Crazy School

Crazy High School Exercise

Problem 1

Prove simply that if v ≤ c (light speed) the mass (m) of a body will increase and the acceleration (a) will decrease as velocity approaching c.

Answer:

Let force F acting on body is constant and motion starts from v = 0.

v = at

v = at ≤ c

a = F/m (Newton II)

v = at = F/m t ≤ c

If F is constant m should go to infinity with increased time to get a finite value less than c.

As m is increasing, a = F/m should go to zero.

Problem 2

What could be an analogy of wave function of quantum mechanics in classical physics? Help: check the units.

Answer:

│Ψ│² has a unit 1/m³ since integrated in space volume gives probability. Dirac function δ has the same unit in 3D space and represents that in classical mechanics matter stays together. Volumetric integral of both are equal to 1 according to normalization equation.

Problem 3

Change the normalization condition based on Gauss theorem even if the answer is stupid.

Answer:

∫│Ψ│²dV = 1, the normalization condition of QM may mean that the particle should remain inside our universe with probability 1.

If a vector-vector function B exists that

div B = │Ψ│², then ∫│Ψ│²dV = ∫ div B dV = ∫ B dA = 1,

with several questions. Looks like volumetric probability density function (scalar) is transformed into a „surface probability density” (vector), seems that probability of being in the volume means probability being on the volume’s surface.

Universe is supposed to be a closed entity inside, then how could we interpret vector B directed outwards? Stupid, indeed.

Problem 4

Richard Feynman about brick: “What about the inside of the brick?”– and I would then point out that no one has ever seen the inside of a brick. Every time you break the brick, you only see the surface. That the brick has an inside is a simple theory which helps us understand things better. The theory of electrons is analogous. So I began by asking, “Is a brick an essential object?”

So we can see the surface and never the interior. Based on Heisenberg’s uncertainty relation give an argument that maybe even the surface of the brick cannot be seen.

Or at least show that a point is always expanding as a consequence of QM.

Give an example that uncertainty relation can serve with an overall statement where Planck constant h is not present.

Answer:

(Will be wrong since velocity is not defined in QM.)

∆p_x ∆x ≥ h/4/π

If p_x = m d(∆x)/dt, then m d(∆x)/dt ∆x ≥ h/4/π,

2∆x d(∆x)/dt ≥ h/(m2/π),

d(∆x)²/dt ≥ h/(m2/π), integrating this inequality by time we get

(∆x)² ≥ √ (h/(m2π)t + (∆x₀)²), that is the size of objects grows slowly in time with this quantum diffusion. Maybe we cannot see even the surface of the brick just this extending quantum skin or boundary layer aura qed. The derivation is false, find it where.

∆E∆t ≥ h/4/π

Let ∆E = h/2/π ∆f and replacing E we get ∆f ∆t ≥ 1/2 which is valid also outside quantum mechanics since h is dropped out. Sampling consequences…

Problem 5

Show a simple mechanical model where it becomes clear that time is not absolute.

Answer:

Mathematical pendulum: a non-stretching, massless rope length L with one end fixed in gravitational field g and a small mass m on the other end and the whole is in air.

In special relativity L decreases as velocity increases, shorter time T is measured.

Similarly if g is increasing with the originating mass M below the pendulum, the time lapse measured will be smaller similar to general relativity.

If the mass m on the pendulum is very small e.g. close to the mass of nitrogen molecule the time measurement with the pendulum becomes irregularly uncertain.

Physically it is not correct, length dilatation occurs but gravity is not supposed in special relativity.

Problem 6

Give a stability criterion for radius of a non-rotating charged (Q) black hole without higher math. Assume that mass is lost in time while charge remains constant. What happens when it explodes?

Answer:

Let us suppose that existing forces are gravity, electrical repulsion and surface tension from nucleus drop model. Then one can come to an incomplete 6th grade algebraic equation easy to resolve. Formula can be easily extended for rotation by repelling pressure of centrifugal force. Calculation draft is below.

Explosion… who knows. Maybe: the black hole, its parts remain inside its event horizon –EH- so the explosion smears its material to the internal surface of the EH. The result is a hollow black hole but we cannot check it.

Problem 7

Write an ordinary differential equation for brain regarding EEG frequency change during sleep. Use the bouncing ball analogy.

Answer:

Problem 8

To how much you have to increase the value of π to compress the whole universe in a ping pong ball to fit in your pocket?

Answer:

About 10⁸¹ from 3.141…

Problem 9

iⁱ = ? if i is the imaginative unit that is i² = -1.

Answer:

i = e^{i(π/2 + 2kπ)}, then iⁱ = e^{-(π/2 + 2kπ)}, so it is a series of real numbers, k is any integer (Z).

Problem 10

Banach-Tarsky paradox and Gauss theorem.

Answer:

Problem 11

What is STHM?

Answer:

"It is probably true quite generally that in the history of human thinking the most fruitful developments frequently take place at those points where two different lines of thought meet. These lines may have their roots in quite different parts of human nature, in different times or different cultural environments or different religious traditions: hence if they actually meet, that is, if they are at least so much related to each other that a real interaction can take place, then one may hope that new and interesting developments may follow." (Werner Heisenberg)

STHM stands for Science Tradition Hybrid Method. An interdisciplinary initiative to get information from traditional texts may be needed to describe soul or psychological phenomena.

Briefly: read tradition like a scientific book and read science text books like religious scripts with analogies and associations.

Problem 12

What is soul or psyche?

Answer:

“The fact that religions express themselves in pictures, analogies and paradox in all times, means just that there is no other tool to grab the reality they refer to.” (Werner Heisenberg)

Soul is still undefined although traditional scripts of several thousand years are supposed to describe it. Usually two main attributes are mentioned: it is awake and sensitive. Applying STHM we try to translate these concepts so awake means something that does not sleep. Sensitivity means that notices and reacts to even the smallest perturbations, that is mechanically very small. Sleep and quantum mechanics and sacred texts treated together.

“In this world, while awake or dreaming, in the other world, while in deep sleep.” (Upanishads)