Category: Physics Notes PPT

In 1925, the Austrian physicist Wolfgang Pauli proposed Pauli’s exclusion principle. Pauli’s exclusion principle states that no two electrons can have the same set of quantum numbers or quantum states. It is known as Pauli’s exclusion principle because it excludes electrons from being in the same quantum state. The Pauli exclusion principle is valid throughout atomic and quantum physics, it is extremely powerful and broadly applicable. Pauli’s exclusion principle is one of the important concepts that help us to understand the atomic structures and arrangement of molecules.

 

What is the Pauli Exclusion Principle?

All atoms except hydrogen atoms are multi-electron atoms. The physical and chemical properties of elements are directly related to the number of electrons in a given atom. The periodic table of the elements groups elements with identical properties into columns. This systematic arrangement is related to the number of electrons in a neutral atom, known as the atomic number, denoted by Z.  The exclusion principle is key to the underlying explanations, and that it applies far beyond the realm of quantum physics.

The Pauli exclusion principle dictates this arrangement effectively forces electrons to take up space in the atom. By recognizing that no two electrons can occupy the same quantum state simultaneously, it effectively stops electrons from piling up on top of each other, thus explaining why matter occupies space exclusively for itself and does not allow other objects to pass through it, while at the same time light and radiation are allowed to pass.

The Pauli exclusion states that no two electrons can have an identical set of quantum numbers. The Pauli principle applies to identical particles with half-integral spin i.e., S = 1/2, 3/2, 5/2  In other words, each electron should have its own singlet state or unique state. The salient features of the Pauli exclusion principle are as follows:

1. In a given orbital only two electrons can occupy different quantum states.

2. The two electrons occupied in a given quantum state must have an opposite spin or in other words, should be antiparallel to one another.

The Pauli exclusion principle isn’t only valid for the electrons but also for other elementary particles with half-integral spin, for example, fermions. The Pauli exclusion principle does not hold good for the elementary particles such as bosons that possess full integer spins. The Bosons can have or share the same quantum states simultaneously, which violates the exclusion principle. Thus the Fermi Dirac distribution follows the Pauli exclusion principle whereas the Bose-Einstein distribution violates the exclusion principle.

 

Define Pauli Exclusion Principle?

The Pauli Exclusion principle is one of the important principles for the arrangement of electrons in an atom along with the Aufbau principle and Hund’s rule. The Pauli exclusion principle states that- no two electrons can have the identical set of quantum numbers or quantum states simultaneously. Every electron should have a unique quantum number and quantum states. 

From the Pauli exclusion principle definition, we understood that No two electrons with up spin can be arranged together at the same time no two electrons with down spin can be arranged in a single quantum state. The Pauli exclusion principle in chemistry is as important as the Pauli exclusion principle in physics.

 

Pauli Exclusion Principle in Chemistry:

In chemistry, the principle is mainly used to explain or determine the electron shell structure of atoms and predict which atoms are likely to have free electrons at the end of configurations. How is the exclusion principle used in chemistry or where does it apply? So, if we have a look at the atoms whenever it gains a new electron or electrons it usually tries to reach their lowest energy state or it shifts to the outermost shell. Now, if the state has one electron then it can either be up spin or down spin. Therefore, now if we consider the Pauli exclusion principle if there are two electrons in a state, then each of the electrons must have either spin up or spin down but can not have the same spin.

 

Pauli Exclusion Principle Example:

For better understanding let us have a look at an example. We can take the helium atom as a common example. The Helium atom has 2 electrons and they occupy the outermost shell with opposite spins. Here, we will see that the two electrons are in the 1s subshell where n = 1, l = 0, ml = 0 where n, l, and ml are the principal quantum number, orbital quantum number, and magnetic quantum number respectively. 

Now, both electrons will have different spins. One will be ml = -1/2  and the other will be +1/2. If we plot a diagram for the distribution, then the subshell of the helium atom will be represented with one up an electron and one down electron. Therefore, 1s subshell will have two electrons, which have opposite spins.

