Why are complex numbers needed in quantum mechanics ? - by Ricardo Karam - Article review

This document contains article review "Why are complex numbers needed in quantum mechanics" by Ricardo Karam
To order to read the article select: https://www.ind.ku.dk/english/research/didactics-of-physics/Karam_AJP_Complex_numbers_in_QM.pdf




However, the situation seems to be different in quantum mechanics, since the imaginary unit appears explicitly in its fundamental equations.
This raises two basic questions: What is the physical meaning of the real and imaginary parts in quantum mechanics. How important are the fundamental equations in order to understand quantum mechanics.
From a learning perspective, this can create some challenges to newcomers.
No it is not. The issue is understanding physics, making physical changes and predicting the future.
All these parts take place in the real world, in which complex numbers don't exist.
Even more strongly in which mathematical numbers don't exist, except objects.

I. Introduction

Complex numbers were invented (or discovered?) in 16th-century Italy as a calculation tool to solve cubic equations.
These specific cubic equations reflect mathematical issues, but no physical issues.
It is particularly helpful to use complex numbers to model periodic phenomena, especially to operate with phase differences.
Another possibility is to treat the real and imaginary parts of a complex number as two related (real) physical quantities.
The most famous case is in Special Realtivity (and GR) were distance is introduced as a real quantity and time an imaginary quantity.
Using the parameters r for distance and t for (imaginary) time we get the following equation: s^2 = r^2 - t^2. The parameter s has no physical meaning.
Using instead of t the parameter ct, with c the speed of light, defined as a constant! we get the following equation: s^2 = r^2 - (c*t)^2. The parameter s is now a distance in complex space.
In both cases, the structure of complex numbers is useful to make calculations more easily, but no physical meaning is actually attached to complex variables.
It suffices to look at some of the most basic equations, both in the matrix XYZ and wave XYZ formulations, to wonder about the presence of the imaginary unit. (See actual document)
What is essentially different in quantum mechanics is that it deals with complex quantities (e.g. wave functions and quantum state vectors) of a special kind, which cannot be split up into pure real and imaginary parts that can be treated independently.
What is not mentioned and what is important, is the wave function and the state vector in an actual situation. What is the wave function for one electron and for one atom consisting of many electrons.
Furthermore, physical meaning is not attached directly to the complex quantities themselves, but to some other operation that produces real numbers (e.g. the square modulus of the wave function or of the inner product between state vectors).
Also here it would be beneficial what it means to calculate the wave function of an atom. This involves a mathematical operation on each the individual wave functions of an electron, proton and neutron.
The problem is that when you try to calculate (measure) one, the two other ones are disturbed.
This complex nature of quantum mechanical quantities puzzled some of the very founders of the theory.
I can understand that because a water wave is a 3D construct. Interference patterns of water waves are 3D structures.

II. Why are Complex Numbers needed in Quantum Mechanics? - page 3

A. No information on position when momentum is known exactly - page 3

The last piece needed for the argument is inspired by Heisenberg's uncertainty principle. It is presented qualitatively in Shankar's textbook by thought experiments like the gammaray microscope.
Such an issue can not be explained as a thought experiment.
One way to state the principle is to say that " it is impossible to prepare a particle in a state in which its momentum and position (along one axis) are exactly known."
That maybe the case, but what does this principle say about the actual position of the particle?
Nothing. Nor how the particle actual moves in 3D.
For more information about uncertainty principle: Reflection 2 - Uncertainty Principle
Now, suppose the electron is in a state of definite momentum.
Why using the word definite? My assumption is that the speed of the particle is constant and moves in a circle at a fixed distance from the center.
When that is the case the particles momentum is constant.
How would the wave function for this state look like?
In that case the wave function is a straight line.
However in both the x and y dimension this can be a sinus or cosines. i.e p = px + py = m*vx^2 + m*vy^2

