Inadequacy of classical mechanics

Classical mechanics is a branch of physics that deals with the motion of macroscopic bodies or objects $(i.e \: the \: size \: range \: greater \: then \: 10^{-8}m)$ under the influence of forces. While it was groundbreaking when first developed by Sir Isaac Newton in the 17th century, it has certain limitations that were discovered over time. In this answer, we will discuss these limitations in more detail:

  1. Classical Mechanics is not applicable to extremely small objects: Classical mechanics assumes that particles have a definite position and momentum, which is not true in the quantum world. This limitation became apparent in the early 20th century with the discovery of quantum mechanics. Quantum mechanics is a branch of physics that deals with the behavior or motion of particles on an atomic and subatomic level $(i.e \: the \: size \: range \: is \: in \: between \: 10^{-8}m \: to \: 10^{-15}m)$ i.e microscopic particles. It has been successful in explaining phenomena such as the photoelectric effect, blackbody radiation, and the behavior of electrons in atoms, which cannot be explained by classical mechanics.
  2. Classical Mechanics is not applicable to objects moving at very high speeds: Classical mechanics assumes that the speed of an object can be infinite,  it is not true in the relativistic world. The theory of relativity which was developed by Albert Einstein in the early 20th century, explains the behavior of objects moving at high speeds (i.e. equal to the speed of light). The theory of relativity has been successful in predicting phenomena such as time dilation, length contraction, and the equivalence of mass and energy.
  3. Classical Mechanics can not well explain the behavior of systems with many particles: Classical mechanics is not well suited for dealing with systems that have many particles. This is because it is difficult to solve the equations of motion for systems with many particles, and the behavior of the system can become chaotic. The theory of statistical mechanics, developed in the late 19th century, addresses this limitation by using probability distributions to describe the behavior of large systems.
  4. Classical Mechanics can not explain the behavior of objects that are very far apart or have very high masses: Newton's law of gravity works well for objects that are close together, but it fails to explain the behavior of objects that are extremely far apart or have very high masses, such as black holes and galaxies. The theory of general relativity, developed by Einstein in the early 20th century, provides a better explanation of the behavior of objects with very high masses and gravitational fields.
  5. Classical Mechanics assumes determinism: Classical mechanics assumes that the universe is deterministic, meaning that the future state of a system can be predicted with complete accuracy if the initial state is known. However, this assumption has been challenged by the theory of chaos, which suggests that small changes in the initial state of a system can lead to unpredictable and chaotic behavior in the future.
In summary, while classical mechanics is still useful for understanding the behavior of macroscopic objects, its limitations have led to the development of new theories, such as quantum mechanics, relativity, statistical mechanics, and chaos theory, which can explain the behavior of the universe at different scales and levels of complexity.

Classical world and Quantum world

Classical world vs Quantum world:

The classical world and the quantum world are two fundamentally different ways of describing the behavior of matter and energy.

In the classical world, the laws of physics are described by classical mechanics, which is based on the concepts of position, velocity, and acceleration of objects. Classical mechanics is deterministic, meaning that if you know the initial conditions of a system, you can predict its future behavior with complete accuracy. This is the world we experience in our everyday lives, and it is characterized by a continuous, smooth flow of events.

In contrast, the quantum world is described by quantum mechanics, which is based on the behavior of particles on a subatomic scale. In the quantum world, particles do not have well-defined positions and velocities but rather exist in a superposition of many possible states. Moreover, measurements of quantum particles do not give deterministic results, but rather give probabilities of various outcomes. This probabilistic nature of quantum mechanics is known as the uncertainty principle.

Another important feature of the quantum world is entanglement, which occurs when two particles become linked in such a way that the state of one particle depends on the state of the other particle, even if they are separated by large distances. This has important implications for the way we understand the nature of reality itself.

While the classical and quantum worlds may seem very different, they are not entirely separate from each other. Classical mechanics can be seen as an approximation of quantum mechanics for macroscopic objects, and quantum mechanics can be used to explain phenomena that cannot be explained by classical mechanics alone.

Overall, the classical world and the quantum world are both valid ways of describing the behavior of matter and energy, and they each have their own unique properties and characteristics.