Let us have a look at another example for the Pauli exclusion principle, if we take a hydrogen atom consisting of one electron, then it will have a 1s subshell with one up-spin electron(1s¹). Moving forward, if we take up another atom such as Lithium it consists of 3 electrons, three electrons will be distributed in 2 subshells, first, two electrons will be distributed among 1s (1s²) subshell as an up-down spin pain and 2s (2s²) subshell will have a single electron with up spin. The distributions can be clearly understood if we draw the diagram as given below.

These are a few examples of how Pauli’s exclusion principle is used. Pauli’s exclusion principle plays an important role in atom physics as the arrangement of electrons is one of the important parts of the atomic structures. In advanced physics, the Pauli exclusion principle is one of the most basic observations of nature. When we try to calculate the probability of the electron in any given state, we need to write the wavefunctions. So, the particles of half-integer spin must have antisymmetric wavefunctions and particles with integer spin must have symmetric wave functions.

 

Applications of Pauli Exclusion Principle:

1. No two electrons in a solid can have the same energy states. This will help us in studying the concept of Fermi levels in the band theory of solids.

2. No two electrons in an atom can have an identical quantum number, this will lead us to model the grouping in the periodic table.

3. Electron degeneracy governs the death of stars to the white dwarf stage of stars.

4. Neutron degeneracy governs the death of stars to the neu
   tron star stage.

 

Did You Know:

• The Austrian physicist Wolfgang Pauli was honored with the Nobel prize for his remarkable contribution to the field of physics. He was honored for the discovery of the exclusion principle. 

• When Neil Bohr proposed his atomic theory, he conveyed that the electrons are revolving around the nucleus in a fixed orbit. After some time, the Bohr model was corrected and every electron was assigned certain quantum numbers corresponding to their distinct states and energy levels. 

• Later in the early 1900s, Pauli came up with Pauli’s exclusion principle by introducing two new quantum numbers and he stated no two electrons in an atom can have the same set of quantum numbers.

• Later it was discovered that quantum numbers can also be assigned to the protons and neutrons present in an atom, and the Pauli exclusion principle is in good agreement with this.

 

Electron Spin Theory

Electron spin is a quantum property of electrons and it also is a form of angular momentum. Further, the magnitude of this particular angular momentum happens to be permanent. Also, the electron spin is a fundamental property of electrons just like charge and mass. The electron spin theory explains the electron as a quantum particle. Electron Spin theory states the electron spin direction and its influence on certain properties like the magnetic properties of an atom. The electron can spin in 2 directions are Spin up and Spindown

The spin-up and spin-down directions are coordinated to the spinning in the +z or –z-direction. These are the particles that have spin ‘s’ equal to 1/2 for electrons. The quantum theory states that the electrons are thought of like the minute magnetic bar. Its spin points to the north pole of the minute bar and if 2 proximate electrons have the same spin direction, the magnetic field formed by the electrons supports and strengthens each other. Therefore a powerful magnetic field is acquired. If the relative electrons have an opposite spin direction, the magnetic field formed by them cancels each other. Thus, no magnetic field is existent.

 

Properties of Principle Quantum Numbers

• Principal Quantum Numbers represent the most likely distance between the nucleus and the electrons of an atom.

• The Principal Quantum Number’s value may be any integer with a positive value that is equal to or greater than 1. The value n=1 represents that the innermost electron shell of an atom is equal to the lowest energy state, which is also called the ground state, of an electron.

• The principal quantum number can never have a negative value or be even equal to zero. This is because an atom can never have a negative value or any value for a principal shell.

• When a given electron is infused with energy which is called the excited state, the electron jumps from one principal shell to a higher shell. This causes an increase in the value of n( Principle Quantum Number). The same happens when electrons lose energy. Principal Quantum Numbers jump back into lower shells and the value of n also drops with it.

• Absorption is the process where the increase in the value of n for an electron is seen. This highlights the photons of energy being absorbed by the electron. In the same way, the decrease in the value of ‘n’ (Principle Quantum Numbers for an electron is called emission. In Emission, the electrons of an atom emit their energy.

