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Today in History – 7 July

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Syskool
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today in history 7 july

today in history 7 july

1763

Britishers declared Mir Jafer as Nawab of Bengal.

1797

For the first time in U.S. history, the House of Representatives exercised its constitutional power of impeachment and voted to charge Senator William Blount of Tennessee with “a high misdemeanor, entirely inconsistent with his public duty and trust as a Senator.”

1799

Ranjit Singh captured Lahore in Punjab from Sikh rulers.

1852

According to Sir Arthur Conan Doyle’s stories, Dr. John H. Watson was born on this day. Coincidentally, the author died on this day in England at the age of 71.

1855

30,000, people went to Calcutta along with weapons to fight the British army.

1865

Mary Surratt was the first women who was executed by the U.S. government for her role as a conspirator in Abraham Lincoln’s assassination.

1901

J. Watsan Harod started a branch of Gramophone Company at Calcutta.

1905

Six Language penel were started on the Rs. five denomination again. The printing colour of note changed to Red with Special text mentioned, ‘At any office of issue not situated in Burma’.

1910

‘Bharat History Research Mandal’ was established.

1917

On this day in 1917, British Army Council Instruction Number 1069 formally established the British Women’s Auxiliary Army Corps (WAAC), authorizing female volunteers to serve alongside their male counterparts in France during World War I.

1930

On this day in 1930, construction of the Hoover Dam began. Over the next five years, a total of 21,000 men worked ceaselessly to produce what would be the largest dam of its time, as well as one of the largest manmade structures in the world.

1937

Jagdish Prasad Goenka, great industrialist, was born in Calcutta.

1943

Rasbehari Bose handed over the command of ‘Azad Hind Fauz’ to Netaji Subhashchandra Bose at Singapore.

1946

Gandhiji addressed the A.I.C.C. meeting at Bombay; Congress accepted the Cabinet Mission plan of May 16.

1948

Damodar Valley Corporation, first public corporation, was established.

1976

For the first time in history, women were enrolled into the United States Military Academy at West Point, New York. On May 28, 1980, 62 of these female cadets graduated and were commissioned as second lieutenants.

1999

The Indian Army recaptured Jubar height in the Batalik sector.

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Today in History – 4 July

Today in History – 2 July

Postulates of Quantum Mechanics: Foundations of the Quantum World

By
Kumar Prafull
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0
postulates quantum mechanics

Table of Contents

  1. Introduction
  2. Why Postulates?
  3. Postulate 1: State Vector and Hilbert Space
  4. Postulate 2: Observables and Operators
  5. Postulate 3: Measurement and Expectation Values
  6. Postulate 4: Time Evolution and Schrödinger Equation
  7. Postulate 5: Collapse of the Wavefunction
  8. Postulate 6: Identical Particles and Symmetrization
  9. Mathematical Framework and Examples
  10. Compatibility and Commutation Relations
  11. Uncertainty Principle from Postulates
  12. Relation to Classical Mechanics
  13. Role in Quantum Computing and Information
  14. Interpretational Challenges
  15. Conclusion

1. Introduction

Quantum mechanics is based on a set of fundamental postulates. These postulates form the axiomatic foundation from which all quantum phenomena emerge. Understanding these is essential for mastering both basic and advanced quantum theory.

2. Why Postulates?

  • They define the rules of the quantum game
  • Not derived — instead, they’re principles inferred from experiment
  • Allow for a consistent mathematical structure and predictive power
  • Serve as the basis for further developments in quantum field theory, quantum computation, and condensed matter physics

3. Postulate 1: State Vector and Hilbert Space

Every isolated physical system is described by a state vector \( |\psi\rangle \), which is an element of a Hilbert space \( \mathcal{H} \).

  • The state encodes all measurable information about the system
  • For composite systems, use tensor product of subsystems
  • The state is normalized:
    \[
    \langle \psi | \psi \rangle = 1
    \]

Examples:

  • Position basis: \( \psi(x) = \langle x | \psi \rangle \)
  • Spin basis: \( |\psi\rangle = \alpha | \uparrow \rangle + \beta | \downarrow \rangle \)

4. Postulate 2: Observables and Operators

Every observable (measurable quantity) corresponds to a Hermitian operator \( \hat{A} \) acting on the Hilbert space.

  • Measurement outcomes are eigenvalues of \( \hat{A} \):
    \[
    \hat{A} |a\rangle = a |a\rangle
    \]
  • Hermitian operators have:
  • Real eigenvalues
  • Orthonormal eigenvectors
  • Complete basis set

Examples:

  • Position: \( \hat{x} \), Momentum: \( \hat{p} = -i\hbar \frac{d}{dx} \)

5. Postulate 3: Measurement and Expectation Values

The probability of obtaining result \( a \) from measuring \( \hat{A} \) in state \( |\psi\rangle \) is:

\[
P(a) = |\langle a | \psi \rangle|^2
\]

The expectation value of \( \hat{A} \) is:

\[
\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle
\]

Measurement alters the state — this is fundamentally different from classical physics.

