This is a note, which tries to simplify the first chapter in Quantum Mechanics: The Theoretical Minimum.

**Classical mechanics:** This is what we as humans can experience. A player can take a quick look at a flying ball, and from its location and its velocity, know where to run to be there just in time to catch the ball.

**Quantum mechanics:** This is what humans cannot experience because it is so small that it is entirely beyond the range of the human senses. Therefore, quantum mechanics is actually more fundamental than classical mechanics, which is merely an approximation.

Quantum mechanics is different from the classical mechanism in two ways.

**Different Abstractions**: It is fundamentally different. An example is the concept of a state, which is conceptually different.**States and measurements**: One can perform an experiment to determine the state of a system. In the quantum world, this is not true. States and measurements are two different things.

### The most simple experiment

Let us try to explain this through a simple deterministic system: a coin that can show either heads (H) or tails (T). This is a two-state system, or a bit, with the two states being H and T.

In classical mechanics then these are the only two states.

Now we introduce A, the apparatus that measures the value of the outcome (). The apparatus is a black box, which has an arrow indicating how it should be placed on a table. There is a circle which shows the two states being either -1 or +1.

We spin the coin and get a result of +1. If the spin (outcome) is not disturbed and the apparatus keeps the same orientation then it should show the same result of the outcome, +1.

The first interaction with the apparatus prepares the system in one of the two states. Subsequent experiments confirmed that state of +1. So far, there is no difference between classical and quantum.

Now let’s turn the apparatus upside down without disturbing the previous coin spin. Now the apparatus shows a result of -1 which makes sense because the apparatus has been flipped upside down.

From this result, we might conclude that the outcome is associated with a sense of direction in space. A simple explanation is that the apparatus measures the component of the vector along an axis embedded in the apparatus.

Now if we put the apparatus on the side (90 degrees turned) we would expect to get a 0 if the outcome represent the component of a vector along the up-arrow.

What really happens is that we get +1 and -1 with a 50% chance of each when we repeat the test over and over. This goes against our previous theory of the vector along the axis.

Now let us try something new where we neither place the apparatus upright or down but at an angle of and now each time we do the experiment we get the result that is statistically based on that the average value is cos .

This can be explained more generally by the following. (NOTE: M and N should be displayed hat accents as in the picture above, but the page won’t load them. They are instead marked with bold.)

We start with either upside or downside as a measure of **M. **We then prepare a spin and logically it will show either +1 or -1. Then without changing the spin, we rotate the apparatus to the direction **N** as shown in the figure above.

A new experiment on the same spin will give random results +-1 but with an average value equal to the cosine of the angle between **N** and **M**. In other words, the average will be **N*M**.

The quantum mechanical notation for this statistical average is called Dirac’s bracket notation. We can now give the following explanation of the test results. If we begin with apparatus oriented along **M **and confirm that Then subsequent measurements with the apparatus oriented along **N** instead will give the following statistical result.

This experiment show that opposed to classical mechanics then quantum mechanical systems are not deterministic. The experiments can be statistically random, but if we repeat an experiment many times, average quantities can follow the expectations of classical mechanics.

This is why classical physics are merely an approximation of quantum mechanics.

**Further reading **

Quantum Mechanics: The Theoretical Minimum