Intensity of a wave

Definition of Intensity of a wave:
In a medium, the energy per unit area per unit time delivered perpendicuar to the direction of the wave propagation s caled the intensity of the wave. It is denoted by $I$.

If the energy $E$ is delivered in the time $t$ rom area $A$ perpendicular to the wave propagation, then

$I=\frac{E}{At} \qquad{1}$

Unit: $Joule/m^{2}-sec$ or $watt/m^{2}$

Dimensional formula: $[MT^{-3}]$

We know that the total mechanical energy of a vibrating particle is

$E=\frac{1}{2}m \omega^{2} a^{2}$

Where $\omega$ is the angular frequency and $a$ is the amplitude of the wave.

$E=\frac{1}{2}m (2\pi n)^{2} a^{2} \qquad \left( \omega=2\pi n \right)$

$E=2 \pi^{2} m n^{2} a^{2} \qquad(2)$

Where $m$ is the mass of the vibrating particle.

Now substitute the value of $E$ from equation $(2)$ to equation $(1)$. So the intensity of the wave

$I=\frac{2 \pi^{2} m n^{2} a^{2}}{At} \qquad(3)$

If the wave travels the distance $x$ in time $t$ with velocity $v$, Then

$t=\frac{x}{v}$

Now substitute the value of the above equation in equation $(3)$

$I=\frac{2 \pi^{2} m v n^{2} a^{2}}{Ax}$

$I=\frac{2 \pi^{2} m v n^{2} a^{2}}{V}$

Where $V$ is the volume of the corresponding medium during the wave propagation in time $t$.

$I=2 \pi^{2} \rho v n^{2} a^{2} \qquad \left(\because \rho=\frac{m}{V} \right)$

It is clear that for wave propagation in a medium with a constant velocity, i.e. wave's intensity is directly proportional to the square of amplitude and frequency both.

$I\propto a^{2}$ and $I \propto n^{2}$

Difference between sound waves and light waves

Sound Waves:

  • Sound waves are mechanical waves in nature.
  • They need a medium to propagate and so cannot be produced in a vacuum.
  • They can move in all types of mediums as solid liquid or gas whether they are transparent or not.
  • They propagate in the form of longitudinal waves.
  • The particle of the medium vibrates along the direction of the propagation and so contraction and rarefaction are formed there.
  • These are three-dimensional waves.
  • These waves do not show a polarization effect.
  • For a normal human being the audible frequency range is $20$ to $20000 Hertz$.
  • The speed of the sound waves is more in a dense medium than in a rare medium.

    • Light waves:

  • light waves are electromagnetic waves in nature.
  • They don't need any medium and so can produce and propagate in a vacuum.
  • Their velocity in a vacuum is the maximum of value $3 \times 10^{8} m/s$
  • They can move in a transparent medium only.
  • They propagate in the form of transverse waves.
  • The electric field and magnetic field vibration are perpendicular to the direction of wave propagation.
  • These wave waves are also three-dimensional waves.
  • These waves source the polarization effect.
  • For a normal human being the visible frequency range is $4 \times 10^{18} Hz$ to $8 \times 10^{14} Hz$.
  • The speed of light is more in rare mediums than in dense ones.

    • Momentum wave function for a free particle

    A non-relativistic free particle of mass $m$ moving in the positive $x$-direction with speed $v_{x}$ has kinetic energy

    $E=\frac{1}{2} m v^{2}_{x}$

    and momentum

    $p_{x}=mv_{x}$

    The energy and momentum are associated with a wave of wavelength $\lambda$ and frequency $\nu$ given by

    $\lambda = \frac{h}{p_{x}}$

    and

    $\nu=\frac{E}{h}$

    The propagation constant $k_{x}$ of the wave is

    $k_{x}= \frac{2\pi}{\lambda}=\frac{2\pi}{\left(\frac{h}{p_{x}} \right)}=\frac{p_{x}}{\left(\frac{h}{2\pi} \right)}=\frac{p_{x}}{\hbar}$

    and the angular frequency $\omega$ is

    $\omega = 2\pi \nu = \frac{2\pi E}{\hbar}=\frac{E}{\hbar}$

    A plane wave traveling along the $x$ axis in the positive direction may be represented by

