Postulate of wave mechanics or Quantum Mechanics

Postulate of Wave or Quantum Mechanics (or Operator formalism in Quantum mechanics) →

The formulation of mathematical equations of quantum mechanics is based on the linear operator. This operator formulation of quantum mechanics is known as postulates of quantum mechanics. These postulates are given below:-

1. For a system consisting of particles moving in a field of a conservative force, there is an associated complex wave function $\psi(x, y, z, t)$ where $x$,$y$,$z$ space coordinates, and $t$ is the time. This wave function enables us to obtain a description of the behavior of the system, consistent with the principle of uncertainty.


2. There is an operator with every observable dynamical quantity. The operator corresponding to the pertinent dynamical quantities is:-
   
   
   

   Dynamical Variable	Symbol	Quantum Mechanical Operator
   Position	$x$ $y$ $z$	$x$ $y$ $z$
   Momentum	$P_{x}$ $P_{y}$ $P_{z}$ Generalised Form $\overrightarrow{P}$	$\frac{\hbar}{i}\frac{\partial}{\partial x}$ $\frac{\hbar}{i}\frac{\partial}{\partial y}$ $\frac{\hbar}{i}\frac{\partial}{\partial z}$ Generalised Form $\frac{\hbar}{i}\overrightarrow{\nabla}$
   Total Energy	$E$	$i\hbar \frac{\partial}{\partial t}$
   Total Energy	$E$	$-\frac{\hbar^{2}}{2m} \nabla^{2}+V(x)$ This is also known as Hamiltonian Operator $H$.
   Kinetic Energy	$K$	$-\frac{\hbar^{2}}{2m} \nabla^{2}$
   Potential Energy	$V(x,y,z)$	$V(x,y,z)$

   
   
   All the operators have eigen functions and eigen values.


3. The wave function $\psi(x,y,z,t)$ and its partial derivatives $\frac{\partial \psi}{\partial x}$, $\frac{\partial \psi}{\partial y}$, $\frac{\partial \psi}{\partial z}$ must be finite, continuous and single-valued for all values of $x$, $y$, $z$ and $t$


4. The product of $\psi(x,y,z,t)$ and $\psi^{*}(x,y,z,t)$ is always a real quantity. The product is called the probability density and $\psi\psi^{*} d\tau $ is interpreted as a probability that the particle will be found in volume element $d\tau$ at $x$,$y$,$z$ and time $t$. Since the total probability of finding particles somewhere in the entire space must be equal to 1.
   
   
   $\int_{-\infty}^{+\infty} \psi \: \psi^{*}\: d\tau = 1$
   
   
   The integral is taken overall space.


5. The average or expectation value of an observable quantity $\alpha$ with which an operator $\hat{\alpha}$ is associated is defined by

$\left< \alpha \right> = \int_{-\infty}^{+\infty} \psi^{*} \hat{\alpha} \psi d\tau$


The integral being taken overall space.

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Eigenfunction, Eigenvalues and Eigenvectors

Eigenfunction and Eigenvalues → If $\psi$ is a well-behaved function, then an operator $\hat{P}$ may operate on $\psi$ in two different ways depending upon the nature of function $\psi$ `

1. When an operator $\hat{P}$ operates on any function $\psi$ then this function $\psi$ changes into another function $\phi$. i.e.
   
   
   $\hat{P} \psi =\phi$
   
   
   Where $\phi$ is a new function linearly depending upon the initial function $\psi$.
   
   
   Example:
   
   
   Let us consider a function $f(x)=x^{2}$ and an operator ie. differential operator $\frac{d}{dx}$ is operate on the function. Then we get
   
   
   $\frac{d}{dx}f(x)= \frac{d}{dx} (x^{2})$
   
   
   $\frac{d}{dx}f(x)= 2x$
   
   
   Now the given function $f(x)=x^{2}$ change into another function $f(x)=x$.


2. When an operator $\hat{P}$ operates on any function $\psi$ then this function $\psi$ does not change into another function but now this function $\psi$ may be with multiples of complex or real numbers(or values).i.e
   
   
   $\hat{P} \psi =\lambda \phi$
   
   
   Where $\lambda$ is Real OR Complex Number. This number or value is known as Eigenvalues.
   