6. Postulate 4: Time Evolution and Schrödinger Equation

Time evolution of a quantum state is deterministic and governed by the Schrödinger equation:

\[
i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle
\]

Where:

  • \( \hat{H} \) is the Hamiltonian (total energy operator)
  • Solution:
    \[
    |\psi(t)\rangle = e^{-i\hat{H}t/\hbar} |\psi(0)\rangle
    \]

7. Postulate 5: Collapse of the Wavefunction

Upon measurement, the system collapses into one of the eigenstates of the measured observable.

  • If \( a \) is observed, then immediately after:
    \[
    |\psi\rangle \rightarrow |a\rangle
    \]
  • Collapse is instantaneous and probabilistic

This postulate introduces non-unitarity and is at the heart of interpretational debates.

8. Postulate 6: Identical Particles and Symmetrization

Particles are either:

  • Bosons: symmetric under exchange
  • Fermions: antisymmetric under exchange

Wavefunctions for identical particles must satisfy:

\[
\psi(x_1, x_2) = \pm \psi(x_2, x_1)
\]

  • Fermions obey Pauli exclusion principle
  • Bosons can occupy the same state (Bose–Einstein condensation)

9. Mathematical Framework and Examples

Example: Quantum harmonic oscillator

  • Hamiltonian: \( \hat{H} = \frac{1}{2m} \hat{p}^2 + \frac{1}{2} m \omega^2 \hat{x}^2 \)
  • Energy levels:
    \[
    E_n = \hbar \omega \left(n + \frac{1}{2}\right)
    \]

Example: Spin-1/2 system

  • Basis: \( |\uparrow\rangle, |\downarrow\rangle \)
  • Pauli matrices: \( \hat{\sigma}_x, \hat{\sigma}_y, \hat{\sigma}_z \)

10. Compatibility and Commutation Relations

Two observables \( \hat{A} \) and \( \hat{B} \) are compatible if they commute:

\[
[\hat{A}, \hat{B}] = 0
\]

  • Compatible observables can be simultaneously measured
  • Incompatible observables obey uncertainty relations

11. Uncertainty Principle from Postulates

From the non-commutativity of \( \hat{x} \) and \( \hat{p} \):

\[
[\hat{x}, \hat{p}] = i\hbar
\]

This leads to the Heisenberg uncertainty principle:

\[
\Delta x \Delta p \ge \frac{\hbar}{2}
\]

Limits precision of simultaneous measurements.

12. Relation to Classical Mechanics

Quantum mechanics recovers classical physics in the correspondence limit:

  • Large quantum numbers \( n \rightarrow \infty \)
  • \(\hbar \rightarrow 0\)
  • Expectation values evolve classically (Ehrenfest’s theorem)

13. Role in Quantum Computing and Information

  • Qubits are represented by state vectors in a two-dimensional Hilbert space
  • Quantum gates are unitary operators
  • Measurement collapses qubits probabilistically
  • Postulates underpin quantum algorithms, entanglement, and teleportation

14. Interpretational Challenges

  • What triggers wavefunction collapse?
  • Is measurement observer-dependent?
  • Realism vs probabilism vs many-worlds
  • Are the postulates fundamental or emergent from deeper theory?

Debates continue in the foundations of quantum theory.

15. Conclusion

The postulates of quantum mechanics provide a rigorous framework that has successfully explained phenomena from atomic spectra to quantum entanglement. While complete in predictive power, they challenge our notions of reality, determinism, and causality — marking a profound departure from classical intuition. Mastery of these principles is the gateway to understanding the quantum world.

.

Today in History – 6 July

By
Syskool
-
0
today in history 6 july

today in history 6 july

1787

Robert founded Indian Botanic Garden near Calcutta on 273 acres of land.

1837

Ramkrishna Gopal Bhandarkar, a reputed oriental scholar and famous Indian Scientist, was born in Malvan in Ratnagiri district. He was a social reformer and an active member of Prathana Samaj of Bombay.

1881

Saint Gulabrao Maharaj was born in Vidarbha.

1901

Dr. Shyama Prasad Mukherjee, great social reformer, politician and leader, was born at Calcutta. He was the founder of the “Bharatiya Janta Sangh”, of which BJP is the successor party.

1917

Bhandarkar Prachyavidya Sanshodhak Mandir’ was established.

