Quantum Mechanics: Electric Fields

• The study notes cover concepts leading to Quantum Mechanics in 2024.

Electric Fields

• Electric fields stem from the presence of electric charges, which can be isolated and are either positive or negative.
• The fundamental unit of electric charge is the electron charge, with a magnitude of 1.602x10-19 C.
• Neutral atoms can become ions with a unit positive charge if an electron is removed through thermal energy. The electron then carries a unit negative charge.
• Electric field strength depends on the quantity of charge at a point.
• Electric potential, quantifies charge, representing work done to bring a unit positive charge from infinity to a point at distance x: 𝑉𝑥 = (𝑄 / 4πε₀) * (1 / 𝑥).
• Electric potential is inversely proportional to distance from the point charge and external work occurs if the moved charge is positive.
• Electric field due to a point charge is: 𝐸𝑥 = (𝑄 / 4πε₀) * (1/ 𝑥²).
• Electric field is inversely proportional to the square of the distance from the point charge.
• Electric field related to the potential by: Ex = -dVx / dx.
• Electric fields are visualized using flux lines that indicate force direction on a unit positive charge.
• Positive charge flux lines radiate outwards, indicating repulsion.
• Negative charge flux lines direct inward showing direction of attraction.
• Electric field lines indicate the measure of amount of charge at a given point.

Magnetic Fields

• Magnetic fields arise from dipoles, consisting of inseparable north and south poles.
• Magnetic monopoles remain theoretical, lacking experimental evidence.
• Magnetic field strength is inversely proportional to the square of the distance from the magnetic dipole.
• Magnetic potential is inversely proportional to distance.
• Magnetic field lines are continuous loops both inside and outside magnetic materials.
• Magnetic field lines indicate the strength of the field at a point.
• Current through a solenoid creates a magnetic field whereby direction dictates polarity.
• Electric and magnetic fields exhibit interplay changing over time.
• Ampere’s law states current through a conductor creates a magnetic field.
• Faraday's law refers to electromagnetic induction to induced current within a closed coil from, placed in a variable magnetic field.

Maxwell’s Equations

• Maxwell's equations (1860) unify electric and magnetic fields, describing radiation as electromagnetic waves.

The ∇ Operator (Nabla Operator)

• The del operator (∇) is given by:. ∇ = î(∂/∂x) + ĵ(∂/∂y) + k̂(∂/∂z)
• î, ĵ, and k̂ are unit vectors along the three orthogonal directions.
• Applying ∇ to a scalar yields the gradient, indicating the field's maximum rate of change.
• For example, the electric field (E ⃗ ) is the gradient of the electric potential (V):. ∇V = î(∂Vx/∂x) + ĵ(∂Vy/∂y) + k̂(∂Vz/∂z) = îEx + ĵEy + k̂Ez = E ⃗
• Dot product of the ∇ operator with a vector field gives divergence, quantifying field change:. ∇ · A⃗ = (∂/∂x)Ax + (∂/∂y)Ay + (∂/∂z)Az = (∂Ax/∂x) + (∂Ay/∂y) + (∂Az/∂z)
• Cross product gives curl field with results in a vector: ∇ × A⃗ = î((∂Az/∂y) - (∂Ay/∂z)) - ĵ((∂Az/∂x) - (∂Ax/∂z)) + k̂((∂Ay/∂x) - (∂Ax/∂y))
• The curl of the curl of a vector equals the gradient of its divergence minus the Laplacian operating on the vector:. ∇ × (∇ × E ⃗ ) = ∇(∇ · E ⃗ ) - ∇²E ⃗
• where the scalar Laplacian operator is: ∇² = (∂²/∂x²) + (∂²/∂y²) + (∂²/∂z²).

Gauss’s Law

• Gauss's Law for electric fields relates divergence of electric field to charge density and is given by: ∇ · E⃗ = ρ / ε₀
• Gauss's Law for magnetic fields states divergence of the magnetic field uniformly zero and is given by: ∇ · B⃗ = 0.
• The law implies absence of magnetic monopoles.

Faraday's Law

• Faraday’s Law of Electromagnetic Induction relates the curl of the electric field to the rate of change of the magnetic field and is given by: ∇ × E⃗ = -(∂B⃗/∂t)

Ampere–Maxwell Law

• The Ampere-Maxwell circuital law (modified by Maxwell) is expressed by: ∇ × B⃗ = μ₀j + μ₀ε₀(∂E/∂t)
• The second term represents displacement current from time-varying electric fields.

