Quantum Mechanics (Revised Edition) 1
I-II
Quantum Mechanics
Re-visited
Ψ Notes Compiled from
David J. Griffiths | Richard L. Liboff | Nouredine Zettili
, Quantum Mechanics (Revised Edition) 2
Contents
1.0 What is Quantum Mechanics?
1.1 Origin of Quantum Mechanics
1.2 Contribution of scientists
1.3 What is a Wavefunction 𝚿(𝒓, 𝒕)?
1.4 What is a Operators 𝐀 ̂?
1.5 What is a Schrödinger Wave Equation?
1.6 What is Hilbert Space 𝓗?
1.7 What is Dirac Notation?
1.8 Define Postulates of Quantum Mechanics?
1.8 What is Commutation Relation?
1.9 What is The Uncertainty Principle?
1.10 What are the Application of One Dimensional SWE?
1.11 What is Angular Momentum?
1.12 What is representation of Angular Momentum in Spherical polar co-ordinates ?
1.13 Eigenvalue and Eigenfunction of 𝑳̂𝒛 Angular Momentum?
1.14 What is Raising 𝑳̂+ and Lowering 𝑳̂− operators?
1.15 Eigenvalue 𝑳𝟐 Angular Momentum?
1.16 Matrix Representation of angular momentum?
1.17 Define the Quantum Mechanics in 3-Dimensions?
1.18 What is Many particle system?
1.19 What are Identical Particles?
1.20 What is Interchange Symmetry of Identical particles?
1.21 What are Approximation Methods for the Solution of SWE?
1.22 What is Perturbation theory?
1.23 What is Time-Independent Perturbation theory?
Time-Independent Non-degenerate Perturbation theory
Time-Independent Degenerate Perturbation theory
1.24 What is Time-Dependent Perturbation theory?
1.25 What is Variational Principle?
1.26 What is WKB Approximation?
1.27 What’s so special in Special theory of Relativity?
1.28 Differentiate between Non-Relativistics & Relativistics QM?
1.29 What is Klein Garden equation?
1.30 What is Dirac Equation?
, Quantum Mechanics (Revised Edition) 3
1.0 What is Quantum Mechanics?
Quantum Mechanics is the study of the microscopic world.
Quantum Mechanics is the theory that describes the dynamics of matter at the microscopic scale.
Quantum Mechanics is the branch of physics in which the bodies whose velocity approaches the speed of light v → c.
e.g. 2 × 107 m/s is the velocity of electron approaches to speed of light 3 × 108 m/s of light.
Quantum mechanics is the founding basis of all modern physics: solid state, molecular, atomic, nuclear, and particle physics, optics,
thermodynamics, statistical mechanics, chemistry and biology
Quantized Quantity
Quantized quantity is certain minimum amounts (or some minimum constant).
Quantized quantity is Integral multiple of certain minimum amounts (or some minimum constant).
e.g. Energy formula 𝐸 = ℎ𝑓, E is quantized quantity because E is integral multiple certain minimum amounts or Planck’s
constant ℎ = 6.63 × 10−34 𝑗𝑠
Quantum (Quanta)
The certain minimum amount that is associated with a quantity is called the Quantum or Quanta of that quantity.
e.g. Light is quantized and its Quanta is called photons.
The word Quantum derives from
the Latin, meaning "how great" or
"how much"
1.1 Origin of Quantum Mechanics
At the the end of the 19th century, many scientists believed that they had completely understand all the phenomena happening
in universe.
But certain phenomenon—such as blackbody radiation, the photoelectric effect, atomic stability, and atomic
spectroscopy— created new problems which were not solved by Newtonian or Classical physics.(19th century physics)
This give birth to new physics branch called Quantum Physics.
1.2 Contribution of scientists
Max Plank’s ___ The Blackbody Radiation
In 1909 Max Plank’s introduced concept of Quantum of energy (photon) to explain -- Blackbody radiation.
He proposed that energy of radiation (light) is Quantized and dependent of frequency as 𝐸 = ℎ𝑓 where h is the plank’s
constant of value ℎ = 6.63 10 × 10−34 𝐽 𝑠
Albert Einstein ____ The Photoelectric Effect
In 1905 Einstein confirmed the Planck’s quantum concept.
Einstein proposed that electromagnetic radiation (light) is quantized and exists in fixed amounts (quanta) or photons.
The introduction of the photon concept enabled Einstein to give explanation to the photoelectric problem, by Hertz in 1887.
Niels Bohr ___ The model of Hydrogen atom
Bohr proposed that atoms can found only in discrete states of energy.
The emission or Absorption of radiation (or light) by atoms, takes place only in discrete amounts of 𝐸 = ℎ𝑓.
, Quantum Mechanics (Revised Edition) 4
Arther Holy Compton ____ Compton’s effect
By scattering X-rays with electrons, He confirmed that the (X-ray) photons behave like particles.