    $\psi(x,t)=A e^{-i\left(k_{x} \: x - \omega t\right)}$

    Now subtitute the value of $\omega$ and $k_{x}$ in above equation then we get

    $\psi(x,t)=A e^{i\left( \frac{p_{x}}{\hbar} \: x - \frac{E}{\hbar} \: t\right)}$

    $\psi(x,t)=A e^{\frac{i}{\hbar}\left( p_{x} \: x - E \: t\right)}$

    The superposition of a number of such waves of propagation number slightly different from an average value traveling simultaneously along the same line in the positive $x$- direction forms a wave packet of small extension. By Fourier's theorem the eave packet may be expressed by

    $\psi(x,t) = \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{+\infty} A (p_{x}) e^{\frac{i}{\hbar}\left( p_{x} \: x - E \: t \right)} \: \: dp_{x} \qquad(1)$

    The function $\psi(x,t)$ is called the momentum wave function for the motion of the free particle in one dimension.

    The amplitude $A(p_{x})$ of the $x$-component of the momentum is given by the Fourier tranform

    $A(p)=\frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{+\infty} \psi (x,t) e^{-\frac{i}{\hbar}\left( p_{x} \: x - E \: t \right)} \: \: dx \qquad(2)$

    In three dimension the wave function is represented by

    $\psi(\overrightarrow{r},t) = \frac{1}{(2 \pi \hbar)^{3/2}} \int_{-\infty}^{+\infty} A (\overrightarrow{p}) e^{\frac{i}{\hbar}\left( \overrightarrow{p} . \overrightarrow{r} - E \: t \right)} \: \: d^{3}\overrightarrow{p} \qquad(3)$

    Where $d^{3}\overrightarrow{p}=dp_{x} \: dp_{y} \: dp_{z}$ is the volume element in the momentum space. In equation $(1)$, equation $(2)$ and equation $(3)$ $\frac{1}{\sqrt{2 \pi \hbar}}$ and $\frac{1}{(2 \pi \hbar)^{3/2}}$ are normalization constants.

    Schrodinger's equation for the complex conjugate waves function

    Derivation:

    The time dependent Schrodinger quation for the wave function function $psi(x,y,z,t)$ is

    $-\frac{\hbar^{2}}{2m}\nabla^{2} \psi + V\psi=i\hbar\frac{\partial \psi}{\partial t} \qquad(1)$

    Since wave function, $\psi$ is complex quantity i.e.

    $\psi=\psi_{1}+i \: \psi_{2} \qquad(2)$

    Where $\psi_{1}$ and $\psi_{2}$ are real functions of $x,y,z,t$. Substituting this form for $\psi$ in equation $(1)$, we get

    $-\frac{\hbar^{2}}{2m}\nabla^{2} \left( \psi_{1}+i \: \psi_{2} \right) + V\left( \psi_{1}+i \: \psi_{2} \right) \\ \qquad = i\hbar\frac{\partial }{\partial t} \left( \psi_{1}+i \: \psi_{2} \right)$

    Equation real and imaginary parts on either side of this equation, we obtain the following two equations:

    $-\frac{\hbar^{2}}{2m}\nabla^{2} \psi_{1} + V\psi_{1}=-\hbar\frac{\partial \psi_{2}}{\partial t} \qquad(3)$

    $-\frac{\hbar^{2}}{2m}\nabla^{2} \psi_{2} + V\psi_{2}=\hbar\frac{\partial \psi_{1}}{\partial t} \qquad(4)$

    Mutiplying equation $(4)$ by $-i$ and adding it to equation $(3)$, we get

    $-\frac{\hbar^{2}}{2m}\nabla^{2} \left( \psi_{1} - i \: \psi_{2} \right) + V\left( \psi_{1} - i \: \psi_{2} \right) \\ \qquad = -i\hbar\frac{\partial }{\partial t} \left( \psi_{1} - i \: \psi_{2} \right) \qquad(5)$

    The complex conjugate of wave function $\psi^{*}$ is

    $\psi^{*}=\psi^{*}_{1} - i \: \psi^{*}_{2} \qquad(6)$

    Therefore, The equation $(5)$ can be written as

    $-\frac{\hbar^{2}}{2m}\nabla^{2} \psi^{*} + V \psi^{*} =-i \hbar \frac{\partial \psi^{*}}{\partial t}$

    This is the equation for complex conjugate wave function $\psi^{*}$.

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