Quantum Mechanics

                                                     Quantum Mechanics

[1] What is quantum mechanics?

- Quantum mechanics is a fundamental theory in physics
   that provides a description of the physical properties of
   nature at the scale of atoms and subatomic particles.

[2] Who are the founder of quantum mechanics?

- Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie,
   Max Born, Paul Dirac, Werner Heisenberg, Wolfgang
   Pauli, Erwin Schrödinger, Richard Feynman

[3] History / Timeline of quantum mechanics –

1. 1900- Max Planck attempted a simple optimization question, “How
    much do I have to heat a lightbulb so that it gives off the maximum
    amount of light?” While he obtained the required formula, he had
    no explanation for it. He also discovered the phenomena of black
    body radiation.

2. 1905- Albert Einstein used the above results to explain the
    photoelectric effect. This helped in postulating that light consisted
    of quantized (discrete) bundles of energy called photons.

3. 1913- Niels Bohr designed the Bohr model of the atom using
    classical physics where electrostatic forces held the nucleus and
    electrons together. This fit with Planck’s relation for photons
    emitted from the hydrogen atom given by- E2 — E1 = hf.

4. 1926- Erwin Schrödinger develops wave mechanics and develops
    the famous Schrödinger Wave equation

5. 1927- Heisenberg’s Uncertainty Principle is developed which states
    that “It is impossible to simultaneously determine the position and
    momentum of a particle”

6. 1935- The famous thought experiment of Schrodinger’s Cat is
    developed

7. 1948- Richard Feynman develops the first reconciliation of photons
    and electron through quantum electrodynamics.

1. Black Body Radiation

[1] What Is Black Body Radiation ?

       - A Black Body Is An Ideal Body Which Allows The
         Whole Of The Incident Radiation To Pass Into Itself
         (Without Reflecting The Energy) And Absorbs Within

         Itself This Whole Incident Radiation (Without Passing
         On The Energy)

 Formula

2. photoelectric effect

[1] what is photoelectric effect ?

    - Photoelectric effect is the process of emitting
      theelectrons from the a metal surface when themetal
      rface is exposed to an electromagneticradiation of
      sufficiently high frequency. Forexample, example,
      ultraviolet ultraviolet light is required required in the
      caseofejection of electrons from an alkali metal.

 Laws of Photoelectric Emission :

- There is no time lag between the irradiation ofthe surface
  and the ejection of the electrons.

- At a particular fixed frequency of incidentradiation the rate of
  the emission of photoelectrons i.e. the photocurrent
  increases withincrease increase in the intensity intensity of
  the incident incident light.

- Photo electric effect does not occur at frequencyless than
  threshold frequency.

- At the frequency above the threshold frequency, the kinetic
  energy of the ejected electronsdepends only on the
  frequency of the exposedradiation and not on its intensity.

 Explanation of Photoelectric Effect :

- The photoelectric effect cannot be explained on thebasis of
  electromagnetic theory.

- In 1905 Einstein proposed that the photoelectric effectcould
  be understood through the idea proposed by theGerman
  theoretical physicist Max Planck in 2000.

- Planck was seeking to explain the characteristics of
  theradiation emitted by hot bodies. Explanation of
  Photoelectric Effect radiation emitted by hot bodies.

- Plank assumed that while the radiation is emittedcontinuously
  as little bursts of energy called quanta butpropagated
  continuously in space as electromagneticwaves.

- Einstein proposed that light not only was emitted asquanta at
  a time but also propagated as individual quanta, sufficiently
  small to be absorbed by theelectron.

-Planck found that the quantity associatedwith a particular
 frequency ν of light all had the same energy and that
 thisenergy was proportional to ν that is E = h ν

- Photoelectric effect can be explained bythe following
  equation E(=hν) = hν0 + Tmax Here E is the total energy of the
  photonincident on the metallic surface, ν is thefrequency of
  the incident radiation, ν0isthe threshold frequency of the
  metal andTmax is the maximum kinetic energy withwhich the
  electron moves after ejectionfrom the surface

3. Bohr model of the atom

 Postulates Of Bohr's Model Of An Atom

- In an atom, electrons (negatively charged) revolve
  around the positively charged nucleus in a definite
  circular path called orbits or shells. Each orbit or
  shell has a fixed energy and these circular orbits are
  known as orbital shells.

 Main Points of the Bohr Model

- Electrons orbit the nucleus in orbits that have a set
  size and energy.

- The energy of the orbit is related to its size. The
  lowest energy is found in the smallest orbit.