Loss in photon energy = gain in electron energy
de Broglie ___ Matter is a wave and wave is a matter too!
In 1923, de Broglie proposed that Matter waves.
ℎ ℎ
He showed that wave nature of matter by relation___ 𝜆 = =
𝑚𝑣 𝑝
In 1927 by Davisson and Germer proved de Broglie hypothesis.
Heisenberg___ The Uncertainty principle ( One thing at a Time!)
ℏ
In 1925 , Heisenberg proposed that it is impossible to determine the exact position and momentum at same time. Δ𝑝𝑥 . Δ𝑥 ≥
2
There is always be the Uncertainty (but multiple) of ℏ. It is open challenge and basis of Quantum Mechanics.
He say that Quantum mechanics is a completely indeterministic theory, no one know the future state of particle. E.g three
wings of rotating fans.
He formulated the Matrix Quantum Mechanics in which Eigen values are represented in matrices.
Erwin Schrödinger___ The Schrodinger Wave Equation of Particle (SWE)
In Quantum Mechanics Schrodinger Wave Equation play a same role like Newton’s Second Law: 𝐹 = 𝑚𝑎
In 1926, Schrodinger describe the dynamics of microscopic particles with a wave equation called Schrodinger Wave
Equation.
Schrodinger Wave Equation is 2nd order differential equation i.e
𝐻Ψ = 𝐸Ψ
2 2
ℏ 𝜕 Ψ 𝜕Ψ
− + 𝑉(𝑟) = 𝑖ℎ
2𝑚 𝜕 2 𝑥 𝜕𝑡
Where H is Hamilton Operator or Total Energy Operator, Ψ is the wavefunction , E is the Energy Operator, V(r) is potential and
ℎ
ℏ is modified formed of Planck’s constant .i.e ℏ = = 1.054573 × 10−34 𝐽𝑠
2
The solution of SWE is a Wavefunction Ψ(𝑟, 𝑡). A wavefunction completely describe the de-Broglie waves in space with
respect to time.
Schrödinger give the wave formulation of Quantum Mechanics
Paul Dirac___ An easy <Bra|Ket> Notation of wavefunction 𝜳 and Relativistic approaches
Dirac then suggested a more general formulation of quantum mechanics using two state vectors i.e Bras and Ket sVectors
The Kets notation of wavefunction Ψ is |Ψ > while Bra notation of Ψ is < Ψ|.
Where Kets |Ψ >= ∫ Ψ𝑑𝑥 and Bras < Ψ| = ∫ Ψ ∗ 𝑑𝑥
I-II
Quantum Mechanics
Re-visited
Ψ Notes Compiled from
David J. Griffiths | Richard L. Liboff | Nouredine Zettili
, Quantum Mechanics (Revised Edition) 2
Contents
1.0 What is Quantum Mechanics?
1.1 Origin of Quantum Mechanics
1.2 Contribution of scientists
1.3 What is a Wavefunction 𝚿(𝒓, 𝒕)?
1.4 What is a Operators 𝐀 ̂?
1.5 What is a Schrödinger Wave Equation?
1.6 What is Hilbert Space 𝓗?
1.7 What is Dirac Notation?
1.8 Define Postulates of Quantum Mechanics?
1.8 What is Commutation Relation?
1.9 What is The Uncertainty Principle?
1.10 What are the Application of One Dimensional SWE?
1.11 What is Angular Momentum?
1.12 What is representation of Angular Momentum in Spherical polar co-ordinates ?
1.13 Eigenvalue and Eigenfunction of 𝑳̂𝒛 Angular Momentum?
1.14 What is Raising 𝑳̂+ and Lowering 𝑳̂− operators?
1.15 Eigenvalue 𝑳𝟐 Angular Momentum?
1.16 Matrix Representation of angular momentum?
1.17 Define the Quantum Mechanics in 3-Dimensions?
1.18 What is Many particle system?
1.19 What are Identical Particles?
1.20 What is Interchange Symmetry of Identical particles?
1.21 What are Approximation Methods for the Solution of SWE?
1.22 What is Perturbation theory?
1.23 What is Time-Independent Perturbation theory?
Time-Independent Non-degenerate Perturbation theory
Time-Independent Degenerate Perturbation theory
1.24 What is Time-Dependent Perturbation theory?
1.25 What is Variational Principle?
1.26 What is WKB Approximation?
1.27 What’s so special in Special theory of Relativity?
1.28 Differentiate between Non-Relativistics & Relativistics QM?
1.29 What is Klein Garden equation?
1.30 What is Dirac Equation?
, Quantum Mechanics (Revised Edition) 3
1.0 What is Quantum Mechanics?
Quantum Mechanics is the study of the microscopic world.
Quantum Mechanics is the theory that describes the dynamics of matter at the microscopic scale.
Quantum Mechanics is the branch of physics in which the bodies whose velocity approaches the speed of light v → c.
e.g. 2 × 107 m/s is the velocity of electron approaches to speed of light 3 × 108 m/s of light.
