This article is just my first thoughts after reading the first few pages of Introduction to Quantum Mechanics¹, Infinite Powers², and Engineering Mathematics³. These thoughts come from learning about quantum technologies, but I’m not an expert in this area yet. I’m not sure if any of these things are actually used in quantum mechanics. It’s just the beginning of my learning journey, but it might be interesting to see how someone just starting out looks at the quantum world.

Let’s Take a Look at the Derivative

Today, I was using numerical methods to calculate difficult or strange functions. While looking at derivatives⁴, I noticed that some parts seem to be missing. Over time, we’ve simplified calculus to make it easier to use, but we may have lost something along the way. Let’s look at a simple example of a derivative:

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

Expanding $(x+dx)^3$:

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

We then drop $3x(dx)^2$ and $(dx)^3$ because they are very small. A small change raised to a power becomes even smaller. If we solve it further using $y = x^3$, we get:

$dy = 3x^2dx$

Here’s my first thought: in quantum mechanics, we work with extremely small scales. In classical mechanics, calculus works well—we can use derivatives to calculate acceleration from velocity and so on. But in the quantum world, we’re not dealing with large masses made up of billions of atoms. We’re working with single atoms. Does this simplification of derivatives still work at such small scales?

Now Let’s Look at Numerical Methods

One of the most common techniques is Newton’s method⁵ to find solutions:

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

This method uses derivatives, but as mentioned earlier, derivatives may miss important information about very small changes. This might be fine for larger, classical problems, but what about the tiny, precise changes we need to track in quantum systems? Could this lack of precision become a problem?

Probability: The Curse of Quantum?

Another issue is probability⁶. It’s hard to analyze what’s happening when everything is based on probabilities. For example, let’s look at five measurements:

For both data sets, the:

• Average is 4.6
• Median is 5
• Sum is 23

But these are two very different sets of numbers. This shows how using probabilities or averages can sometimes hide important differences in data. Could this be a problem in quantum mechanics?

Maybe I’m Wrong

These are just basic math examples. The formulas are designed to work for any situation, big or small. But I wonder if the mathematical methods we use are really good enough for quantum mechanics. It feels like we’re using old tools and pretending they’re fine for new problems. Are we missing something? Could there be new types of math that would be better for quantum physics?

Questions to Myself for the Future:

• Do imperfections in derivatives affect quantum mechanics?
• Do numerical methods only show the direction, instead of solving the problem fully?
• Is probability the only way to work with quantum mechanics?
• Are there advanced math methods that work better for this?