- Radiation is absorbed or emitted when an electron
  moves from one orbit to another.

 Problems With the Bohr Model

- It violates the Heisenberg Uncertainty
  Principle because it considers electrons to have both
  a known radius and orbit.

- The Bohr Model provides an incorrect value for the
   ground state orbital angular momentum.

- It makes poor predictions regarding the spectra of
  larger atoms.

- It does not predict the relative intensities of spectral
   lines.

- The Bohr Model does not explain fine structure and
   hyperfine structure in spectral lines.

- It does not explain the Zeeman Effetc

Bohr's Model Of An Atom :

4. Schrödinger Wave equation

[1] what is Schrödinger Wave equation?

- Schrodinger wave equation is a mathematical expression
  describing the energy and position of the electron in space
  and time, taking into account the matter wave nature of the
  electron inside an atom.

- Schrödinger Wave Equation based on three considerations

• Classical plane wave equation,
• Broglie’s Hypothesis of matter-wave
• Conservation of Energy.

- There are two equations

1. Time-Dependent Schrödinger Equation
2. Time-Independent Schrödinger Equation.

1.Time-dependent Schrödinger equation :

i = imaginary unit
Ψ = time-dependent wavefunction
h2 is h-bar
V(x) = potential and
Ĥ = Hamiltonian Operator

2.Time-Independent Schrödinger Equation :

5. Heisenberg’s Uncertainty Principle

[1] What Is Heisenberg’s Uncertainty Principle

- Heisenberg uncertainty principle or indeterminacy principle, statement,
  articulated (1927) by the German physicist Werner Heisenberg, that the
  position and the velocity of an object cannot both be measured exactly,
  at the same time, even in theor

- If ∆x is the error in position measurement and ∆p is the error
  in the measurement of momentum, then

- If ∆e is the error in energy measurement and ∆t is the error in
  the measurement of time, then

ħ = value of the Planck’s constant divided by 2*pi
∆X = uncertainty in the position
∆p = uncertainty in momentum
∆E = uncertainty in the energy
∆t = uncertainty in time measurem

6. Schrodinger’s cat experiment

[1] What Is Experiment?
- Schrödinger's cat is a thought experiment that illustrates a
  paradox of quantum superposition

Experiment –

Schrödinger's cat is a thought experiment that illustrates a
paradox of quantum superposition. In the thought experiment, a
hypothetical cat may be considered simultaneously both alive and
dead

[1]  What did Schrodinger's Cat experiment prove?

- In quantum theory, quantum particles can exist in a
  superposition of states at the same time and collapse down
  to a single state upon interaction with other particles. Some
  scientists at the time that quantum theory was being
  developed (1930's) drifted from science into the realm of
  philosophy, and stated that quantum particles only collapse
  to a single state when viewed by a conscious observer

- The cat ends up both dead and alive at the same time.
   Because the existence of a cat that is both dead and alive at
   the same time is absurd and does not happen in the real
   world, this thought experiment shows that wavefunction
   collapses are not just driven by conscious observers.

7. Richard Feynman - Quantum Electrodynamics

[1] What is Quantum electrodynamics?

- Quantum electrodynamics, commonly referred to as QED, is
   a quantum field theory of the electromagnetic force. Taking
   the example of the force between two electrons, the classical
   theory of electromagnetism would describe it as arising from
   the electric field produced by each electron at the position of
   the other. The force can be calculated from Coulomb's law

 Operator:

1) What is operator ?

- An operator is a generalization of the concept of a function
  applied to a function. Whereas a function is a rule for turning
  one number into another, an operator is a rule for turning
  one function into another.

2) Types of operators

- In quantum mechanics, we deal with two types of operators.

 Linear operators

- The operator is linear if it satisfies two conditions:

1. For the functions being added or subtracted, the function
    can be applied to all functions individually.

  - Â ( m + n ) = Âm + Ân

2. Constants are not affected by the application of linear operators.

   - Â (cm) = cÂm

 Hermitian operator

- The operator is said to be hermitian if it satisfies the
   following conditions.

1. A hermitian operator can be flipped over to the other side. In
    other words, it justifies the complex conjugate transpose of
    matrices.

 - If  is hermitian, {g| .f} = {f| .g}

2. The eigenvalues of a hermitian operator are always real.

  - from above example, {f|Â .f} must be a real value.

3.The eigenvalues are orthonormal by convention for a

    hermitian operator. in other words, they have a complete set
    of orthonormal eigenfunctions (eigenvectors).

Table :