Since the advent of the photoelectric effect, a remarkable concept theorized by Sir Albert Einstein opened a new dimension where we started considering lights as an accumulation of energy packets. These energy packets are called photons. In this section, we will study photons and its exceptional features. We will discuss its features, how it is formed, emitted, and absorbed. Here, we will also discuss biophotonics and how it has been developed over the years. Let us know more about these subatomic particles with no mass and charge.

What is a Photon?

Photons are explained as energy packets emerging from any source. They are defined as packets of energy emitted from a source in the form of electromagnetic radiation. Sir Albert Einstein proposed and explained these particles in the year 1905. It was his theory of photoelectric effect that properly described the existence and emission of photons of light.

Before this theory, Max Planck described how heat energy is radiated and absorbed in the form of units. It was his theory that showed how packets of heat energy are absorbed. He explained these units as quanta. The term ‘photon’ came into light when it was explained and coined in the year 1926.

If you observe the nature of all the electromagnetic radiation, we will find that these energy packets exist in all of them. Despite the fact, why do these gamma rays, X-rays, infrared rays, etc vary in energy? All these rays do not have any electric charge but have a particular level of energy to disperse. It all depends on the frequency of the photons of light that decide the energy of the electromagnetic rays. As per the insights into quantum mechanics, these particles show duality behavior. They behave like a wave of energy but also as a particle in some aspects.

What is a Dark Photon?

The advancement in this section of physics also revealed the presence of a heavier particle, a carrier of forces hidden from our eyes. This particle is called dark photon and can be connected to the presence of dark matter. They exhibit properties similar to photons but remain hidden.

The gravitational effects that cannot be explained by visible matter can easily be demonstrated by the presence of dark photons. Its presence has been detected by the Large Hadron Collider.

What is Biophotonics?

If you observe this term, you will find that it is linked to the emission of photons in a living matter. Biophotonics is linked with the emission of photons, units of light, detected by a machine to understand the processes and other aspects of life. The energy packets emitted by a living source naturally or due to the presence of a fluorescent marker are called a biophoton. It can be used in different fields of scientific detection required in the biological aspects.

Scientists are working on this subject for decades to make the imaging techniques of different medical processes better. When a marker of a fluorescent substance is injected to mark a particular tissue or organ, the researchers get a canvas to check the emission level. This is where biophotonics is used to create a clearer image of an organ system, a particular tissue, etc where the fluorescent marker will be accepted. These optical techniques can be used in different biological fields and will make research better.

What is Photo Gas?

This is a segment of quantum physics where photons are collected in a particular space showing properties similar to general gases. This collection of photons in the form of gas is called photon gas. The thermodynamic properties of these gases can be measured using various formulas and equations.

Photon Double Slit Experiment

As we have understood that photons are energy packets that emerge from a source. They also behave like waves, as well as, particles showing duality in characteristics. Hence, photons will also display the nature of light and will pass the double-slit experiment. The photon double slit experiment shows that it has both wave-like and particle natures.

This experiment was conducted in the first decade of the 1800s and has still significantly proved the duality of photons in light rays. The proper explanation of this simple yet effective experiment can only be given when photon particles exhibit both natures.

What is the difference between Photon and Electron?

Electrons are the negatively-charged subatomic particles that remain outside the nucleus rotating in their respective suborbital. Every electron is defined by its energy level and bound with the nucleus via electrostatic force of attraction. They also have a negligible mass and can also be found in cathode rays. Electrons can also move in the form of waves.

Photons, on the other hand, are energy packets that show both wave-like and particle nature. They have no mass but have high energy. They move at the speed of light. Their energy level is determined by the frequency of the electromagnetic radiation waves.

This is the basic difference between a photon and electron. To understand these two particles, you need to observe their properties and differences aptly.

Plasticity refers to the quality of an object to transform to any shape and size.

When an elastic material is stretched beyond its elastic limit, the material becomes permanently deformed. This permanent deformation is termed plasticity.

