Introduction to Quantum Mechanics

Introduction to Quantum Mechanics

• Newtonian mechanics started as the thinking of science, capable of resolving almost all problems
• Newtonian mechanics are applied to visible systems
• Quantum mechanics started as formulating science and aims to resolve problems in chemistry and physics
• Quantum mechanics is applicable to all systems
• Newtonian mechanics is philosophical approach
• Quantum mechanics takes a probabilistic approach
• Quantum mechanics exhibit dual nature characteristics
• State of the particle = position and momentum in Newtonian mechanics
• State of the system = wave function in quantum mechanics
• Elementary quantum mechanics is a transition from the old to modern era

The Beauty of Quantum Theory

• Without quantum mechanics, specific technologies would not have been developed including:
  • Semiconductor devices
  • Computers
  • Cell phones
  • Lasers
  • CD/DVD players
  • MRI technology
  • Nuclear reactors
  • Atomic clocks (GPS navigation)

Quantum Theory

• Quantum theory has many diverse applications:
  • Evolution of the universe
  • Medical uses
  • Power and bombs
  • Materials and technology
  • Nuclear physics
  • Atoms and molecules
  • Optics
  • Quantum computer
  • LASER
  • Communication
  • Quantum cryptography

Speed and Size

• Relativistic mechanics involve high speeds
• Classical mechanics involves larger sizes
• Quantum mechanics involves smaller sizes
• Relativistic Q M could involve smaller sizes
• There are challenges to classical mechanics on the relativistic and microscopic domains

Origin of Quantum Mechanics

• Blackbody radiation, atomic structure and photoelectric effect were crises in physics
• Quantum mechanics was demanded to explain these physics crises

Crises in Physics: Blackbody Radiation

• How the intensity of electromagnetic radiation emitted by a body depends on the frequency of the radiation and the temperature of the body
• Kirchoff described this in 1859

Crises in Physics: Photoelectric Effect

• Metals shed electrons when certain light is incident on it
• Heinrich Hertz observed this in 1887

Wave Model of Light

• Increasing light intensity should increase the kinetic energy of emitted photoelectrons
• Brightness should produce more electrons
• Increasing the frequency should increase measured current

Photoelectric Effect Experiments

• Showed that increasing the light frequency increased the kinetic energy of the photoelectrons
• Increased light amplitude increased the current

Photoelectric Effect: Three Challenges

• There is no time interval between light arrival at a metal surface and emission of photoelectrons
• The energy in an EM wave is supposed to spread across the wavefronts, a period should elapse before an individual electron accumulates enough energy to leave the metal
• A bright light yields more photoelectrons than a dim one of the same frequency, but the electron energies remain the same
• The EM theory of light predicts the more intense the light, the greater energies of the electrons
• The higher the frequency of light, the more the energy the photoelectrons have
• At frequencies below a certain critical frequency, which is characteristic of a metal, no electrons are emitted

Atomic Structure

• Atomic structure produced data that could not be explained through classical Physics

Origin of Quantum Mechanics Highlights

• Plank: Quanta of energy discovered in 1900
• Einstein: Photon of light proposed in 1905
• Bohr: Atom's discrete states discovered in 1913
• Davisson and Germer: Wave behavior of electrons discovered in 1927
• De Broglie: Dual nature of electrons discovered in 1923
• Compton: Particle behavior of radiation discovered in 1923
• Heisenberg: Uncertainty in measurements discovered in 1923

Plank's Quanta of Energy

• Planck considered that energy exchange between radiation and matter must be discrete
• The radiation of frequency v emitted by the matter must come only in integer multiples of hv, defined as 𝐸ᵥ = nhv, n = 1, 2, 3, 4 …

Einstein: Photon of Light

• Einstein thought that light is equivalent to wave packets called photons, i.e. particle nature
• Energy of each packet = hv. 'h' is Planck's constant
• If hv is larger than the metal's work function (W), the electron will then be knocked out of the metal
• No electron can be emitted from the metal's surface unless hv > W

Breakthroughs in Science: Light

• The wave theory of light explains diffraction and interference
• The quantum theory explains the photoelectric effect
• the wave theory cannot account for it

Bohr: Atom's Discrete States (1913)

• If an electron jumps one orbit closer to the nucleus, it must emit energy equal to the difference in the energies of the two orbits
• If an electron jumps to a larger orbit, it must absorb a quantum of light equal in energy to the difference in the orbits.