Quantum mechanics is the founding basis of all modern physics: solid state, molecular, atomic, nuclear, and particle physics, optics,
thermodynamics, statistical mechanics, chemistry and biology
Quantized Quantity
Quantized quantity is certain minimum amounts (or some minimum constant).
Quantized quantity is Integral multiple of certain minimum amounts (or some minimum constant).
e.g. Energy formula 𝐸 = ℎ𝑓, E is quantized quantity because E is integral multiple certain minimum amounts or Planck’s
constant ℎ = 6.63 × 10−34 𝑗𝑠
Quantum (Quanta)
The certain minimum amount that is associated with a quantity is called the Quantum or Quanta of that quantity.
e.g. Light is quantized and its Quanta is called photons.
The word Quantum derives from
the Latin, meaning "how great" or
"how much"
1.1 Origin of Quantum Mechanics
At the the end of the 19th century, many scientists believed that they had completely understand all the phenomena happening
in universe.
But certain phenomenon—such as blackbody radiation, the photoelectric effect, atomic stability, and atomic
spectroscopy— created new problems which were not solved by Newtonian or Classical physics.(19th century physics)
This give birth to new physics branch called Quantum Physics.
1.2 Contribution of scientists
Max Plank’s ___ The Blackbody Radiation
In 1909 Max Plank’s introduced concept of Quantum of energy (photon) to explain -- Blackbody radiation.
He proposed that energy of radiation (light) is Quantized and dependent of frequency as 𝐸 = ℎ𝑓 where h is the plank’s
constant of value ℎ = 6.63 10 × 10−34 𝐽 𝑠
Albert Einstein ____ The Photoelectric Effect
In 1905 Einstein confirmed the Planck’s quantum concept.
Einstein proposed that electromagnetic radiation (light) is quantized and exists in fixed amounts (quanta) or photons.
The introduction of the photon concept enabled Einstein to give explanation to the photoelectric problem, by Hertz in 1887.
Niels Bohr ___ The model of Hydrogen atom
Bohr proposed that atoms can found only in discrete states of energy.
The emission or Absorption of radiation (or light) by atoms, takes place only in discrete amounts of 𝐸 = ℎ𝑓.
, Quantum Mechanics (Revised Edition) 4
Arther Holy Compton ____ Compton’s effect
By scattering X-rays with electrons, He confirmed that the (X-ray) photons behave like particles.
Loss in photon energy = gain in electron energy
de Broglie ___ Matter is a wave and wave is a matter too!
In 1923, de Broglie proposed that Matter waves.
ℎ ℎ
He showed that wave nature of matter by relation___ 𝜆 = =
𝑚𝑣 𝑝
In 1927 by Davisson and Germer proved de Broglie hypothesis.
Heisenberg___ The Uncertainty principle ( One thing at a Time!)
ℏ
In 1925 , Heisenberg proposed that it is impossible to determine the exact position and momentum at same time. Δ𝑝𝑥 . Δ𝑥 ≥
2
There is always be the Uncertainty (but multiple) of ℏ. It is open challenge and basis of Quantum Mechanics.
He say that Quantum mechanics is a completely indeterministic theory, no one know the future state of particle. E.g three
wings of rotating fans.
He formulated the Matrix Quantum Mechanics in which Eigen values are represented in matrices.
Erwin Schrödinger___ The Schrodinger Wave Equation of Particle (SWE)
In Quantum Mechanics Schrodinger Wave Equation play a same role like Newton’s Second Law: 𝐹 = 𝑚𝑎
In 1926, Schrodinger describe the dynamics of microscopic particles with a wave equation called Schrodinger Wave
Equation.
Schrodinger Wave Equation is 2nd order differential equation i.e
𝐻Ψ = 𝐸Ψ
2 2
ℏ 𝜕 Ψ 𝜕Ψ
− + 𝑉(𝑟) = 𝑖ℎ
2𝑚 𝜕 2 𝑥 𝜕𝑡
Where H is Hamilton Operator or Total Energy Operator, Ψ is the wavefunction , E is the Energy Operator, V(r) is potential and
ℎ
ℏ is modified formed of Planck’s constant .i.e ℏ = = 1.054573 × 10−34 𝐽𝑠
2
The solution of SWE is a Wavefunction Ψ(𝑟, 𝑡). A wavefunction completely describe the de-Broglie waves in space with
respect to time.
Schrödinger give the wave formulation of Quantum Mechanics
Paul Dirac___ An easy <Bra|Ket> Notation of wavefunction 𝜳 and Relativistic approaches
Dirac then suggested a more general formulation of quantum mechanics using two state vectors i.e Bras and Ket sVectors
The Kets notation of wavefunction Ψ is |Ψ > while Bra notation of Ψ is < Ψ|.
Where Kets |Ψ >= ∫ Ψ𝑑𝑥 and Bras < Ψ| = ∫ Ψ ∗ 𝑑𝑥