It means that when a material is subjected to a high magnitude of external force, its interatomic particles leave their old lattice points and become distant from each other.

Here, we will discuss elasticity, plasticity, types of plasticity, and plasticity psychology in detail.

Plasticity

Plasticity is the ability of solid materials to go with a flow or to change orientation permanently when they are subjected to stresses of intermediate magnitude between those producing temporary deformation and elastic behaviour, and those causing failure of the material to its original shape. 

Plasticisation is the influence of external forces that leads a material to undergo permanent deformation without rupture or damage. 

Elasticity enables a solid to return to its original orientation after the load or external force is removed. Plastic deformation occurs in many metal-formation procedures like rolling, pressing, forging, and in geological processes like rock folding and rock flow under the earth at very high pressures and at rising temperature.

So, the meaning of plasticity of a material is a material that can be moulded to any desired shape and size when subjected to high temperature and pressure.

Cortical Plasticity

We refer to cortical plasticity as neuroplasticity. It refers to the extraordinary ability of the brain to reorganize itself by forming new neural connections depending on their experiences, lifestyle, and environment. 

To collect information about the sensory experience and practised movements is a universal property of all cortical areas; and this capacity of the brain is known as cortical plasticity.

We observe that cortical plasticity is observed in variations that rely on experiences and in the functional attributes of cortical neurons and in the alteration of cortical circuits of the brain.

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Neuroplasticity Meaning

Neuroplasticity is the capacity of neural networks and neurons in the brain to change their connections and behaviour in response to the following:

1. Gaining new information

2. Sensory stimulation

3. Development

4. Damage, or dysfunction

Though some neural functions appear to be hard-wired in particular, localized regions of the brain, certain neural networks possess modularity and perform specific functions while retaining the capacity to divert from their usual functions and to reorganize themselves. Hence, we consider neuroplasticity as a complex, multifaceted, fundamental property of the brain. 

Neuroplasticity

Neuroplasticity was forwarded by well-known neuroscientists to study only for childhood, but after the research in the latter half of the 20th century, a study showed that many aspects of the brain can be changed or are plastic even through adulthood as well.

We call neuroplasticity both neural plasticity and brain plasticity. It is the ability of neural networks in the brain to bring alterations through growth and reorganization. 

These alterations range from individual neuron pathways making new connections, to systematic adjustments like cortical remapping. 

Neuroplasticity examples are circuit and network variations that result from learning a new ability, environmental influences, practice, and psychological stress.

Developing the brain (by elasticity) exhibits a higher degree of plasticity than the adult brain. However, Activity-dependent plasticity can have significant implications like healthy development, learning, memory, and recovery from brain damage.

Types of Neuroplasticity

There are two types of plasticity; these are as follows:

• Structural plasticity

• Functional plasticity

Structural plasticity is known as the ability of the brain to change its neuronal connections. 

New neurons are produced constantly and integrated into the central nervous system (CNS) throughout the life span based on this type of neuroplasticity. 

At present, researchers use multiple cross-sectional imaging methods (that is a magnetic resonance imaging (MRI), and computerized tomography (CT) to study the structural alterations of the human brain.

This type of neuroplasticity studies the effect of various internal or external stimuli on the anatomical reorganization of the brain. The changes of grey matter proportion and synaptic strength in the brain are considered as the source of study of structural neuroplasticity. 

Do you know that structural neuroplasticity is currently investigated more in the field of neuroscience in current academia?

Functional Neuroplasticity

Functional plasticity refers to the ability of the brain to alter and get used to the functional properties of neurons. The alterations may happen in response to activity-dependent plasticity in order to acquire memory or in response to malfunction or damage of neurons, i.e., reactive plasticity to overcome a pathological event. In the latter case, the functions from one part of the brain transmit to another part of the brain relying on the demand to produce recovery of behavioural or physiological processes. 

Plasticity Psychology

Talking about the psychological forms of activity-dependent plasticity, these involve synapses, i.e, synaptic plasticity. The strengthening or weakening of synapses often results in an increase or decrease in the firing rate of the neurons. This rising/decreasing is the long-term potentiation (LTP) and long-term depression (LTD), respectively, and they are considered as examples of synaptic plasticity that are directly linked with memory.