   
   In this case, the function $\psi$ is a member of the class of physically meaningful functions called the eigen function of the operator $\hat{P}$. The number $\lambda$ is called the eigen value of operator $\hat{P}$ associated with eigen function $\psi$ and this equation is known as the eigenvalue equation.
   
   
   Example: Let us consider a function $f(x)=e^{2x}$ and an operator ie. differential operator $\frac{d}{dx}$ is operate on the function. Then we get
   
   
   $\frac{d}{dx}f(x)= \frac{d}{dx} (e^{2x})$
   
   
   $\frac{d}{dx}f(x)= 2e^{2x}$
   
   
   Now the given function $f(x)=e^{2x}$ change into another function $f(x)=x$.

Eigenvalues and Eigenvectors →

Let a linear transformation equation

$AX=\lambda X \qquad(1)$

Where
A → Square matrix of $n$ order (where $n=1,2,3,.....$)
$\lambda$ → Scalar Factor

The equation $(1)$ may be written as

$AX=I\lambda X$

$AX-I\lambda X=0$

$(A-I\lambda)X=0 \qquad(2)$

Where $I$ is unit matrix

Any value of $\lambda$ for which equation $(1)$ or equation $(2)$ has non zero (i.e $X \neq 0$) solution is called eigenvalues or characteristic roots or latent root of the matrix $A$ and corresponding non zero solution of $X$ is called eigenvectors or characteristic vectors or latent vectors of the matrix $A$

The matrix $\left| A- \lambda I \right|$ is called characteristic matrix of $A$. The determinant $\phi(\lambda) = \left| A- \lambda I \right|$ is called the characteristic polynomial of $A$. So

$\phi(\lambda) = \left| A- \lambda I \right| =0$

And $\phi(x)= a_{0}+a_{1} \lambda a_{2} \lambda^{2}+ .......+a_{n} \lambda^{n}=0 $

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Kepler's Laws of Planetary Motion

Kepler's Laws →

Kepler's found important uniformity in the motion of the planets. This uniformity is known as "Kepler's Laws of planetary motion". There are basically three laws →

1. First law (Law of orbits)


2. Second Law (Law of Areal Speed)


3. Third Law (Law of Periods)

First law (Law of orbits) →

All the planets move around the sun in elliptical path and the sun is at the one foci of the ellipse.

Second Law (Law of Areal Speed) →

A line joining the any planet to sun sweeps out equal areas in equal interval of of times. i.e The areal speed of the planets remains constant.

According to the second law, When the planet is farthest from the sun, then its speed is maximum, and when it is nearest the sun, then its speed is maximum.

Planetary Motion

From the above figure, A planet is moving around the sun from $A$ to $B$ in a given time interval, and from $C$ to $D$ in the same time interval, then according to the law of areal speed, the areas $ASB$ and $CSD$ will be equal with respect to the same time-interval

Proof →

Let us consider, A planet moves the angle $d\theta$ from $A$ to $B$ in the infinitesimal interval of time $dt$, then the area swept out by radial line $SA$ is

$dA$= Area of the curved $\Delta$ $SAB$

$dA \approx \frac{1}{2} \left( AB \times SA \right)$

$dA \approx \frac{1}{2} \left( r \: d\theta \times r \right)$

$dA \approx \frac{1}{2} \left( r^{2} \: d\theta \right)$

Thus, the instantaneous areal speed of the planets is

$\frac{dA}{dt}=\frac{1}{2} r^{2} \frac{d \theta}{dt}$

$\frac{dA}{dt}=\frac{1}{2} r^{2} \omega \qquad(1)$

Where $\omega$ → The angular speed of the planet.

We know that

$J=I \: \omega$

Where $I$ is the instantaneous moment of inertia of the planet about the Sun $S$.

$J=m \: r^{2} \: \omega \qquad(2)$

Where $m$ is the mass of the planet. Now substitute the value of $r^{2} \: \omega$ from equation $(2)$ in equation $(1)$.

