In one of the previous episodes we learned how quantum mechanics is fundamentally different (check here). Now let’s take that idea one step further.

Most of the time, order feels irrelevant.

If I add salt to a glass and then pour water, or pour water first and then add salt, I still end up with salty water. The final result is the same. Life often works like this, and mathematics agrees. 4 × 2 is the same as 2 × 4. We are simply rearranging groups. Four groups of two apples or two groups of four apples contain the same 8 apples.

But not everything in life behaves that way. If you are into cooking you would soon realise

That sentence hides a deep truth. The order matters. The ingredients are the same, but the outcome is not.

A more serious example is a Rubik’s Cube. In the first glance it is a colorful toy. When you twist a Rubik’s Cube, the order of your moves matters. Rotate the top face first and then the right face, and the cube ends up in one configuration. Do the moves in the opposite order, and the cube ends up in a completely different arrangement. The same pieces, just a different outcome. So the order matters!

Quantum mechanics lives entirely in this order sensitive world.

In an earlier episode (check here), we learned something unsettling about the quantum world. Particles do not have a well defined path.

In classical physics, a particle has a position and a velocity at every moment. Draw a line through space and time and you get its trajectory. Its path exists whether you look or not.

Quantum mechanics breaks this picture. Position and velocity cannot be measured at the same time. If you measure position very precisely, information about velocity is lost. If you measure velocity precisely, position becomes fuzzy. Without both, the idea of a continuous path collapses. There is no single line the particle follows.

So what do we do instead?

We measure one thing after the other. Maybe position first, then momentum. Or momentum first, then position.

And suddenly a new problem appears.

Suppose you measure the position of a particle and then immediately measure its momentum. Now swap the order. Measure momentum first, then position.

In classical physics, the order does not matter. The measurements reveal properties that were already there.

In quantum mechanics, the first measurement changes the system. The second measurement is performed on a different state than before.

So the outcome depends on the order. Position then momentum is not the same experiment as momentum then position.

This is the essence of non commutativity.

So this brings us to a new question!

How do we describe order dependent quantities mathematically?

We already know that numbers will not work. Numbers commute. Quantum observables like position and momentum clearly care about order. So physicists needed a new mathematical language.

Something that keeps track of sequence.

Scientists found their answer in matrices.

A matrix is not a single number. It is a table of numbers arranged in rows and columns. For example, a two rows and two columns matrix has four entries (written as 2×3), similarly a two row three column matrix has (2×3) has 6 entries.

Matrices can be added, but more importantly, they can be multiplied and here is the key point:

Matrix multiplication is order dependent.

In general,
AB ≠ BA

Two matrices multiplied in one order give a different result than when multiplied in the reverse order (watch matrix multiplication here). At last, mathematics had an object that behaves like quantum measurements.

This raised a bold question.

The answer turned out to be yes.

In quantum mechanics, physical quantities like position, momentum, and energy are represented by matrices. These matrices act on something called the quantum state. Sometimes people call these matrices operators, but that is for a different episode.

Now comes a natural confusion. If I measure position, I get one number. But a matrix contains many numbers. So what do the other numbers mean?

They are not multiple measurement outcomes. Instead, they encode how the observable acts on all possible states of the system. Each entry tells us how one state is transformed into another when we measure or interact with that quantity.

Think of a matrix as a machine. You feed in a state and the matrix produces a new state. The full table is needed to describe how the machine behaves in every situation, not just the one measurement you perform today.

At first glance, non commutativity feels like a problem. Measurements disturb each other so order matters. But this strange feature is not a flaw. It is the reason quantum technologies exist.

Quantum computing relies on applying operations in a precise order. Swap two quantum gates and the result changes. Quantum cryptography uses non commuting measurements to detect eavesdroppers. Quantum sensing exploits measurement back action to reach sensitivities impossible in classical devices.

All of this traces back to one simple idea.

Just like cooking steps, and human conversations, the sequence shapes the outcome. Numbers were too simple to describe this reality. Matrices stepped in and revealed a deeper structure of nature. Once we accepted that AB is not the same as BA, a new technological era began.