaginary numbers. Here you can see the particular differential equation and its solutions that was developed to make a theoretical physics proposal for the quantum mechanical ad hoc solution at the time, the Bohr model.

The wave equation developed by Erwin Schrodinger in 1926 shows some similarities in its one-dimensional form:

shcrod

Note that the complex numbers are inherent in a quantum mechanical wave equation.

Physics theories use mathematics with its theorems and axioms as a tool, to develop calculations that will be descriptive and predictive, i.e. fit experiments and observations. To pick up the correct solutions from the infinity of mathematically possible ones, one uses new axiom like statements, called laws, postulates , principles, ever since the time of Newton (see link for postulates of quantum mechanics).

So by the postulates of quantum mechanics, the theory that describes the small dimensions, the only possible predictions are probabilistic. One can only predict the probability of measuring a particle at (x,y,z,t). This is the result of using complex numbers for the wave equation, which means that the wavefunction itself is not measurable, only $Ψ^*Ψ$ measurable , and gives the probability of finding a particle at (x,y,z,t).

So the answer is No, real functions as solutions of a wave equation cannot predict what the theoretical construct of quantum mechanics predicts for data.

To the observation by @mmesser that :

an imaginary number can be represented by two real numbers. You can turn the Schrodinger equation into two coupled real-valued equations.

my answer is that then extra postulates for quantum mechanics would be needed on how to determine the probability from two coupled differential equations. Complex algebra keeps the formats compact.

$\endgroup$
2
  • 1
    $\begingroup$ I think you are answering a different question. You cannot do it by using a real number in the Schrodinger equation. But as knzhou said, an imaginary number can be represented by two real numbers. You can turn the Schrodinger equation into two coupled real-valued equations. There just isn't much point. $\endgroup$
    – mmesser314
    Aug 24, 2020 at 4:29
  • $\begingroup$ @mmesser314 in your form, you would need an extra postulate in quantum mechanics, how to get probabilities out of two real functions of independent differential equations. In a sense the complex number formalism is compact in postulates. I am trying to point out, as always, that physics theories are driven by data. $\endgroup$
    – anna v
    Aug 24, 2020 at 5:15

Not the answer you're looking for? Browse other questions tagged or ask your own question.