$\frac{dA}{dt}=\frac{J}{2m} \left( Constant \right)$

From the above equation, The areal speed $\frac{dA}{dt}$ of the planet is constant which is Kepler's Second Law. The above equation shows that the angular momentum of the planet is constant. i.e. The angular momentum of any planet is conserved.

Third Law (Law of Periods) →

The square of the time period of one complete revolution of any planet around the sun is directly proportional to the cube of the semi-major axis of its elliptical orbit.

Path of the Planet

Proof →

Let $a$ and $b$ be the semi-major and semi-minor axes of the ellipse, then the area of the ellipse will be $\pi ab$. Let $T$ is the period of one complete revolution of any planet, then

$T=\frac{Area \: of \: the \: ellipse}{Areal \: Speed}$

$T=\frac{\pi a b}{\frac{J}{2m}}$

$T=\frac{2\pi a b m}{J}$

$T^{2}=\frac{4 \pi^{2} a^{2} b^{2} m^{2}}{J^{2}}$

Let $l$ be the semi-latus rectum of the elliptical orbit i.e. $l=\frac{b^{2}}{a}$. Then we get the above equation which can be written as

$T^{2}=\frac{4 \pi^{2} m^{2} a^{3} l }{J^{2}}$

$T^{2} \propto a^{3} $

Where $\frac{4 \pi^{2} m^{2} l }{J^{2}}$ is constant.

From the above equation, It is clear that the larger the distance of a planet from the sun, the larger will be its period of revolution around the sun. The period of revolution of the nearest planet to the sun i.e. Mercury is $88$ days, while that of the faraway planet from the sun i.e. Pluto is $248$ years.

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Relation between gravitational acceleration and gravitational force

Relation between $g$ and $G$ →

Let us consider:
The mass of earth = $M_{e}$
The radius of earth = $R_{e}$
The mass of the object = $m$

If the object is placed on the surface of the earth then the gravitational force on the object is →

$F=G \frac{M_{e} m}{R_{e}^{2}} \qquad(1)$

The force on the object due to gravitational acceleration is →

$F=mg \qquad(2)$

From equation $(1)$ and equation $(2)$

$mg=G \frac{M_{e} m}{R_{e}^{2}}$

$g=G \frac{M_{e}}{R_{e}^{2}}$

$G M_{e}=g R_{e}^{2}$

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Newton's law for Gravitational Force

Gravitational Force →

Newton's Gravitational Law statement is a combination of three individual statements. These are

1. The force between the two-particle is directly proportional to the product of their masses i.e.
   
   
   
   $F \propto m_{1} \: m_{2} \qquad(1)$
   
   
   Where $m_{1}$ & $m_{2}$ are the masses of the particles.


2. The force between the two-particle is inversely proportional to the square of the distance between them i.e.
   
   
   
   $F \propto \frac{1}{r^{2}} \qquad(2)$
   
   
   Where $r$ is the distance between the particles.


3. This force always acts between the line joining the masses.

From the above the equation $(1)$ and equation $(2)$

$F\propto \frac{m_{1} \: m_{2}}{r^{2}}$

$F=G \frac{m_{1} \: m_{2}}{r^{2}}$

Where $G$ is Newton's gravitaional constant and its experimental value $6.67\times 10^{-11} \frac{N-m^{2}}{kg^{2}}$

Properties of Newton's law for Gravitational force →

There are the following properties of Newton's law for gravitational force

Gravitational force is always an attractive force.

Gravitational force is action and reaction pair and follows Newton's third law.

A Gravitational force is a conservative force.

Gravitational force is central force i.e. it is always acting along the line joining between two particles.

Unit and Dimensional formula of $G$

The unit of $G$ is $\frac{N-m^{2}}{kg^{2}}$

The dimensional Formula of $G$ is $[M^{-1}L^{3}T^{-2}]$

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Frame of References (Inertial Frame and Non Inertial Frame)

It is assumed that space is continuous and the motion of particles in space can be described by their position at different instants of time. The position of a particle is known as a point in space. These points are described by the coordinate system in space.

