The basic claim that there is no clear boundary between classical mechanics and Quantum mechanics is already completely wrong.

In classical mechanics waves are not quantized and equally the wave nature of all matter is not accounted for. So if you have both of these things, you are in quantum theory, if you don't have them, you are classical. That is the answer to your question and it's not that hard.

When it comes to "erroneous applications": I don't know, my USB drive works pretty well and that requires a quantum mechanical effect to operate.

Side note: Niels Bohr (not Nil Bohr) is not the source of "Shut up and calculate", that was Mermin decades later.

If you want to put forward an alternative which people seriously consider, you have to stop promoting them with endless trivially false claims.

I certainly can, but don't need to, it was defined by Heisenberg, Dirac, Schrödinger between 1925 and 1926 and in a totally independent way by Feynman in 1942.

The boundary between classical and quantum physics is defined by Planck's constant, which sets the scale of the quantum fluctuations. So, when energy x time or position x momentum are large compared to Planck's constant, quantum fluctuations are negligible and classical physics is a good aporoximation; when these products become comparable to Planck's constant, classical physics isn't a good approximation and quantum effects must be taken into account. The most convenient way is then a matter of taste.

We believe that, contrary to popular belief, quantum mechanics constitutes the limit of classical physics, and not the other way around.

Quantum mechanics operates in infinite free space with a closed control volume subject to Dirichlet boundary conditions.

When fluctuations in time, space, and energy density are large relative to the distance between free nodes, classical physics should be replaced by quantum mechanics.

Quantum fluctuations are significant, and therefore, classical physics is a good approximation to quantum mechanics only when these fluctuations are sufficiently small.

However, when these products become comparable, classical physics is no longer a good approximation, and quantum effects must be taken into account.

We also believe, contrary to popular belief, that the Heisenberg-Dirac-Schrödinger matrix of 1925-1927 is incomplete and misleading because it operates in an incomplete R^4 space (3D+t as the external controller).

The Heisenberg matrix is ​​neither a transition matrix nor a statistical matrix. The chains of transition matrices B, derived from Cairo's statistical theory, constitute a much more meaningful statistic.

Planck's constant h, or one of its multiples, plays no role in the definition of quantum mechanics. It is simply a transformation constant between energy and frequency E=hf.

Now let's return to the question: can quantum mechanics be defined?

Quantum mechanics is defined by too many conditions or constraints, not just one or two. Let's start with the first two:

1- Quantum mechanics must obey the sawtooth geometry in 1D, 2D, and 3D, whether in a vacuum or in other materials such as aluminum and steel.

2- The entanglement of the transition matrix (between the elements of the quantum system and between the elements of the quantum system and the walls) and the transition matrix itself must be composed of irrational numbers (e.g., square roots such as √2, √3, √5, √10 etc.).

Below, we present three examples of generating 1D, 2D, and 3D quantum matrices from the exterior product of two sawtooth vectors u⊗v.

Or, in index notation:

u⊗vij = [ui vj]

Furthermore, we show that the quantum matrix u⊗v exhibits a pattern reflecting the sawtooth shape of the input vectors. Furthermore, its product with any initial state vector produces sawtooth vector properties.

Example 1,

1D quantum sawtooth matrix with 9 free nodes Q:

Q=

1

2

3

4

5

4

3

2

1

⊗

1 2 3 4 5 4 3 2 1

=

1 2 3 4 5 4 3 2 1

2 4 6 8 10 8 6 4 2

3 6 9 12 15 12 9 6 3

4 8 12 16 20 16 12 8 4

5 10 15 20 25 20 15 10 5

4 8 12 16 20 16 12 8 4

3 6 9 12 15 12 9 6 3

2 4 6 8 10 8 6 4 2

1 2 3 4 5 4 3 2 1

Note that the matrix Q satisfies three conditions or constraints:

1- Q is a double statistical matrix, which means that

ΣQij = 1 for all rows i.

ΣQij = 1 for all columns j.

2- The determinant Q is zero in all 1D, 2D, or 3D spaces.

3- The matrix Q is invertible, which means that Q^-1 exists.

The above 1D quantum matrix, when multiplied by an arbitrary initial quantum state matrix, will produce a sawtooth-shaped quantum matrix, namely,

*1 2 3 4 5 4 3 2 12 4 6 8 10 8 6 4 23 6 9 12 15 12 9 6 34 8 12 16 20 16 12 8 45 10 15 20 25 20 15 10 54 8 12 16 20 16 12 8 43 6 9 12 15 12 9 6 32 4 6 8 10 8 6 4 21 2 3 4 5 4 3 2 1

=100000001

2

4

6

8

10

8

6

4

2

In arbitrary dimensionless units.

What remains is a sawtooth shape.

This is the case for the energy density distribution in a one-dimensional infinite potential well, as shown in Figure 1.

📷

We begin by defining and presenting the statistical theory of Cairo techniques and its B-transition matrix chains, and assert that it is the universal unified field theory.

There are no partial differential equations, no classical mathematics, classical differentiation/integration, or anything else from classical R^4 space (3D geometry with real-time as an external controller) involved here.

All classical and quantum physics, as well as classical mathematics, are replaced by statistical matrix mechanics.

In other words,

It is true that Cairo intelligence techniques = natural intelligence = artificial intelligence in the strict sense = unified field theory.

The statistical theory of Cairo techniques has been working effectively since 2020, that is, for four years, to numerically solve:

1- Time-dependent PDEs of classical physics in their most general form, such as LPDE PDEs, PPDE PDEs, and heat diffusion PDEs;

2- Time-dependent PDEs of quantum physics in 1D, 2D, and 3D;

3- Pure mathematics of differentiation/integration and classical and quantum statistical distributions, etc.

We show that the quantum physics transition matrix Q is the square root of the classical physics transition matrix B.

Q=√B.

It follows that, Q is essentially a complex matrix, but its energy eigenvalues are all real.

1- The elements of a quantum transfer matrix are all non-rational, but their square roots are such that √2, √3, √5, √10, etc.

2- The solution to the quantum system follows a sawtooth structure.

3- The geometry of a right-angled sawtooth structure implies that:

1111111 x 1111111 = 1234321 and 1111 x 11111 = 12321, etc.

The only flaw in this theory is that it reveals the errors of the scientist Niels Bohr, which displeases both Bohr's iron guards and their followers.

They were nursed and weaned by Bohr and his followers, and the probability of their recovery is less than a few percent.

You can also obtain the quantum transition matrix Q without resorting to the formula Q=√B:

For 17 free 1D QM nodes, start with the sawtooth form of the outer product of,

[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1]T

⊗

[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1]

You finally obtain the 1D quantum transition matrix with the sawtooth form,

The geometry of a right-angled sawtooth structure implies that,

1111111 x 1111111 = 12345654321

and

1111 x 1111 = 12321, etc.

Note that all the above numerical values are expressed as dimensionless arithmetic numerical values.