1927

Prabhjot Kaur, great poet and author, was born at Langrayal (Pakistan).

1935

Dalai Lama, the 14th incarnate and religious leader of Tibet, was born. He was born to Tenzin Gyotso (His Holiness the VIVth Dalai Lama) in Taksten Village, Chhija Nangso, Tibet. He was awarded with Nobel Prize for Peace in 1989.

1942

Urmila Devi Shastri, freedom fighter and editor of “Janma Bhoomi”, died due to non-medical facility and mal-nutrition in Jail. She has also written “Karagaar Ke Anubhav” in 1930. She was arrested for Satyagrah and forcefully closing of Foreign Goods.

1944

Mahatama Gandhi for the first time was addressed as ‘Father of Nation‘.

1955

Marshall Tito and Nehru‘s joint declaration in Belgrade.

1957

On this day in 1957, Althea Gibson claimed the women’s singles tennis title at Wimbledon and became the first African American to win a championship at London’s All England Lawn Tennis and Croquet Club.

1960

The President promulgates the ESMA, 1960.

1976

In Annapolis, Maryland, the United States Naval Academy admited women for the first time in its history with the induction of 81 female midshipmen. In May 1980, Elizabeth Anne Rowe became the first woman member of the class to graduate. Four years later, Kristine Holderied became the first female midshipman to graduate at the top of her class.

1986

Babu Jagjivan Ram, valiant freedom fighter, passed away. He held many prominent ministerial portfolios such as Communication Minister (1952-56), Railway Minister (1956-62), Food and Agriculture Minister (1967-70), Defence Minister (1970-74). After 1977, he became the Deputy Prime Minister (Defence) in Janta regime. Thereafter, he established Congress (J). His uninterrupted representation in the Parliament from 1936 to 1986 is a world record.(5-4-91).

1992

The centre decided to allow TV and Radio time slots for private producers.

2000

India, Nepal signed agreement to set up a specialist joint unit to counter terrorism and share information about criminals operating along the border.

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Today in History – 5 July

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Today in History – 1 July

Bohr Model of Hydrogen: Quantizing the Atom

By
Kumar Prafull
-
0
Financial Inclusion
Financial Inclusion

Table of Contents

  1. Introduction
  2. Historical Context and the Failure of Classical Models
  3. Basic Postulates of Bohr’s Model
  4. Quantization of Angular Momentum
  5. Energy Levels of the Hydrogen Atom
  6. Spectral Lines and the Rydberg Formula
  7. Radius of Bohr Orbits
  8. Velocity and Time Period of Electron
  9. Limitations of the Bohr Model
  10. Successes of the Bohr Model
  11. Bohr–Sommerfeld Extension
  12. Comparison with Quantum Mechanics
  13. Experimental Verification
  14. Legacy and Influence on Modern Theory
  15. Conclusion

1. Introduction

The Bohr model of hydrogen was a revolutionary step in atomic theory. Introduced by Niels Bohr in 1913, it merged classical mechanics with new quantum ideas to explain the discrete energy levels and spectral lines observed in hydrogen. Though ultimately superseded by quantum mechanics, the Bohr model remains an essential pedagogical tool.

2. Historical Context and the Failure of Classical Models

Classical electromagnetism predicted that electrons orbiting a nucleus would radiate energy continuously and spiral into the nucleus. However, atoms were observed to be stable, and hydrogen emission spectra showed discrete lines — not a continuous range.

These contradictions called for a new model.

3. Basic Postulates of Bohr’s Model

  1. Electrons orbit the nucleus in fixed circular paths without radiating energy.
  2. Only orbits with quantized angular momentum are allowed.
  3. Radiation is emitted or absorbed when an electron jumps between energy levels.

4. Quantization of Angular Momentum

Bohr postulated that angular momentum is quantized:

\[
L = n\hbar = n\frac{h}{2\pi}, \quad n = 1, 2, 3, \dots
\]

Where:

  • \( h \) is Planck’s constant
  • \( n \) is the principal quantum number

This condition selects discrete orbits.

5. Energy Levels of the Hydrogen Atom

From Coulomb’s law and centripetal force:

\[
\frac{ke^2}{r^2} = \frac{mv^2}{r}
\]

Combine with quantized angular momentum:

\[
E_n = -\frac{13.6\ \text{eV}}{n^2}
\]

Where:

  • \( E_n \) is energy of the \( n^\text{th} \) level
  • \( 13.6 \ \text{eV} \) is the ground state energy

6. Spectral Lines and the Rydberg Formula

Transition between levels emits a photon:

\[
h\nu = E_{n_i} – E_{n_f}
\]

The corresponding wavelength is given by the Rydberg formula:

\[
\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} – \frac{1}{n_i^2} \right)
\]

Where:

  • \( R_H \approx 1.097 \times 10^7 \ \text{m}^{-1} \) is the Rydberg constant

Explains Lyman, Balmer, and Paschen series.

