Quantum Mechanics

It is one of the branches of physics that studies the phenomena that occur at the atomic and subatomic levels, where classical physics, such as Newton’s laws, is unable to explain the behavior of tiny particles. In quantum mechanics, particles such as electrons, protons, and photons are treated as having dual properties, meaning they behave as both particles and waves at the same time.

Quantum mechanics developed thanks to the work of Max Planck, who introduced the quantum theory, followed by Einstein’s research on the photoelectric effect, then Niels Bohr, Heisenberg, Schrödinger, and other scientists who advanced the theory more deeply.

There are key concepts in quantum mechanics:
(Uncertainty Principle, Wave Function, Quantum Superposition, Quantum Entanglement)

Theory of (Soft Hair on Black Holes)

In the past, the scientist Stephen Hawking had a hypothesis that a black hole could never retain the information of the body or anything it attracts.

But in 2016, Stephen Hawking presented his last scientific contribution in collaboration with some scientists, stating that a black hole can in fact retain the information of everything it attracts.
But how?
And why was the first hypothesis modified?

In reality, his first hypothesis contradicted the principles of quantum mechanics when it claimed that information is destroyed..!

For this reason, Stephen Hawking presented this modification: information is preserved in the event horizon of the black hole, but not inside it. This also matches the theory of relativity when it states: everything that enters a black hole cannot escape from it — including information, of course!

Therefore, all the information of the objects attracted by the black hole remains preserved in its event horizon (stored in a quantum way).

Differentiation

Differentiation is a mathematical process that aims to calculate the instantaneous rate of change of a function, or in other words, the slope of the tangent line to the curve of the function at a specific point.

The instantaneous change of a function means the change that occurs in the function at a particular moment.

For example, if you calculate the speed of a car by dividing the total distance it traveled by the total time (this is called the average rate of change).
But if you calculate the distance the car traveled in a very small period of time to know the exact speed at a “specific point” (this is called the instantaneous rate of change), it is measured using differentiation.

Before differentiation was developed, scientists like

Archimedes

studied equations in different ways to reach results. But in the 17th century, the two scientists

Newton

and

Leibniz

developed differentiation (and also integration) in many ways and for different applications.

To understand how differentiation is studied in its original form:

1. First, the concept of the Limit, which is fundamental in differentiation. To calculate the slope of the tangent line, we use this definition:
f'(x) = lim(Δx → 0) [f(x + Δx) - f(x)] / Δx
2. Second, the rules of differentiation, which are six main rules.

From these basic concepts, many advanced methods have been developed, but these remain the foundation.

Differentiation is also used in many fields such as physics, artificial intelligence, engineering, and many others.

Integration

Integration is a mathematical process used to calculate the area under a curve, the volume, the mass, or any varying quantity — in other words, “any quantity that gradually accumulates over time until it becomes something final.”

Now let us explain the phrase “calculate the area under a curve.” Imagine you want to calculate the distance a car travels in a certain amount of time. For example, how far can the car travel in half an hour? If the car’s speed is constant, we can simply use the formula:

Distance=Speed*Time

But in the case where the car’s speed is increasing and not constant, we need to use integration to calculate the distance it covered in that time (half an hour).

Integration originated from the need of scientists to study the volumes of irregular bodies and the areas of curved shapes.

Integration gradually developed through the work of Newton and Leibniz. Before them, Archimedes used the idea of dividing objects into small parts to calculate areas and volumes — a concept similar to integration, but in a primitive way without modern symbols. Later, integration advanced further when Riemann introduced what is called Riemann sums.

Now, let us see how integration is studied. There are several stages:

1.Multiple integrals: used to calculate volumes or quantities in higher dimensions.
2.Indefinite integrals: the reverse of differentiation, they give you the original function before it was derived.
3.Definite integrals: used to calculate the area under a curve between two points.
4.Methods of integration: such as substitution, integration by parts, and numerical integration.

Integration is applied in various fields such as physics, engineering, statistics, medicine, and many others.

Quantum Entanglement

It is a strange phenomenon in quantum mechanics, where it states that when particles are formed, they can exist in overlapping states such that their conditions become inseparably connected regardless of the distance between them. This means that measuring the state of one particle immediately reveals the state of the other particle, even if the distance between them is extremely large.

Quantum Superposition

It is a principle in quantum mechanics that indicates particles have the ability to exist in multiple states at the same time before being measured. This means that a particle can simultaneously be in different energy states, positions, or spin states. Once the particle’s state is measured, the quantum superposition collapses. An example of this is Schrödinger’s cat. After measurement, the particle settles into one of the possible states.