Page 3

According to de Broglie's relation, there is a wavelength associated with the electron.
This assumes that de Broglie has observed a certain experiment which demonstrates the wave function. Primarily the particle moves in a circle around the center but this can also be an ellipse. From a mathematical point of view the difference is a wave function (superimposed).
Therefore, it seems plausible to assume a classical/real oscillating wave function of the form
psi.p(x) = A cos(2pix/lambda) = Acos( px/h) (1)
where p(x) denotes a state of definite momentum. psi.p(x), as well as |psi.p(x)|^2, are plotted in Fig. 1.
The problem is that psi.p(x) is a description of a water wave in one dimension. In reality we have a particle which moves in a circle (like a planet around the Sun).
The graph of |Psi.p(x)|^2 (probability distribution) illustrates an apparent contradiction.
It should be understood that there also exists a |Psi.p(y)|^2 in the y direction
If you know the momentum of the electron exactly, you should have no information about its position.
That is a complete wrong interpretation of the Uncertainty Principle
See also Reflection 2 - Uncertainty Principle
From a physical point of view all parameters like position, speed, mass and momentum of a particle, at each moment have a certain specific value. The problem is to measure these parameters twice, if required because the first measurement will influence the second. This becomes worse if time is involved.
Shankar argues, the probability distribution should be "flat", meaning that it is equally likely to find it anywhere.
He is correct.
Thus, one cannot accept the wave function as described in (1).
Yes and No. To describe the movement of a particle as a wave is wrong. Mathematical as a wave function is 'okay'.

Page 4

Is there a function that has a wavelength and, yet, a constant absolute value? The complex exponential has this property. Therefore, instead of (1), if the wave function looks like
psi.p(x) = Ae^(ipx/h) (2)
From a mathematical point this is correct. But from a physical point not. There exists no wave. The problem is we should not try to catch out incapability to measure the fact that we cannot measure the parameters of a particle as a physical reality
In sum, the wave function needs to be complex so that no information about the position is obtained for a state of definite momentum.
Wrong logical argumentation by using the uncertainty principle
Now the contradiction is eliminated, since there is no information about the electron's position for a state of definite momentum.
From a physical point of view there never was a contradiction.
One advantage of this argument is that it has the structure of a reductio ad absurdum proof, i.e., it starts by assuming a real wave function and reaches a contradiction, which can be psychologically more appealing to students.
This is scientifically unaceptable
It also nicely stresses a formal difference between a cosine and a complex exponential, both are periodic functions but the latter has a constant absolute value, and therefore is more suitable to describe the intended physical properties.
On the other hand, the argument relies on providing plausible reasons for why one cannot have full information about position and momentum simultaneously, and it postulates the square modulus of (x) as probability density, which can leave students wondering where this comes from.
As I said before: To measure both is physical impossible. Even to measure the position of a single elementary does not make sense. It is much more important to understand the structure of elementary particles by carefull designed experiments.

B. Because i appears explicitly in the Schrödinger equation - page 5

The second justification stems from the textbook "Quantum Theory" written by David Bohm and is based on a structural difference between the classical wave equation and Schrödinger's time-dependent equation.
It is very important to know what the physical difference between the classical wave equation and the time-dependent equation. In a certain situation (or process) can you always use both or is each equation relevant to a specific type of situations?

Page 5

For simplicity reasons, let us first consider the wave equation in one dimension
d^2y/dt^2 = c^2 d^2d/dx^2 (3)
Can we allow the solutions to this equation to be complex valued?
The most important question is: what is the physical situation or process that this equation describes?

C. Sx, Sy and Sz in sequential Stern-Gerlach experiments - page 8

D. To enable continuous transitions - page 12


Reflection 1 -

Reflection 2 - Uncertainty Principle

The Uncertainty Principle is not a Natural Law. Natural Laws are descriptions of physical processes and each law describes a specific subject. Natural Laws should always be backed up by actual experiments.
For more detail see: Uncertainty Principle
The reason why the Uncertainty Principle is not a Natural is because it does not describe a specific physical process, but the difficulty to perform measurements by humans. For example: it describes the limitations to measure both simultaneous the position and the momentum of a particle. The advantage (accuracy) of one goes to the disadvantage (inaccuracy) of the other. That is true, but that does not address the deeper problem: You can only measure any parameter only once. The reason is that when you perform a measurement the original state of the particle is disturbed, making a second measurement 'invalid'.

The simplest example is the calculation of the speed of a particle. In the first measurement you are supposed to measure both the position and the time of the particle. The problem is this measurement will always influence the position and direction of the particle. The result is, that at the moment when the particle is measured for the second time, the position of particle will be different compared with the supposed position, as if the first measurement did not took place. That means the calculated speed is wrong.

To say this in a different way: The value of the calculated speed is uncertain, caused by the measurement process. But, and that is important, that does not mean that the positions and velocity at any moment are uncertain. It is the other way around: At any moment in time, both the position and the speed of any particle have a certain value. Except we humans are not capable to measure or calculate these parameters, acurate.

Even more problematic is the calculation of momentum of a particle. The momentum is the product of the speed of the particle times the mass of a particle. To calculate the speed is discussed previous. To calculate the mass of a particle or an object requires Newton Law. That means you have to trace the trajectory over 'a long period of time', which involves many measurements.

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Created: 18 January 2022

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