You might’ve observed that wrestlers pick up the heavy mass in very little time because they have the power to perform such an activity.

So, what is power?

Power is the rate of doing an activity or work in the minimum possible time. It is the amount of energy transferred or converted per unit time where large power means a large amount of work or energy.

For example, when a powerful car accelerates speedily, it does a large amount of work which means it exhausts large amounts of fuel in a short time.

On This Page, we will Learn About the Following :

What is power with example?

Example1: Suppose, person A and B are assigned the task of picking up an equal number of boxes to the top floor of the building.

Let’s say there are 10 boxes of 10 kg each to be picked up by both A and B. Each time they walk a distance of 5 m. Since the work done by both is in the form of potential energy mgh is given by,

W =  mgh = 10 x 10 x 5 = 500 J is the work done by you both.

Suppose, A finishes his task in 50 s and B in 100 s.

As you can see the relationship of Work done per unit time is nothing but the Power.

The rate at which work is done is referred to as power. It is always dependent on the work done. It is defined as the amount of energy that is converted per unit of time. Its International System of Unit is Watt which is equal to one joule per second. It is a scalar quantity. 

For instance, a large amount of work is done and a large amount of fuel is consumed in a short period of time when a powerful car is accelerated rapidly.

Definition of Power

The power is the time taken by you to complete any task or activity.

The power doesn’t remain constant, but how?

Let’s consider Example.1, 

Suppose the boy A walks at a pace (high power), he slows down (less power), continues with this speed, takes rest in between (P= 0 as W = 0), then walks with the pace.

Here, when he makes variations in the speed, the work done varies too, at an instant, the power delivered is different at the different instant.

We can conclude that at different instants, the power (Example.1)  doesn’t remain the same.

So the power delivered in a certain period of time is called instantaneous power.

If the  Δt approaches to zero then power will be instantaneous and given by,

ΔW is the work done in a short interval of time Δt (instant time).

Power, the least possible time required by a person or an object to do the work.

Power  Formula

The power is a time-based quantity that is related to the pace at which work is done. The power of an object can be given by-

[textrm{Power}=frac{textrm{work done}}{Time}]

[textrm{Power}=frac{W}{t}]

SI Unit of Power

The standard metric unit of work is represented in the terms of Joule and the unit for standard metric time is represented in seconds. Thus, the unit for standard metric power is given as-

[textrm{Power}=frac{textrm{Joule}}{textrm{second}}=Js^{-1}=Watt]

The multiples of power: KW, MW, GW…

Watt: When a body does work of one joule in one second it is called one-watt power.

Another unit of power (In British engineering) is Horsepower (hp).

Where 1hp = 746 W

Dimensional Formula of P: [[M^1][L^2][T^{-3}]]

When a body does work of 550 foot-pounds per second (746 W) is called its one horsepower.

Average Power

The ratio of the total amount of work done in the total amount of time is called the average power.

There are certain instruments used to compute average power. If we talk about Fibre optic power instruments, they measure the average power of a continuous light beam that is used to test signal power in fiber-optic networks.

         

Note: If the work is done at a uniform rate, then the average and instantaneous power becomes equal, and the common equation comes out to be,

Electric Power

Electric power is defined as the rate, per unit time at which energy is transformed from the electrical energy of the moving charges to some other form, e.g. heat, mechanical energy, usually created by electric generators.

Electric generators convert mechanical energy obtained from an external source (the power of motion) into electrical energy.

Electric Power Formula is Stated as,

P = [frac{W}{t}]

states that higher the electrical current (I) the higher the heat generated, and so the higher the power/ energy loss since electrical energy is transformed into heat.

Work, Energy, and Power

Suppose, you want to displace a body (do some work W) to some distance S by applying your energy (Force F).