The Theory of Electrodynamics in the Framework of Relativity

Before Albert Einstein and before the theory of special relativity, electricity and magnetism appeared as if they were separate phenomena. But after Einstein introduced his theory and the Lorentz transformations, it was revealed that electricity and magnetism are two sides of the same coin, called electromagnetic fields.

Let us take an example to clarify the idea further:
Suppose you are an observer watching a wire with moving electrons. If you, the observer, are at rest relative to the wire, you will see a magnetic field around it.

However, if you move (run) at the same speed as the electrons, you will see the electrons as stationary and the positive ions moving (relative to you). In this case, you will observe an electric field.

In other words, depending on the reference frame, you observe something different.

But what does relativity have to do with all of this?
Relativity plays the fundamental role here because of Lorentz transformations, which showed the change in length measurements (contraction). This contraction alters the distribution of charges, and therefore, if an electric field appears in one frame of reference, it will appear as a magnetic field in another.

In summary, those who established and developed this theory were James Clerk Maxwell, Albert Einstein, and Hendrik Lorentz, and finally Hermann Minkowski, who expressed everything in the language of spacetime, making it simpler within relativity.

Principle of Causality

The principle of causality states that in all reference frames, the cause must occur before the effect, and nothing can travel faster than the speed of light.

((It is the foundation of relativity))

In special relativity, spacetime and the constancy of the speed of light appear, and events are ordered here. Since the speed of light is the fastest speed in the universe and cannot be exceeded, the cause will occur before the effect. If this were violated (exceeding the speed of light), the effect would happen before the cause, which means a break of causality. This cannot happen (the light cone explains this matter in more detail).

In general relativity, some mathematical equations seem to allow causality to be violated. For example, some results of general relativity equations indicate that when an object approaches a region of very high spacetime curvature, like a black hole, it might return to the past. This breaks the principle of causality because the effect would occur before the cause. Many scientists rejected this result (such as Stephen Hawking) and pointed out that even if the mathematical equations give such results, they cannot happen in reality. It is only a theoretical matter on paper.

Proper Time and Coordinate Time

First, let us define each concept, then take an example, and finally explain the main equation.

Proper Time:
Proper time is the time recorded by a clock that moves along with a specific object in space, or in any region with strong gravity and high velocity.

Coordinate Time (or Measured Time):
Coordinate time is the time recorded by a clock that is located at a fixed place relative to the observer (for example, a clock on Earth).

Each of these times will give different results, and they differ from one another.

Example:
Suppose there is an observer on a spaceship approaching a region with strong gravity, for example, a black hole. This observer has a clock.
Meanwhile, there is another observer on Earth, also with a clock, but in a fixed location (Earth).

Now, when the first observer on the spaceship looks at his clock, he will see time passing normally. But the second observer on Earth will see that many years have passed.
The time that passed for the person on Earth is called coordinate time, while the time that passed for the person in space is called proper time.

The Main Equation of Proper Time:

ΔT=Δt√1-v^2/c^2


ΔT= Proper time
Δt= Coordinate time
v = Velocity of the moving object
c = Speed of light

By applying the equation, we can determine how many years have passed on Earth and how much time has passed for the moving object.

This idea was introduced by Albert Einstein in 1905, and in 1908 the scientist Hermann Minkowski formulated the geometry of spacetime (Minkowski space), which further developed from the fundamental equation above.

Tensors

Tensors are fundamental mathematical objects in General Relativity.

But what are they?
Tensors are quantities that remain unchanged no matter how the reference frames or coordinates are transformed.

They are often explained as if they were vectors. Normally,

a vector is a number plus a direction (for example,

a velocity

of

20 km/h

north

).

But tensors are more complex — they are a number plus multiple directions and more intricate relationships.

Example:
Imagine you have a grid, such as a soccer net. Any change happening in one part of the grid is recognized by the tensor, which can determine whether a certain point of the grid is flat or curved.

Now, going back to relativity, tensors measure every point in the universe in both space and time. They essentially define the

geometry

of

spacetime

at

a

specific point. From this, a tensor can determine whether that region of spacetime is curved (meaning there is mass, energy, or something like a black hole) or flat.

The first significant use of tensors was in Maxwell’s equations after the development of Special Relativity. However, these tensors are fundamentally different from those in General Relativity. In Maxwell’s formulation, tensors were used to combine the electric and magnetic fields. Previously, these equations were expressed only as differential equations.

One major reason for this shift was that the old formulation treated space and time as separate entities, and also considered the electric and magnetic fields as separate phenomena. The tensor formulation, however, unified them.