Compton: Particle Behavior of Radiation (1923)

• When a high frequency photon hits electrons, it gets scattered
• There is a decrease in the energy of the photon
• The lost energy from the photon is transferred to the recoiling electrons

Double Slit Experiment with Electron Beam

• Slit 1 open gives wave properties
• Slit 2 open gives wave properties
• Both slits open gives Interference pattern of a wave
• This indicates the wave behavior of the electron

Breakthrough: Davisson, Germer - Wave Behavior of Electrons (1927)

• G.P. Thomson in 1928 performed experiments with thin foil of gold
• This was in place of nickel crystal and a diffraction pattern was observed

De Broglie: Dual Nature

• Plank discovered quanta
• Einstein discovered the photoelectric effect
• Compton described Compton scattering
• Scientists experimentally performed the double slit experiment
• Davisson and Germer did experiments
• Thomson demonstrated gold foil experiment

De Broglie: Dual Nature (1923) equations

• Wavelength of a particle - λ = h/p
• Momentum of a wave - p=h /λ
• These equations represent wave-particle duality
• Matter waves

De Broglie: Equations

• For a cricket ball of mass 160 gm and velocity of 150 km/h is λ=0.98X10⁻³⁴ m.
• For an electron accelerated by applied potential of 1 volt is λ=1.228 nm.

Mechanical Analogy

• Describes properties of particles
• Involves electron in potential V
• Involves mechanical, electromagnetic, probability, and pilot waves

Breakthroughs with Electrons (1927)

• Davisson and Germer studied the the wave behavior of electrons
• For constant accelerating voltage, observed angles of 31 to 62 degrees
• Derived key formula, 2dSin θ = nλ

Heisenberg: Uncertain Measurements (1923)

• Slit 1 open or Slit 2 open gives wave indications
• Both slits open show an interference pattern of a wave
• There arise questions for whether an electron is a particle or a wave

Heisenberg: If There is no Observer

• "Somehow, nature knows whether we have the information of which slit the electron passed through."
• "If the particle is marked in some fashion, an interference pattern will not appear at the screen."

Nature Disturbance

• Nature behaves one way when looked at and a completely different way when not observed
• Observation disturbs the natural behavior

Heisenberg: Describing Observation

• Observation plays an important, decisive role in events, so reality varies, depending upon whether events were observed or not; -Heisenberg
• Observation depends on wavelength of the illuminating light
• Smaller the wavelength, the better the resolution
• Should use gamma rays (extremely small wavelength) to see the electron

Heisenberg: Problems With Measurement

• Illuminating electron with gamma rays consists really of bombarding it with photons
• Photons impart momentum and disturb the position of the electron through Compton scattering
• Observation disturbs position
• Heisenberg concluded one cannot simultaneously measure with infinite precision both the position and momentum of a particle

Heisenberg Uncertainty Principle

• It is impossible to simultaneously describe the position and momentum of a particle with absolute accuracy
• Represented as Δpₓ. Δx ≥ ħ/2
• Heisenberg's uncertainty principle is true only for conjugate pairs

Predictions or Results of Uncertainty Principle

• Explains nonexistence of free electron in nucleus
• Estimates the radius of Bohr's first orbit
• Predicts zero point energy of simple harmonic oscillators

Nonexistence of Free Electron in Nucleus

• The diameter of a nucleus is ~10⁻¹⁴ m
• Maximum uncertainty in position if inside = 10⁻¹⁴ m
• Minimum uncertainty in momentum and energy lead to electrons not existing freely inside

Estimate of Bohr's First Orbit

• Potential energy + Kinetic energies are used to explain energy in relation to the separation
• Estimated Bohr's first orbit

Zero Point Energy of Simple Harmonic Oscillators

• A total energy is derived at certain oscillations
• The value of a at which total energy is minimum and related equations

Formulation of Quantum Mechanics

• λ =h/p
• • Matter waves Δ𝑝. Δ𝑥 ≥ ℏ/2

Q: How will you define a system?