The mathematical relationship to describe the above scenario is given by,

W = F S …(1)

P = [frac{W}{t}]…..(2)

Using the above two equations,

P = [frac{Fs}{t}]

We know that V = [frac{s}{t}]

So   

This is the relation between power and force.

Consumption of Energy and Power

The power consumption rate is higher as the appliance is used for a long time which results in a greater cost of that appliance. Therefore the power consumption rate for an appliance is given by-

P = [frac{W}{t}] = [frac{E}{t}]

Where the energy supplied is represented by E 

Therefore, the energy which is consumed in time t will be given by-

E
= Pt

Power – Solved Examples

Problem1: Riya has a mass of 60 kg and runs up to 13 m high in 50 seconds. Compute her power. (Take g = 10m/s²)

Solution: Given h = 13 m, m = 60 kg and t = 50 s

P = W/t = mgh /t

= 60 x 10 x 13/ 50

On solving we get,

P = 156 W

Problem2: If the current and voltage of an electric circuit are 3 A and 15 V respectively. Calculate the electrical power.

Solution: Given  I = 3 a and V = 15 V

Since P =VI 

=  3 x 15

We get, P = 45 W

Problem3- If the mass of a person climbing a tree which is 5 meters high is 60 kg and he climbs up the tree in 10 seconds. What is the power required for him to climb up the tree? g=10 m/s²

Solution- Given

Mass of the person m = 60 kg

Height of the tree h = 5 meters

Acceleration due to gravity g = 10 m/s²

Time interval t = 10 seconds

Therefore work done will be,

Work = m.g.h

Work = 60 × 10 × 5 = 3000 Joules

Thus the power will be work done per unit time,

Power = [(frac{Watt}{t})] = [(frac{3000}{10})] = 300 Joule/second

Hence the power required to climb up the tree will be 300 J/s.

Conclusion:

Electricity is a type of power which is produced by electric means and it has wide applications in everyday life. It has become an essential part of modern life and is used by people for cooling, heating, and refrigeration, and for operating appliances such as machinery, electronics, public transportation systems, and so on. 

We all know from Newton’s Universal Laws of Gravity that any object on Earth is constantly influenced by gravity and pulled towards the center of the gravitational field, that is the center of the Earth. Gravity influences any object suspended in the air to fall downwards, as in the case of fruits from trees. Also, if you launch an object up into the air, it travels into the air due to its inertia of motion. But since the object is also under the influence of gravity, it also dips down in the ground. Due to the influence of gravity, that object takes a curve path forward towards the ground before it eventually stops after touching the ground. The object which is launched into the air is called projectile, and this kind of motion is called projectile motion.

When we throw an object up, due to the force of gravity it eventually comes down. An object which is in motion in the air with no other force but gravity influencing it is a projectile. We can define a projectile as an object which is projected into the air with only downward gravitational force influencing it and continues its motion by the virtue of its own inertia.

 Projectiles can roughly be classified into three kinds:

•   A projectile which is left to free fall from a considerable height

•   A projectile which is launched directly upwards

•   A projectile upwards at an angle towards the horizontal

What Is Projectile Motion?

Projectile motion is the motion experienced by an object in the air only under the influence of gravity. A projectile, that is launched into the air near the surface of the Earth’s and moves along a curved path, or in other words a parabolic path, under the action of gravity, assuming the air resistance is negligible. Study of such motion is called ballistics. The only force acting upon a projectile is gravity, which imparts a downward acceleration to the projectile. Because of the object’s initial inertia, an external force is unnecessary to maintain the horizontal velocity of the object. It also requires considering other external forces like friction from aerodynamic drag or internal propulsion when the projectile motion is at a larger scale.

When you throw a particle in an oblique path near the surface of the earth, it follows a curved route with constant acceleration. In such a situation, the particle’s path is known as a projectile, and its motion is known as the projectile motion.

In a projectile motion, two separate rectilinear motions occur at the same time:

A particle’s horizontal and vertical projectile motions are accelerated in the following ways: When a particle is sent into the air at a certain speed, gravity’s acceleration is the only force acting on it during that period (g). Along the x-axis is uniform velocity, which is responsible for the horizontal particle, also known as the forward motion.