Maxwell’s Eqautions In Free Space

• Maxwell's equations reduce in free space to:

  • ∇ · E⃗ = 0
  • ∇ · B⃗ = 0
  • ∇ × E⃗ = -∂B⃗/∂t
  • ∇ × B⃗ = +μ₀ε₀(∂E/∂t).

• The curl of the curl of the electric field can be written as: ∇ × (∇ × E⃗ ) = ∇ × (- ∂B/∂t)

• This reduces to: ∇(∇ · E⃗ ) - ∇²E ⃗ = (-∂(∇ × B)/∂t)

• Under the condition of ∇ · E⃗ = 0, -∇²E ⃗ = (- ∂(∇ × B)/∂t).

• The substitution leads to: ∇²E ⃗ = (μ₀ε₀) * (∂²E /∂t²).

• Maxwell observed μ₀ε₀ = 1/c², where c is the speed of light.

• This equation infers the wave should be an electric field in free space traveling at c.

• Similarly, the same can be done with magnetic field:. ∇²B⃗ = (1/c²) * (∂²B⃗/∂t²)

• Light can be treated as electromagnetic waves with perpendicular electric and magnetic vectors.

• Considering a 1D electric wave following this equation describes radiation in the Z direction: 𝐸𝑥 = 𝐸𝑥₀ sin(ωt + kz).

• The electric field vector only has an x component, therefore Ey and Ez are zero

• The third Maxwell equation evaluates the curl of the electric field via:. ∇ × E⃗ = ĵ k̂ Ex₀ cos(ωt + kz)

• Integration with respect to time results in: 𝐵 = ĵ Ex/c.

• This illustrates the magnitude of the wave is 1/c times that of the electric component.

• We conclude EM waves are results of magnetic and electric field components that demonstrate perpendicularity to each other and to the source of radiation.

Energy of EM Waves

• Classically, wave energy is equivalent to its intensity, which is the square of the amplitude.
• The energy per unit volume with an electric field, En = ½ ε₀E².
• The electric component of the wave has the energy content= ½ ε₀Ex².
• This equal to ½ ε₀Ex₀²cos²(ωt + kz).
• The magnetic component equal to (1/2c²)ε₀Ex².
• These sum to equal the total energy content of the wave ε₀Ex².
• EM waves transport energy with vectors described by the Poynting vector with the equation: S = (1/μ₀) E × B = cε₀E².
• Average energy transmitted by these waves:
• Average energy proportional to square of electric or magnetic amplitude and independent of frequency
• Classical EM wave frequency independence could not account for some observed interactions of light with matter.

Polarization

• Light is a transverse wave that's generally unpolarized
• Light in a plane wave format is linearly polarized by adding horizontal and vertical waves
• Electromagnetic waves as superposition of two plane waves of equal amplitude.
  • These differ in phase at 90°, then polarized as a circle
  • If other amplitudes, that relative phase other than 90° are polarized as an ellipse

Blackbody Radiation in Equilibrium

• Classically, radiation reveals a material's color and wavelength absorptions

• Gustav Robert Kirchhoff described absorption in materials, discovering that when heated, they emit wavelengths as its absorption

• A material that absorbs all incident rays is defined as a black body

• The black body also emits frequency radiation that escalates with temperature but amount is restricted by the rate at which they're absorbed .

• Emissions of a frequency follow thermodynamic equilibrium of the absorption and emission levels.

• Equilibrated cavities emit with radiation density is based on temperature of the walls despite the structure.

• The distribution and wavelengths decrease to lower side

• Blackbody emissions are modeled after harmonic oscillators on the walls of the cavities

• Physical dimension in cavities emit harmonic oscillator and wavelengths.

• Standing waves are made this way, and a cubicle cavity can be defined with the equation: a = nλ/2

• Sustained frequencies are derived: ν = c/λ = n(c/2a)

• Additional frequencies are defined when "n" changes: dν= dn(c/2a)

• The amount of 3D cavity frequency is: V = r(c/2a)

• The surface area is the value of the frequency.

• Additional no. of frequencies changes to is; dν= dr (c/2a)

• The intensity increases given, intensities are infinite with temperature due to Rayleigh Jeans law.

• Max plank identified that energy is a harmonic of the oscillator is frequency(v= times.

