# Initial thoughts about precision problem in quantum mechanics

February 16, 2024This article is only my initial thoughts after reading a few of the first pages of Introduction to Quantum Mechanics

^{1}, Infinite powers^{2}and Engineering Mathematics^{3}. My thoughts are the result of learning about quantum technologies, but I’m not an experienced specialist in this area yet. I’m not sure any of these things are really used in quantum mechanics. It’s further in my learning journey. Those thoughts might be interesting, how someone just starting sees the quantum world.

# Let’s take a look at derivative

Even today, I was using numerical methods to calculate “hard/weird” functions. After looking at derivatives^{4}, I found they are missing some parts. During the evolution of calculus, we lost some parts of it to make it easier to use.

To simplify everything, let’s take a look at sample derivative:

$y + dy = (x + dx)^3$

$(x+dx)^3 = x^3 + 3x^2dx + 3x(dx)^2 + (dx)^3$

And now we should drop $3x(dx)^2 + (dx)^3$ as they are really small. Power of small change is an even smaller thing. Solving it later using $y=x^3$, we can simplify till we get $dy = 3x^2dx$.

Here comes my first thought. Working with quantum, we are working on super small scales. In classical mechanics, everything works fine, we can take the derivative from velocity and get acceleration etc. All calculus methods look like they are designed for this. But in the quantum world, we don’t have classical things, we don’t have big masses, built from billions of atoms. We are working on single atoms.

# Now let’s take a look at numerical methods

One of the most common methods is to use Newton’s method^{5} to find solutions.

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Now, we use a derivative which is already not perfect, as it is missing some information about super small changes, which might be crucial for e.g. movement on atomic level. Calculating “big/classical” things should be fine. But what about precise, “micro” changes, which we need to catch?

# Probability, curse of quantum?

On top of it there is probability^{6}. It’s really hard to analyze what is happening using probability. The problem I can see, are the outcomes e.g. for average or median. Let’s take a look at 5 measurements.

1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|

4 | 5 | 5 | 5 | 4 |

1 | 3 | 5 | 5 | 9 |

If we calculate average, we get 4,6. If we calculate median, it’s 5. If we calculate sum, it’s 23. And we are looking at totally two different cases, when we consider all the numbers seperately.

# I’m probably wrong

All of these examples are rather basic math problems. All formulas are designed to work on any factor or magnitude. Math formulas are universal for problem solving. But mathematical methods might be different for quantum computing. I’m just wondering, is the current version of math capable of working with quantum. Maybe we need to get different types of math to be able to calculate new types of physics. I have a feeling, we are applying old methods, and we want to pretend they are working. But there are so many mental tricks to prove they are alright.

Question to myself in the future:

- Does imperfection in derivatives might have some impact on quantum?
- Does using numerical methods to solve problems only show direction instead of solving them?
- Is probability the only way to work with quantum mechanics?
- What are higher math methods for this?