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Quantum Mechanics

The quantum theory is the theoretical description of modern physics. It involves the study of the nature and behavior of matter and energy on the atomic and subatomic level.

Quantum mechanics is also called quantum physics or quantum theory.

 

What is Quantum Mechanics?

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Quantum mechanics is the branch of physics which involves the study of discrete and indivisible units of energy. These indivisible units are known as quanta. Thus it shows that some physical quantities can alter in some discrete units like angular momentum, energy etc.

The dual behavior and the uncertainty principle describe the behavior of atomic scale objects like photon and electrons etc. The basic principle of Quantum mechanics is that particles like electrons have dual nature of particles and waves. The particles are randomly moved. It is not possible to know the position and the momentum of a particle at a time with accuracy. The atomic world is like the world where we live.

So Quantum mechanics is a kind of physical phenomena which is in the range of plank constant. Quantum mechanics is different from classical mechanics. It includes the mathematical evaluation of dual behavior of particles and the energy and matter interaction.

Advanced quantum mechanics involves the study of macroscopic behavior which emerges at a specific energy and temperature.

History of Quantum Mechanics

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  1. The concept of black body radiation was given by Gustav Kirchhoff in 1859. A black body is a perfect emitter and absorber. He proved that the emitted energy E of a black body is the function of temperature T and the frequency v of the emitted energy. So E = j (T,v) but it was not possible for him to find the function j.
  2. For this, Ludwig Boltzmann used the theoretical considerations of thermodynamics and Maxwell's electromagnetic theory but it was not successful. It does not specify the specific wavelengths. Then a formula was proposed by Planck in 1900 for Kirchhoff’s function.
  3. This formula was proved by all the experimental observation for wavelengths. For giving the theoretical derivation of the formula, he assumed that the total energy is made by quanta of energy which are indistinguishable energy elements. This also involved the Boltzmann’s statistical method.
  4. Ricci and Levi-Civita gave the general formulation of a tensor in 1901 but that is not proved by the quantum theory. After the photoelectric effect of Einstein, he gave the quantum theory of light and used some of the concepts of Planck's theory which was proved useful for the light quantum hypothesis.
  5. In 1906 Einstein showed that the change in energy in quantum material oscillator is multiples of v where is Planck's constant and v is the frequency.
  6. In 1913 Niels Bohr discovered the important laws of the spectral lines. Arthur Compton gave the derivation for the relativistic kinematics for the scattering of a photon or quantum of light of an electron.

In 1924 Bohr and other scientist published theoretical proposals relating to the interaction between light which eject the photon and matter. Bohr made some conclusions which determined

  • How the conservation of energy is possible when some energy changes takes place in quantum amounts.
  • How it is possible for an electron to emit radiation.

At that time, quantum theory was set up with Cartesian tensors of linear and angular momentum. After that in 1924, Bose proposed various stages of the photon. According to him, photons are not conserved. The Louis de Broglie’s thesis was proposed the particle-wave duality behavior.

Then in 1926, Schrodinger published his equation for the hydrogen atom. This gave the introduction of wave mechanics. He also introduced operators of each dynamical variable. In the year 1926, the Planck's law was completely solved by Dirac. In 1925 Heisenberg published for quantum mechanics and gave the uncertainty principle. The Heisenberg equation is $\Delta$ p.

$\Delta$ x = h/4p

where $\Delta$ x is the uncertainty of the position and ?p is the uncertainty of the momentum and h is Planck's constant. Heisenberg used matrix methods. Thus the 'rival' matrix mechanics of Heisenberg and wave mechanics of Schrodinger was proved a good work for quantum mechanics.

In 1927 Bohr proposed that space-time coordinates and causality is complementary to each other. The mathematics of new kind of tensor was given by Eli Cartan. He introduced a spinor.

In 1928, the expression of quantum theory was given by Dirac. He expressed wave equation in operator algebra terms. In 1932 von Neumann finally put quantum theory on theoretical basis in the setting of operator algebra.


Quantum Mechanics Equations

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  • The mathematical function of quantum mechanics is known wave function.
  • The probability amplitude of various properties of particles like position and momentum etc is measured by wave function.
  • Mathematical formulations of the wave function involve the bracket notation.
  • It is necessary to know about the complex numbers and linear functions for explaining the bracket notation.