• That amount is h = 6.6𝑥10−34 𝐽𝑠

• Radiations emerge from multiple frequencies.

• This energy can be found by determining that it could evaluated: 〈𝐸〉= hv∗e^(−hv/kT)/1−e^(−hv/kT)

• Energy can evaluated w/ energy density = 8π/c3ν2 𝑑ν kB T , thus with Planck it becomes: (E)dN 8 π hv3/c3*1/ehv/kT − 1 dv.

• Planck identified the energy quantum as a milestone.

Atomic Spectra

• Different elements have unique atomic spectra, so spectroscopy can identify and quantify a sample’s composition.
• Atomic absorption lines, were noted by Robert Bunsen and Gustav Kirchhoff.
• Classical physics considered emission from electrons orbits that accelerate that light, but with the model electrons give up energy over small intervals.

Dual Nature of Radiation

• Radiation is electromagnetic and displays interference with interference.
• Einstein considered the photo electric effect a particle-particle interaction so radiation interacts over a subatomic level.

Compton Effect

• Arthur H. Compton discovered it as he experimented the scattering of X-ray by identifying additional emissions with varying wavelengths.
• Classic treatment didn't account higher rays, Compton understood the problem to be collision of particles with equations derived as, Δλ = h/mec (1 - cos θ).
• The Compton Effect another confirmation of the particle make up of radiation.

Dual Nature of Matter

• Louis de Brogile thought that matter with energy display wave patterns.
• The de Brogile wavelength is then derived as: λ = h/mv
• A concept verified after unusual observations of electron scatter by Davisson and Germer.

Young’s Double Slit Experiment

– Double-slit experiment outcomes:. - Electrons are expected to arrive at a location. - A second electron arrives with the first. - Overall, if that occurs.

Wave Packets

• Matter's wave packets mathematically show position along w/ measurement.
• Formula:𝑦= 𝑦1 + 𝑦2 = 2𝐴𝑠𝑖𝑛(𝑤𝑡 + 𝑘𝑥). cos( Δ𝑤𝑡+Δ𝑘𝑥/2)
  • 𝑘 is angular frequency
• The amplitude is based on frequency with formula.

Velocity

• It’s possible define both phase and group velocity for the wave packet.
• Wave packet (wave group) velocity has the formula, = dω/dk.
• The relation between them that velocity involves angular velocity and frequency:
• dω = dE/ h = Group velocity of waves = vg =dω/dk
• 𝑣𝑔 = 𝑣𝑝ℎ − λ d vp/ dλ Evaluate the condition under which the group velocity of a wave packet is
• ½phase velocity with equations , 𝑣𝑔 = 𝑣𝑝ℎ/2.
• If twice the phase velocity the evaluation is, 𝑣𝑔 = 2𝑣𝑝ℎ.

Uncertainty Princible

• The uncertanity in position and wave Is intrinsically connect, has defined by, ∆𝑥. ∆𝑝 ≥ /ℏ2
• gamma rays may show momentum
• Electrons can't exist w/in the nucleus.

Wave Functions

• A wave described the 𝛙(𝑥, 𝑦, 𝑧, 𝑡) of a particular packet.
• If it expresses how is must be functions.
  • ψmust be finite
  • continuous, single valued in the regions of interest, also following equation: ∫⬚ 𝛙∗. 𝛙 𝒅𝒙 = 1
• linear superposition of wave funcitons, is determined with equation - 𝝍𝟑 = 𝝍𝟏 + 𝝍𝟐.

Operators

• momentum can be represented in with function = 𝑖/ ℏ(𝑝𝑥−𝐸𝑡).
• A value of a wave can then inferred when operating a defined function.

Expectation Values

• Observables with probability = a values of a physical system
• Operation can evaluate extracted, as function value.
• Equations derive a new value for momentums or what value are extracted.

Superposition of States

• Wave function stores information about the physical observables associated with the state of the system.
• Any quantum mechanical system can be superposed to state with propabilities. |0⟩ = [1/0] State |1⟩=0/1.
• Equations with state may written by these with complex: ψ = α|0⟩ + β|1⟩ subject to the condition that α2 + b2 = 1
• Sensitive with with n quibits as measured for ascertain the systems
• Schrödinger with the help of differential equations relates this equation
• Time can independent equation = to solving this system. Eψ(x, t) = KEψ(x, t) + Vψ(x, t)