In General Relativity, tensors are entirely different in their role. They are used in objects such as the Ricci Tensor, the Riemann Curvature Tensor, and the Einstein Tensor, which describe the curvature of spacetime itself.

Lorentz Transformations

Lorentz transformations are a set of equations upon which Special Relativity (by Albert Einstein) is based. They are used to transform coordinates between two observers moving at a constant velocity relative to each other. Lorentz transformations were originally introduced to explain electromagnetic phenomena, and their study revealed many important insights, including the fact that the speed of particles has a significant effect on measuring the physical dimensions of an object.

γ = 1 / √(1 - v²/c²)
Their equations show that both time and space vary depending on the observer’s velocity, meaning that time is relative. Time can dilate or contract (if it is close to the speed of light, then time contracts).

The reason for using Lorentz transformations is that Newtonian physics failed to explain velocities close to the speed of light, while Lorentz transformations succeeded in doing so.

What is the Purpose of Lorentz Transformations and Why Did They Appear?

Let us go back in time to the era of Newton and after. At first, before the rise of relativity and modern physics, Newton’s laws governed physics.

Newton believed that time was absolute and unchangeable.

Moving forward a bit into the 17th century, the scientist Huygens proposed the existence of the ether.

The reason behind this assumption was that scientists thought light must propagate through a medium, so they imagined the ether: an invisible substance in space that carries light waves.

But in 1887, the scientists Michelson and Morley conducted their famous experiment, which disproved the existence of the ether. The experiment went as follows:
1_ They used a light source that emits a beam.
2_ The beam was split into two beams at a 90-degree angle.
3_ Each beam was reflected by a mirror and then returned to meet.
4_ The time difference taken by each beam to return had to be measured.

The expected result, if the ether existed, was that one beam should travel faster or slower than the other, producing an interference pattern.

The result: no difference was observed, and no interference pattern appeared. This meant that there was no ether, and that the speed of light is constant regardless of the motion of the source or the observer.

This experiment marked the beginning of the road to Special Relativity.

But now, what does the constancy of the speed of light have to do with making time and space relative?
First, let us recall the formula for speed:

Speed =Distance/Time

When the speed of light is constant, both distance and time must adjust to preserve that constancy. That is why space and time become relative.

For example, in General Relativity, why does time slow down near a massive object like a black hole?
The answer is that when light passes near a black hole, it travels through the curvature of its spacetime, taking longer because the path becomes stretched. Here, time must adjust: in order for the speed of light to remain constant (as dictated by the law above), a shift occurs in time. Meanwhile, distance is already altered due to the curvature of spacetime caused by the black hole’s mass.

The Wave Function

A fundamental concept in quantum mechanics, the wave function is used to indicate the state of a quantum particle and determines the probability of its presence at a specific position and time. The particle’s existence is based on probability, not certainty as in classical physics. The wave function relies on the Schrödinger equation, which defines the evolution of the particle’s quantum state and describes how the wave function changes over time. The importance of the wave function lies in its ability to describe many quantum phenomena such as interference and quantum entanglement, where multiple particles have linked wave functions that can lead to their states being interconnected and dependent on each other.

We always see this equation, but what is it? What does it state, and why is it important in physics?

Before this equation appeared, physics relied on classical physics (Newton’s laws), which stated that mass is constant, space and time are absolute (unchanging under any circumstance), and energy 1/2mv^2 is given by

However, with the advent of special relativity, these concepts changed, introducing more precise ideas that explained many phenomena Newton’s laws could not, the most important being the constancy of the speed of light. Among the concepts that changed are:
Spacetime (time and space) is relative.
Energy, motion, and mass are two sides of the same coin (the same thing).

Why did Einstein change these laws?

Because he observed that objects moving near the speed of light cannot have their kinetic energy correctly described by classical laws. Also, the mass of an object increases as its speed increases, meaning its energy increases.

Here, Einstein concluded that even stationary objects carry enormous energy, which he called “mass energy.”

An example in the universe:
When two atoms, such as helium and hydrogen, fuse in a reaction like those in stars, the mass after the reaction is less than the mass before. This shows that the lost mass has been converted into thermal energy. In other words, when mass loses energy, it also loses some of its mass, and vice versa.
In short: “mass = rest energy.”

Energy is also related to spacetime, meaning time and space determine the form of interaction between mass and energy (thermal, radiative,...).

As Einstein famously said:
“Spacetime tells energy how to move, and energy tells spacetime how to curve.”

Now the question arises: We can produce energy from mass, but can we convert energy into mass?
The answer is yes. This is what the principle of “pair production” states, where energy transforms into electrons and positrons.