The wave equation is given by schrodinger which is as below.

$ih\frac{\partial }{\partial t}\psi (r,t)$ = $-\frac{h^2}{2m}\triangledown^2\psi (r,t) + V(r,t)\psi (r,t)$

Where,

i is imaginary number
h is plancks constant divided by 2p
$\Psi$ (r,t) is the wave function, defined over space and time
m is the mass of the particle
$\triangledown^2$ is the Laplacian operator
V(r,t) is the potential energy influencing the particle


For the solution of the Schrodinger equation for the hydrogen atom, the wave function for hydrogen atom by separating the variable is

Schrodinger Equation

It gives the three equations for the three spatial variables and hence, three quantum numbers of the hydrogen energy levels. The spherical polar coordinates are used to develop the Schrodinger equation for hydrogen atom as electrons are in spherically symmetric potential. So the potential energy is

U(r) = $-\frac{e^2}{4\pi \varepsilon _or}$

After solving the equation by separating the variables,
$\Psi (r,\Theta, \Phi )=R(r)P(\Theta )F(\Phi )$

$\frac{ħ^{2}}{2\mu }\frac{1}{r^{2}sin\Theta }\left [ sin\Theta \frac{\partial \Psi }{\partial r} \left ( r^{2}\frac{\partial \Psi }{\partial r} \right )+\frac{\partial }{\partial \Theta }\left ( sin\Theta \frac{\partial \Psi }{\partial \Theta } \right ) +\frac{1}{sin\Theta }\frac{\partial ^{2}\Psi }{\partial \Phi ^{2}}\right ] $

$+ U(r)\Psi (r,\Theta ,\Phi ) = E\Psi (r,\Theta ,\Phi )$

So after separating the variable, the three quantum numbers are

Principle Quantum Number
Principle quantum number

Orbital Quantum Number
Orbital quantum number

Magnetic Quantum Number
Magnetic quantum number

After partial differentiation of the equation, the equation is radial equation which is;

$\frac{1}{R}\frac{d}{dr}\left [ r^{2}\frac{dR}{dr} \right ]+\frac{2\mu }{h^{2}}\left ( Er^{2}+ke^{2}r \right )=l(l+1)$

So the colatitude equation is;

$\frac{sin\Theta }{P}\frac{d}{d\Theta }\left [ sin\Theta \frac{dP}{d\Theta } \right ]+C_{r}sin^{2}\Theta = -C_{\Phi }$

And the azimuthal equation is
$\frac{1}{F}\frac{d^{2}F}{d\Phi ^{2}}=C_{\Phi }$

Postulates of Quantum Mechanics


The main postulates of quantum mechanics are given as below.
  1. The wave function which is related to moving particle in a conservative field of force is a kind of wave function by which, everything about the system can be known.
  2. An operator Q is always associated with every physical observable q. When the operator Q is operated with the wave function, the definite value of Q will yield the value times for the wave function.
  3. An operator Q will be Hermitian which associates with every physical observable q.
  4. The Eigen functions set of operator Q gives a complete set of functions which are linearly independent.
  5. The expected value of property q can be evaluated by measuring the integral expectation value with respect to that wave function for which the property q is measured.
  6. The wave function’s time evolution is measured by the Schrodinger equation which should be time dependent.

The Wave Function Postulates

  1. This postulate of quantum mechanics is particular for a physical system containing a particle. For this there is a wave function associated.
  2. This wave function gives all the information about the system. The wave function is based on the assumption that it’s single-valued function of position and time.
  3. The wave function can be a complex function because it produces the product with its complex conjugate. This complex conjugate specifies the probability of finding the particle in a specific state.
  4. Single valued probabilty amplitude at (x, t) = $\Psi$ (x, t) and $\Psi$ (x, t) $\Psi$ ’(x, t) = Probabilty of finding the particle at x at time t provided the normalized wave function.

Wave Function Constraints


For a physically observable system, the wave function should satisfy some constraints.
  1. The wave function should be solution of the Schrodinger equation.
  2. The wave function should be normalized. This means that wave function approaches to zero as x approaches to infinity.
  3. The wave function should be continuous function of x.
  4. The slope of the function should also be in continuous form. Like $\frac{\partial \psi (x)}{\partial x}$ the function should be continuous.