Along the y-axis: uniform acceleration, which is responsible for the particle’s vertical (downwards) motion. This downhill acceleration has a vertical component. There is no acceleration in the horizontal direction, implying that the particle’s velocity in that direction remains constant.

Maximum Projectile Height

Let us know the maximum height of a projectile after grasping what a projectile is and how it moves. The highest vertical location along the object’s flight determines its maximum height. The object’s starting velocity determines the projectile’s range.

If v is the beginning velocity, g is gravity’s acceleration, and H is the most significant height in metres, then = the initial velocity’s angle from the horizontal plane (radians or degrees).

The formula that has been derived  for calculating the maximum height of a projectile is:

H=[frac{usintheta }{g}]

Ballistics, the study of projectile motion:

The science of mechanics that deals with the flight, behaviour, and effects of projectiles are called ballistics. The word ballistics comes from the word ballista, which was a siege machine used for war during the late 18th century. Ballista was a weapon that is used to hurl boulders or spears against the enemy troops or walls. The boulder or spears used against the enemy were launched by the ballista, after which it would travel in a parabolic path and land among the enemy. The operators of the ballista have to be concerned about the angle to launch the projectile and the path they had to travel to gauge the landing position of the ballista. This is the reason the study of projectile motion is called ballistics. The path travelled by the projectile after it is launched is called the ballistic trajectory. The study of ballistics is important in modern warfare since it involves calculating the trajectories of modern warfare equipment like bullets, unguided bombs, rockets, and such other equipment. It also involves the designing of projectiles to attain desired acceleration and performance.

The elementary equations of ballistics only take into account the initial velocity of the projectile and an assumed constant gravitational acceleration. Practical applications of ballistics have to consider a variety of other factors like air resistance, crosswinds, target motion, the varying acceleration due to gravity, and in the case the target is significantly far away, even the rotation of the earth.

The weapon with the highest potential for destruction is named the ballistic missile since it involves delivering warheads to a target by a missile which follows a ballistic trajectory and falls on the target under the influence of gravity.

Some Important One-Liners to Remember About Projectile Motion

1. The kinetic energy is , and the linear momentum is  m.u.cosፀ at the highest point.

2. The horizontal displacement of the projectile after t seconds is

3. The vertical displacement of the projectile after t seconds is

4. is the equation for the projectile’s path.

5. A projectile’s course is parabolic.

6. The kinetic energy is at the lowest position.

7. The linear momentum is equal to m.u at the lowest position.

8. The projectile’s acceleration is constant during the motion and operates vertically downwards, equal to g.

9. The angular momentum is equal to where h is the height.

10. In the case of angular projection, the angle between velocity and acceleration might range from 0 to 180 degrees.

Key components which are to be remembered while deal
ing with projectile motion:

In the given diagram

The key components which we should remember when calculating projectile motion and solving problems are

• Initial launch angle, or angle of projection

• The initial angle of projection varies from 0° to 90° And it heavily influences the motion            of a projectile. 

• Initial velocity, u

If the path of the projectile, its launching point, and its target is plotted on a graph, initial velocity can be expressed as x and y components. The initial velocity of the projectile, in that case, can be given as

uxor uy = u sinፀ

or we can use the Pythagoras theorem to find the velocity

The time of flight is the time period from when the object is launched to the time it reaches the ground. The time period the projectile is in the air depends on the initial velocity and the angle of projection. It is given as 

T = [frac{2usintheta }{g}]

There is no acceleration in the horizontal direction in case of projectile motion. The only acceleration present is the downward acceleration, that is acceleration due to gravity, also known as free fall. It is given as

ax = 0 or ay = -g

The value of g is equal to 9.8 m/s

• Horizontal velocity, ux, and Vertical velocity, uy

At any point in the projectile motion, the horizontal velocity remains constant. On the other hand, vertical velocity varies linearly. This is because acceleration is constant at 9.8 m/s

At time given by t, the displacement components in a graph plotted with the origin of the projectile as the origin, the displacement components are

X = u.t.cosፀ and y = u.t.sinፀ-gt²

The maximum height of the projectile is the highest height the projectile can reach

It is given by

H = [frac{u^2sin^2theta }{2g}]

Range, R

The range of a projectile is the distance between the launch point and the target in a straight line.

