Quantum Mechanics Notes

The Postulates of Quantum Mechanics

• The dynamics of microscopic systems (the Schrödinger equation): This equation describes how the quantum state of a physical system changes over time.
• The Born interpretation of the wavefunction: This interpretation relates the wavefunction to the probability of finding a particle in a specific location.
• QM principles: These principles include the concept of probability density and the Heisenberg uncertainty principle.

Introduction to Schrödinger Equation

Classical Mechanics vs. Quantum Mechanics

• Classical Mechanics (Newtonian mechanics): Explains the motion of macroscopic objects. Knowing the initial position (x) and velocity (v) of an object allows us to determine its exact location at any future time (t).
  • Initial state: (x, v) at time 0
  • Final state: (x', v') at time t
• Schrödinger Equation: Used to describe the future state of quantum mechanical systems, such as atoms. It provides:
  • Allowed energy levels
  • Wavefunction: Gives the probability of finding a particle at a specific position.

Time-Dependent Schrödinger Equation (1D)

For a particle of mass m moving in one dimension with energy E:

iℏ \frac{\partial}{\partial t} \Psi(x,t) = \hat{H} \Psi(x,t)

Where:

• V(x) = Potential energy of a particle at point x
• E = Total energy (sum of potential and kinetic energy)
• ℏ = Reduced Planck constant (\frac{h}{2\pi}, modification of Planck constant)

The Wavefunction

• The state of a quantum particle is given by the wavefunction, denoted as ψ.
• QM acknowledges wave-particle duality, suggesting particles are distributed through space like a wave rather than following a definite path.
• The wavefunction (ψ) replaces the classical concept of trajectory.

3D Systems

• Time-dependent Schrödinger equation for three-dimensional systems includes the Laplacian operator, denoted as ∇^2 ('del squared').
• General form of the Schrödinger equation:
  iℏ \frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \hat{H} \Psi(\mathbf{r},t)
  Where Hamiltonian is the operator corresponding to the total energy of the system.

Application of Schrödinger Equation

• By applying the Schrödinger equation to an atom (with its mass and charge), we can obtain information about the shapes of atoms through mathematical calculations.

Shapes of Hydrogen Atom (H)

• Hydrogen (H) is the smallest atom, with 1 proton and 1 electron.
• Solving the Schrödinger equation for the H atom was crucial for understanding the shapes of all other atoms.
• Electrons fill the space in an atom not by spinning around, but with blobby-shaped wavefunctions called orbitals.

Visualizing Orbitals

• Orbitals represent position probability distributions defined by non-spherical stationary states (e.g., 2p0, 3p0, 3d0, 4p0, 4d0, 4f0, 5d0, 5f0, 5g0).
• The images depict the fuzzy relative position between the electron and the nucleus in various stationary states of atomic hydrogen, not the nucleus or the electron themselves.
• The proton is at the center, and the electron can move to different orbitals when energy is supplied.

Example: H Atom in 2p Orbital

• When an H atom is prepared in a 2p orbital by adding energy, the electron takes the shape of the wavefunction if left undisturbed.
• Measuring the electron's position collapses the wavefunction into a single point (e-).
• Repeated measurements show a higher probability of finding the electron in regions where the wavefunction is denser.

Disconnection of 'Blobs'

• There are areas where the electron cannot be found, leading to the question of how the electron transitions from one 'blob' to another.

QM as a Link Between Physics and Chemistry

QM serves as the link between physics and chemistry, leading to implications such as:

• Lasers
• MRI scanners
• Electron microscopes
• Molecular design
• Quantum electrodynamics
• Computers
• Transistors
• Interstellar molecular spectroscopy

Schrödinger’s Cat – Thought Experiment

A cat is placed in a steel chamber with a device containing radioactive hydrocyanic acid (HCN).

• If a single atom of HCN decays, a relay mechanism will break the vial, killing the cat.
• Since the observer cannot know if an atom has decayed, the cat is considered both dead and alive (a superposition of states) according to quantum law.
• Superposition is lost when the box is opened and the cat's condition is observed, making the cat either dead or alive.

Observer's Paradox

• The act of observation or measurement affects the outcome, meaning the outcome does not exist unless measured.
• There is no single outcome unless it is observed.
• Superposition occurs at the subatomic level, with observable interference effects demonstrating a single particle in multiple locations simultaneously.

The Born Interpretation of Wavefunction

• The wavefunction contains all dynamical information about the system it describes.
• Max Born interpreted the wavefunction in terms of the location of the particle.
• The value of |ψ|^2 at a point is proportional to the probability of finding the particle in a region around that point.
• The probability of finding a particle in the region dx located at x is proportional to |ψ|^2dx.
  • |ψ|^2 = Probability density
  • ψ = Probability amplitude

3D Systems

• If the wavefunction of a particle has the value ψ at some point r, the probability of finding the particle in an infinitesimal volume dτ = dxdydz at that point is proportional to |ψ|^2dτ.

Quantization

• The Born interpretation imposes restrictions on the acceptability of wavefunctions:
  • ψ must not be infinite anywhere.
  • The wavefunction must be single-valued.
  • The wavefunction must be continuous, have a continuous slope, be single-valued, and be square-integrable.
  • An acceptable wavefunction cannot be zero everywhere because the particle must exist somewhere.
  • The energy of the particle is quantized; it can only possess certain energies, otherwise, its wavefunction would be physically unacceptable.