7. Radius of Bohr Orbits

From angular momentum and centripetal force:

\[
r_n = \frac{n^2 \hbar^2}{k e^2 m_e} = n^2 a_0
\]

Where \( a_0 = 0.529 \ \text{Å} \) is the Bohr radius (radius of ground state orbit).

8. Velocity and Time Period of Electron

Velocity in the \( n^\text{th} \) orbit:

\[
v_n = \frac{ke^2}{\hbar} \cdot \frac{1}{n}
\]

Time period of revolution:

\[
T_n = \frac{2\pi r_n}{v_n}
\]

Reinforces quantization of motion.

9. Limitations of the Bohr Model

  • Applies only to hydrogen-like atoms (one electron)
  • Cannot explain fine structure, Zeeman effect, or electron spin
  • Circular orbit assumption conflicts with quantum principles
  • Violates Heisenberg’s uncertainty principle

10. Successes of the Bohr Model

  • Explained hydrogen spectrum accurately
  • Predicted quantized energy levels
  • Introduced key concepts of quantum theory
  • Inspired future development of wave mechanics

11. Bohr–Sommerfeld Extension

  • Introduced elliptical orbits
  • Included quantization of radial and angular motions
  • Provided improved spectral accuracy

Still semi-classical and later replaced by Schrödinger’s theory.

12. Comparison with Quantum Mechanics

  • Bohr: electrons in fixed orbits
  • Schrödinger: electrons as wavefunctions in orbitals
  • Bohr’s energy levels match the quantum mechanical hydrogen atom, but wave mechanics provides deeper and more general framework

13. Experimental Verification

  • Hydrogen spectral lines matched precisely
  • Balmer series: visible spectrum lines
  • Lyman and Paschen series: UV and IR, respectively
  • Rydberg constant confirmed via spectroscopic measurements

14. Legacy and Influence on Modern Theory

Bohr’s model:

  • Bridged classical and quantum physics
  • Influenced the Copenhagen interpretation
  • Demonstrated necessity of quantization
  • Provided stepping stone toward full quantum theory

15. Conclusion

The Bohr model of hydrogen was a revolutionary yet transitional model in atomic physics. While it fell short of explaining complex atoms, it captured the essence of quantization and paved the way for modern quantum mechanics. Its elegance and predictive power continue to make it a cornerstone of physics education.

Today in History – 5 July

By
Syskool
-
0
today in history 5 july

today in history 5 july

1658

Aurangzeb arrested Murad.

18574

Baigum Hazrat Mahal again attacked on Lucknow.

1882

Hazrat Inayat Khan (1882-1927), classical singer and Muslim mystic, was born. He was instrumental in bringing Sufism to the music.

1901

B. N. Sircar, producer and founder of International Film Craft and New Theaters Ltd., Calcutta was born at Bhagalpur. His films are Boker Bojha, Chorekanta, Chasar Meye (1931).

1905

Lord Curzon joined the part of Bengal to Assam and divided Bengal.

1916

Indian Women’s University was established at Hingne near Poona by Dr. Dhondo Keshav Karve with just four students. After 1920, it was renamed as Shreemati Nathibai Damodar Thackersey Indian Women’s University.

1916

Archik Venkatesh Gopalkrishna, great poet, journalist and orator, was born at Dharwar, Mysore.

1917

`Hindumashinary Society’ was established by Gajanan Bhaskar Vaidya.

1947

Indian Act, 1947 was presented in British Parliament, which was then accepted by the emperor on 18th July.

1957

Dr. Anugrah Narain Singh, great leader, freedom fighter, educationist and Minister of India, died..

1968

Naval Submarine wing in Indian Naval Force came into existence with the arrival of the first submarine from the Soviet Union.

1981

Rajan Mahadevan recites 31,811 digits of ¦(pie) from memory.

1990

G. C. Saxena, J&K Governor, promulgated J&K Disturbed Areas Act, 1990, declaring several districts as disturbed area.

1992

Centre scraps octroi in Union Territories.

1993

Supreme Court dismisses the stay petition of Union Government regarding lifting of ban on the RSS and the Bajrang Dal by the Babri Tribunal.

1995

Tarapur nuclear station controlled the leakage and soil contamination within the natural background radiation levels.

2000

Rachita Mistry breaks P.T. Usha’s long-standing record in 100m (11.39s) clocking 11.26s in the National circuit meet in Bangalore.

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Today in History – 4 July

Today in History – 2 July

Today in History – 1 July

Today in History – 30 June

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