Quantum State

• A system is not defined classically
• Use mathematical expression called wave function
• The wave function is a quantum state
• All info is contained in in a particle's wave function

Wave Function

• The wave function (ψ) itself has no physical interpretation
• Its square (ψ²) at a particular place at any time is the probability of finding the particle there at that time
• Goal is to determine ψ for a body when its motion is limited by action of external forces.

Wave Function Properties

• Wave function may be any function, real or complex, akin to the behavior of the real physical system
• If ψ is a complex function then ψ² = ψ* ψ
• ψ* is complex conjugate of ψ
• Probability cannot be negative

Well-Behaved Wave Function properties

• ψ→0 at r = ∞
• ψ is Finite, continuous, Single valued
• dψ/dr is Finite, continuous, Single valued

Extracting Information

• Applying certain operations/instructions on wave function, the information/observable can be extracted
• These "information specific operations" = the physical operator

Schrödinger Equation

• A basic physical principle cannot be derived from anything else.
• It is a basic principle

Postulates of Quantum Mechanics

• Wavefunction
• Operator
• Measurement
• Probabilistic
• Evolution

Wave Function solutions

• Schrodinger equation may have multiple solutions as a wave function
• Superposition is the characteristic property of the waves so as of a wave function of the quantum state
• Linear combination of solutions of Schrödinger’s equation for a system is also a solution

Schrodinger’s Cat

• Thought experiment on superposition of quantum states
• Can be either be in both a state of alive/dead

Application: Superposition of Wave Function

• Linear combination of solutions of Schrödinger's equation for a given system is a solution
• The formation of molecular orbitals: ss, pp , and pp

Application: Quantum Superposition of Wave Functions

• Two quantum states that has both spin-up and spin-down states
• Has Pauli Exclusion principle
• Can have related entangled states

Quantum Entanglement

• "Spooky Science"
• Measuring entangled photons

Quantum Entanglement Definition

• Quantum entanglement occurs when a group of particles are generated, interact, or share spatial proximity
• They share relation such that quantum state of each particle of a group not be described independently
• Occurs even with large distances

Entanglement: Key Characteristics

• Non-locality: particles seem "connected" over large distances
• Quantum Correlation: measurement of one particle affects the state of the other

Finding Entangled States

• Superconductors and the Cooper pair
• Superconducting qubits such as IBM’s and Google’s quantum computers
• Trapped Ions and Atoms (electromagnetic traps)
• Beam Splitters such as photon pairs with polarizers

Key Quantum Features

• Quantum entanglement can lead to different aspects
  • secure communication via Quantum Key Distribution (QKD) for Quantum Cryptography
  • Faster Computation
  • Quantum Teleportation
  • Future Quantum Networking

Free Particle

• Application of entanglement is shown through free particles
• Free particles have unrestricited energy
• May be anywhere in space
• Associated with wave
• Free particle indicates has no force and no potential, and k may have continuous energy values; lacks discreetness

Particle In A Box

• The particle's motion is restricted
• Can be used in Electrons in Atoms with the potential energy with energy levels
• Quantum Dots: Semiconductors particles with the quantum dot determining dimensions to energy and other properties
• Molecular Vibration
• Optical Cavities

Free Electron

• Is always moving
• Has kinetic energy
• The energy can be calculated depending on the material and applied field

Particle Passing Potential Barrier

• Radioactive substances undergo radioactive decay
• Caused by Heisenberg’s Uncertainty
• Results in transmission of particles
• This behavior is known as quantum mechanical tunneling