When point (position of particle in space) and the time are taken together then it is called an Event.

The coordinate system of a particle which describe the position of any particle relative to it, then such coordinate system is known as Frame of Reference or System of Reference.

Absolute Space:

The absolute space is those frame of reference relative to which every motion and position should be measured.

Types of Frame of Reference:

According to the motion of particles frame of reference is divided into two categories

1. Inertial frame of reference
2. Non-inertial frame of reference

Inertial frame of reference:

Those unaccelerated frame of reference, in which Newton's first and second laws hold, are called inertial frames.

In an inertial frame, body does not experience any force so according to second law of Newton-

$F=ma \qquad (1)$

$F=0$

so from equation $(1)$

$ ma=0$

$ a=0$

$ \frac{d^2r}{dt^2}=0\qquad (2)$

In component form, above equation, can be written as:

$ \frac{d^2x}{dt^2}=0;\:\:\:\frac{d^2y}{dt^2}=0;\:\:\:\frac{d^2z}{dt^2}=0\qquad (3)$

Discussion of Inertial Frame in terms of relative frame of reference:

Let us consider an inertial frame $S$ and another frame $S'$ which is moving with constant velocity $v$ relative to frame $S$. If the position of the origins of the two frames coincide, then in the two frames the position vector of any particle $P$ at any instant $t$ can be related by the following expression-

Inertial Frame of Reference

$ \overrightarrow{r}=\overrightarrow{oo'}+\overrightarrow{r'}$

$ \overrightarrow{r}=\overrightarrow{v}t+\overrightarrow{r'} \qquad \left \{ {\overrightarrow{oo}'=\overrightarrow{v}t} \right \}$

Differentiate the above equation

$\frac{d\overrightarrow{r}}{dt}=\overrightarrow{v}+\frac{d\overrightarrow{r'}}{dt}$

Again differentiate the above equation

$\frac{d^2\overrightarrow{r}}{dt^2}=\frac{d^2\overrightarrow{r'}}{dt^2}$

$a=a'$

Now we conclude that "If a frame is an inertial frame then all those frames which are moving with constant velocity relative to the first frame are also inertial"

Non-inertial Frame:

Those frames of reference in which Newton's law of inertia does not hold are called non-inertial frames.

All the accelerated and rotating frames are the non-inertial frames of reference.

According to Newton's Second Law, the force $F$ applied on a body of mass $m$ is given by

$ F_{i}=ma_{i}\qquad (1)$

Newton's Second Law is not valid when a body of mass $m$ self accelerated and accelerated body will observe the acceleration $a_{N}$. Hence

$ F_{i}\neq ma_{N}\qquad (2)$

If no external force is acting on a particle. Even then in the accelerated frame, It will appear that a force is acting on it. This force is called pseudo force or fictitious force. The direction of force is opposite to acceleration.

$ F_{o}\neq ma_{o}\qquad (3)$

Discussion of Non-inertial Frame in term of relative frame of reference:

Let us consider, an inertial frame $S$ and another frame $S'$ is moving with an acceleration $a_{0}$ relative to frame $S$.

Non-inertial Frame of Reference

$ \overrightarrow{r_{i}}=\overrightarrow{r_{N}}+\frac{1}{2}a_{0}t^{2}$

Differentiate the above equation with respect to $'t'$

$ \frac{d\overrightarrow{r_{i}}}{dt}=\frac{d\overrightarrow{r_{N}}}{dt}+a_{0}t$

Again differentiate the above equation with respect to $'t'$

$ \frac{d^{2}\overrightarrow{r_{i}}}{dt^{2}}=\frac{d^{2}\overrightarrow{r_{N}}}{dt^{2}}+a_{0}$

$ a_{i}=a_{N}+a_{0}$

$ a_{i}-a_{0}=a_{N}$

$ ma_{i}-ma_{0}=ma_{N}$

$F_{i}+F_{0}=F_{N}$

This formula gives the observed force $F_{N}$ in the accelerated system.

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