It is calculated by R = [frac{u^2sin2theta }{g}]

In theoretical Physics, Quantum field theory (QFT), incorporates the classical field theory, the theory of quantum mechanics, and the theory of special relativity. The combination of these theories explains the behaviour of subatomic particles and their interactions through various force fields. 

The two examples of modern QFT are Quantum Electrodynamics and Quantum Chromodynamics.

In particle physics, Quantum field theory uses the physical models of sub-atomic particles and condensed matter physics to establish models of quasiparticles.

On this page, we will understand what is quantum field,  all about the quantum field theory, and the difference between quantum mechanics and quantum field theory.

What is the Quantum Field? 

For understanding the quantum field theory, we’ll start with the quantum field.

Quantum field is a quantum-theoretical generalization of classical fields. the 2 archetypal classical fields are:

1.  Maxwell’s electromagnetic field and 

2. Einstein’s metric field of gravitation

A method to believe the method of quantization is that we first reformulate the (still classical) field equations in terms of mathematical operators replacing some numerical quantities (this part is pure algebra/calculus, with no introduction of physics).

On the other hand, we “solve” the resulting operator-valued equations, including solutions that don’t appear within the classical theory. 

We also make the assertion (validated by observation) that these new, “nonsensical” (in imagination, not during a mathematical sense) solutions accurately describe nature, including all the observed quantum behaviour that contradicts the classical theory.

What is Quantum Field Theory?

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There are several rationales for employing a quantum field theory. 

• First, QFT is a basic generalization of classical field theories, which are our most successful non-quantum/ natural theories. 

• Second, a scientific theory can account for the observed, well-studied creation and destruction of particles, processes that do not exist in physics. 

• Third, a scientific theory is innately relativistic, and “magically” (not really, just elegant math) resolves problems with causality that plague even relativistic quantum particle theories.

But no, quantum fields don’t interact with matter. Quantum fields are matter during a quantum theory, what we perceive as particles are excitations of the quantum field itself.

Theory of Quantum Mechanics

The term “quantum mechanics” is usually utilized in a minimum of two distinct ways.

It often refers to the overall structure of all “quantum” theories – during which observables correspond to self-adjoint operators on a Hilbert space, and changes of the observer’s point of view (like time evolution) refer to unitary operators.          

In terms of this usage, quantum mechanics isn’t such a lot different as just a special case of quantum field theory. Now, let us understand the difference between the two in a tabular format:

Difference Between Quantum Mechanics and Quantum Field Theory

Field Theory

A field theory essentially explains all the physical phenomena in terms of a field and the way in which it interacts with the matter/fields.

For instance, Euclidean field theory is considered a very useful tool for the study of quantum field theory. 

Where the Euclidean Quantum Field Theory talks of the relativistic quantum field theory in which time is supplanted by a purely formal imaginary time, causing the replacement of Lorentz covariance by the Euclidean group covariance.

So, we encountered the two terms, i.e., Lorentz covariance and the Euclidean group covariance. Now, we will understand these two terms. 

1. Lorentz Group Covariance:

In relativistic physics, Lorentz symmetry, named after Hendrik Lorentz, is an equivalence of observation or observational symmetry thanks to the special theory of relativity implying that the laws of physics stay an equivalent for all observers that are moving with reference to each other within an inertial reference frame

2. Euclidean Group Covariance:

Position operators (p.o.) for relativistic fundamental quantum systems are constructed as operator-valued integrals with reference to Euclidean systems of covariance (ESC), i.e., positive operator-valued (POV) measures being covariant under the Euclidean group, and are expressed in terms of the generators of the inhomogenous Lorentz Transformation/Lorentz Tensor. 

This p.o. is partly well-known within the literature where
it is found by other methods.