Heisenberg Uncertainty Principle

It is impossible to specify simultaneously, with arbitrary precision, both the momentum and the position of a particle. It is impossible to measure two properties of a quantum object, such as its position and momentum (or energy and time), simultaneously with infinite precision.

• If we know the location of a particle to within a range Δx, then we can specify the linear momentum parallel to x to within a range Δpx subject to the constraint:
  • ΔxΔpx > ℏ
  • ΔEΔt > ℏ
• The uncertainty principle arises from the wave properties inherent in the quantum mechanical description of nature, not from the accuracy of instruments.
• The principle explains why atoms don't implode, how the sun shines, and that the vacuum of space is not actually empty.

Quantum Mechanics Postulates

• The wavefunction, ψ, contains all dynamical information and is found by solving the Schrödinger equation.
• The probability of finding a particle in an infinitesimal volume dτ = dxdydz at point r is proportional to |ψ|^2dτ.
• Acceptable wavefunctions must be continuous, have a continuous first derivative, be single-valued, and square-integrable.
• Observables, Ω, are represented by operators built from position and momentum operators.
• The Heisenberg uncertainty principle states that it is impossible to specify simultaneously, with arbitrary precision, both the momentum and the position of a particle.

Practical Applications of Quantum Mechanics

Computers and Smartphones

• Modern semiconductor-based electronics rely on the band structure of solid objects, a quantum phenomenon dependent on the wave nature of electrons.
• Understanding the wave nature of electrons allows us to manipulate the electrical properties of silicon.
• Stacking layers of silicon doped with different elements allows the creation of transistors on the nanoscale.
• Millions of transistors packed together in a single block of material make computer chips that power technological gadgets like:
  • Desktops
  • Laptops
  • Tablets
  • Smartphones
  • Household appliances
  • Kids' toys

Lasers and Telecommunications

Used in fiber optic telecommunications.

Atomic Clocks and GPS

• NIST (National Institute of Standards and Technology) physicists developed a clock with laser-cooled single ions confined in an electromagnetic ion trap.
• This clock is 37 times more precise than the existing standard.

Magnetic Resonance Imaging (MRI)

• Spins, an intrinsically quantum phenomenon, cause electrons, protons, and neutrons to behave like magnets.
• A specific arrangement of magnetic fields allows the measurement of hydrogen concentration in different parts of the body, distinguishing between tissues that cannot be visualized with X-rays.

Quantum Mechanics Impact on Nanoscience

Scanning Probe Microscopy (SPM)

• Increased demand for small digital electronic devices has driven the design of smaller microprocessors.
• SPM is a collection of techniques used to visualize and manipulate objects as small as atoms on surfaces.
• SPM offers better resolution than Electron Microscopy.
• Types of SPM:
  • STM (Scanning Tunneling Microscopy)
  • AFM (Atomic Force Microscopy)
• SPMs measure local properties (e.g., height, friction, magnetism) with a probe by raster-scanning the probe over a small area of the sample.

STM (Scanning Tunneling Microscopy)

• A Pt–Rh or Pt-W needle is scanned across the surface of a conducting solid.
• When the needle tip is close to the surface, electrons tunnel across the space.
• In constant-current mode, the stylus moves up and down to match the surface, mapping the topography, including any adsorbates, on an atomic scale.

AFM (Atomic Force Microscopy)

• A sharpened stylus is scanned across the surface.
• The force exerted by the surface deflects a cantilever.

Advantages:

• Quantitative topographical information at high lateral resolution.
• Little to no sample preparation.
• Minimal harm to the sample.
• Applicable to conductive and insulating materials.

Typical Applications:

• High-resolution surface profilometry.
• Surface roughness measurements.
• Microstructural studies.
• Defect and failure analysis.
• Pit analysis for optical disk storage media.
• Magnetic domain and surface roughness analysis for computer hard-disks.
• Semiconductor device structural analyses.
• Surface cleaning and polishing studies.
• Phase separation in polymers.
• Critical Dimension Measurements.
• Investigation of local mechanical properties.
• High-resolution imaging of biological samples.
• Studies of nano-scale forces.

Quantum Dots (QDs)

• A quantum dot is a small piece of material exhibiting different properties than the bulk material due to its small size.
• QDs emit and absorb light at specific wavelengths (different energy levels).
• They are crystals a few nanometers wide, containing a few dozen to a few thousand atoms.
• Made from semiconductors like silicon, CdSe, CdS, ZnSe, ZnS.

Biological and Chemical Applications

• Medical applications include:
  • Drug delivery in anticancer treatment: QDs accumulate in specific body parts to deliver anti-cancer drugs, targeting single organs and reducing side effects seen in conventional chemotherapy.
  • Visualization within cells: Used in place of organic dyes in biological research to light up and color specific cells under a microscope.
  • Sensors for chemical and biological agents (e.g., anthrax).
• Advantages over organic dyes:
  • Brighter
  • Can produce any color of visible light.
  • Photostable (theoretically last indefinitely).

QDs in LCD Screen Technology

• Main Benefits:
  • Higher peak brightness
  • Better color accuracy
  • Higher color saturation
  • Improved battery life
• Main Drawback:
  • High toxicity of Cd-based QDs