All the above constraints are applicable to the boundary conditions of the solutions and the method help to determine the Eigen values of energy.

Probability in Quantum Mechanics

The wave function shows the probability amplitude. The probability amplitude is used for finding a particle at a particular point in space at a particular time. Thus the probability of finding the particle is determined by the multiplication of the wave function and it is complex conjugate.

$\Psi (x,y,z,t)=$ probability amplitude    $\Psi \ast \Psi =$ probability

The probability of finding the particle and sum of the probabilities should be equal to one. The wave function must be normalized which is shown by the integral. The value of any normalized physical observable can be determined by taking it normalized so that the probability integrated is equal to one over space.

$\int \Psi \ast \Psi  dr = 1$

Relativistic Quantum Mechanics

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This is also known as theory of quantum interactions. It involves the evaluation of the theory of special relativity. It incorporates the theory of general relativity into quantum mechanics. The non-relativistic quantum mechanics is explained by Schrodinger Equation.

$H(t)\mid \psi (t) = ih\frac{d}{dt}\mid \psi (t)$

  • Now we would like to extend quantum mechanics to the relativistic domain. For a relativistic single-particle wave equation, the relativistic equation depends on the spin of the particle.
  • Like Klein-Gordon equation for spin-0, Dirac equation for spin-1/2, and proca equation for spin-1 etc. these are useful for study the one particle equation.
  • But the one-particle relativistic quantum theory is up to a limit only. Sometimes it fails because only energy is conserved in special relativity, not mass.
  • So the many-particle theories where particles can be created and destroyed are evaluated by single-particle relativistic wave equations.
  • Special relativity is an event which is defined by its coordinates which takes place at a single point in space-time.
  • The coordinate is $x^{\mu }$ where $\mu$ = 0, 1, 2, 3 …The interval between two events $x^{\mu }$ and $x^{\mu- }$ is equal to s. So by using metrix method, $s^{2}=g_{\nu \mu }$$\Delta x_{\mu }$$\Delta x_{\nu }$


Lorentz Transformation


The s2 is invariant under transformations from one frame to other. These transformations are known as Lorentz transformations. Generally the homogeneous Lorentz transformations are used in which the origin is not shifted. The relativistic Schrodinger Equation is known as klein-gordon equation for zero spin is

$\triangle - \frac{1}{C^2}\frac{\partial }{\partial t^2}$ = $\binom{m_o.C}{h}^2$


Dirac equation for spin-1/2

$\begin{pmatrix}
ħ\frac{\partial }{\partial x_{4}}& -iħ\vec{\sigma }.\vec{\triangledown } \\ iħ\vec{\sigma }.\vec{\triangledown }
& -ħ\frac{\partial }{\partial x_{4}}
\end{pmatrix}$
$\binom{\psi _{A}}{\psi _{B}}+mc\binom{\psi _{A}}{\psi _{B}}=0$

$\psi =\binom{\psi ^{A}}{\psi _{B}}\equiv \begin{pmatrix}
\psi _{1}\\\psi _{2}
\\ \psi _{3}
\\ \psi _{4}

\end{pmatrix}$


$ħ\gamma _{\mu }\frac{\partial }{\partial x_{\mu }}\psi +mc\psi =0$

Proca wave equation for spin-1



$div \vec{g}=\frac{\rho _{m}}{\varepsilon _{g}}\left ( \frac{m_{Gravitation}c}{ħ} \right )^{2}.\phi _{g}$
$div\vec{B_{g}}=0$
$rot \vec{g}=-\frac{\partial \vec{B_{g}}}{\partial t}$
$rot \vec{B_{g}}=\mu _{o}\rho _{m}\bar{\nu }+\frac{1}{c^{2}}\frac{\partial \vec{g}}{\partial t}-\left ( \frac{m_{Gravitation}c}{ħ} \right )^{2}.\vec{A_{g}}$



So the general proca wave equation is

$\triangle \psi -\frac{1}{C^2}\frac{\partial ^2\psi }{\partial t}-\left ( \frac{m_o.C}{h^2} \right )\psi$ = f
More topics in Quantum Mechanics
Max Planck Quantum Theory
Quantum Gravity Centistokes
Centipoise Harmonic Oscillator
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