Quantum Mechanics: Electric and Magnetic Fields

• Unit I: Review of Concepts Leading to Quantum Mechanics

Review of Electric and Magnetic Fields

• Electric fields arise from electric charges, which can be isolated as positive or negative types.
• The electron's charge is the fundamental unit of electric charge, approximately 1.602x10^-19 C.
• Atoms are neutral, however gaining enough energy, an electron can be removed by thermal energies from the nucleus.
• The atom becomes a positive ion, and the electron is a negative ion.
• Electric charges can be isolated, creating electric fields, with the electric potential quantifying the charge's strength, representing the work needed to bring a unit positive charge from infinity.
• Electric potential at a distance x from a charge: Vx = Q / (4πε₀x).
• Electric potential is inversely proportional to the distance from the test charge.
• External work is required to move a positive charge in a positive field.
• Electric field due to a point charge: Ex = Q / (4πε₀x²).
• Electric field is inversely proportional to the square of the distance from the test charge.
• Electric field in terms of electric potential: Ex = -dVx / dx.
• Electric fields can be visualized using flux lines indicating the force direction on a positive charge, lines directed outwards for positive charges and inwards for negative charges.
• The flux of electric field lines indicates the electric field strength and the amount of charge present.
• Magnetic fields are typically dipolar (north and south poles) and currently cannot be isolated, though magnetic monopoles are theoretically possible.
• Magnetic field strength is inversely proportional to the square of the distance from the magnetic dipole, mirroring electric fields.
• The magnetic potential is inversely proportional to the distance from a reference point.
• Magnetic field lines form continuous loops both inside and outside magnetic materials, with flux indicating field strength.
• Magnetic fields are created by current carrying solenoids and the current's direction determines polarity.
• Electric and magnetic fields show a robust interplay, especially with time-dependent quantities.
• Ampere's and Faraday's laws confirm strong electrical and magnetic forces; current via a straight conductor creates a magnetic field and electromagnetic induction produces current in a coil within a varying magnetic field.

Maxwell's Equations in a Medium

• Maxwell summarized the existing ideas of electric and magnetic fields and their related phenomena into four equations in 1860.
• Describing radiation with these equations paved the way as an electromagnetic wave.
• Ñ operator (nabla operator): Ñ = i(∂/∂x) + j(∂/∂y) + k(∂/∂z), where i, j, and k are unit vectors.
• V operating on a scalar produces a vector, giving rise to the gradient of the field.
• For example, the electric potential at a point is the result of the V operator operating on an electric potential. - Vv = i(∂V/∂x) + j(∂V/∂y) + k(∂V/∂z) = iEx + jEy + kEz = Ē.
• Dot product of the V operator with a vector field yields a scalar, the divergence, representing the field's rate of change in three orthogonal directions.
• Applying the cross product of the V operator gives the curl of the field, a vector perpendicular to both V and the original vector.
• An additional identity of the V operator shows that the curl of the curl of a vector is the same as the gradient of the divergence of a field minus the Laplacian acting on the vector.
• This is VxVxĒ = V(V.Ē) – V²Ē.
• V² = V.V= (∂²/∂x²) + (∂²/∂y²) + (∂²/∂z²) and is the scalar Laplacian operator.

Gauss's Law

• For electric fields, the divergence of the electric field is equal to the charge density divided by ε₀ (Gauss' law for electric fields), due to a system of charges enclosed by surface,.
• For magnetic fields, the divergence of the magnetic field is zero.
• V. B = 0 (Gauss' law for magnetic fields) implies the absence of magnetic monopoles.

Faraday's Law of Electromagnetic Induction

• The curl of the electric field is equal to the rate of change of the magnetic field, VxẺ = -∂B/∂t

Ampere-Maxwell Circuital Law

• The curl of the magnetic field is given by the current density and the displacement current, VxB = μ₀J + με₀(∂E/∂t).
• This law extends Ampere's law, where the second term is the displacement current with time-varying electric fields from Maxwell.

Maxwell's Equations in Free Space

• Maxwell's equations are rewritten in free space where there are no sources of charges and currents.
• Free Space Equations
• V.E = 0
• V.B = 0
• VxẺ = -∂B/∂t
• VxB = με₀ (∂E/∂t)
• Taking the curl of the curl of the electric field can be written as ∇ × (V x Ē) = ∇ × (-∂B/∂t. This process reduces to V(V.E) – V²E = (-∂/∂t)(∇ × B).
• A simplified equation with curl of B is given by ∇² = (μ₀ε₀)(∂²E/∂t²), through substitution.
• The general wave equation form comes after observing that με₀ = 1/c², with c as the speed of light.
• Light can be treated as electromagnetic waves, with perpendicular electric and magnetic vectors to the direction of radiation propagation.
• A 1D electric wave can be written as Ex = Exo sin(wt + kz), associated with radiation propagating in the Z direction.
• This states the electric field vector has only an x component where all other components equal zero..
• Maxwell's third equation is needed for the associated magnetic component of the EM wave to be evaluated. Evaluating the curl of the electric field VxE = ijk [∂/∂x ∂/∂y ∂/∂z] [Ex 0 0] = i x 0 + j * (∂Ex/∂z) + k*0 = ĵ * (∂/∂z) [Exo cos(wt + kz)].
• Integration yields the EM wave's magnetic component. This is expressed as B = ĵ * k * Exo cos(wt + kz) * (-1/w) = ĵ * Exo cos(wt + kz) * (-1/w).
• EM waves have coupled electric and magnetic field components mutually perpendicular and also perpendicular to radiation direction

Energy of EM Waves

• Classically, wave energy corresponds to intensity, which is equivalent to the square of wave amplitudes.
• The associated energy of an electric field per unit volume of free space is En = (1/2)ε₀E².
• Electric component's energy content = (1/2) ε₀Exo² cos² (wt + kz)
• Magnetic component's energy content = (1/2μo)Byo² = (1/2)(1/c²μ₀)Exo² = ε₀/(2)Exo²
• Total energy content of the wave is the sum of two components = ε₀Exo².
• EM waves transmit energy in a perpendicular direction to E and B field variations, calculated via the Poynting vector, s = (1/μo)E × B = c.ε₀E² .
• Average energy of the wave transmitted over time and area can be expressed.

Polarisation of EM Waves

• Light is a transverse EM wave; but natural light is typically unpolarized where propagation planes are equally probable.
• Linearly polarized light is a plane wave with alignment, when the light is a combination of a wave that has horizontal polarization and a wave with vertical polarization, with similar amplitude.
• A superposition of two plane waves with a 90° phase difference is circularly polarized, when an electromagnetic wave is the result of the superposition of two plane waves that differ in relation to amplitude or is other than 90°, the light is said to be elliptically polarized.
• The classical EM wave model gives frequency independence regarding for energy for some phenomena of light with matter.

Blackbody Radiation

• The interaction of radiation and matter includes the ways a material absorbs radiation, and wavelengths giving it color.
• Gustav Robert Kirchhoff observed materials absorbing all rays and emitting all radiations absorbed if heated, called a black body. Blackbody radiation is modeled as cavity radiation emission of every frequency rate at particular temperatures of the body and energy limited by the absorption rate.
• The thermodynamic equilibrium limited the energy emitted at a particular frequency based on absorption and emission processes.
• The radiation density in an equilibrium depends on the cavity temperature, regardless of structure, shifting to lower wavelengths.
• Modeling blackbody emission involves emissions from harmonic oscillators on cavity walls. The harmonic oscillators emit at particular wavelengths.
• Emission is restricted to radiations forming standing waves within, and at wavelengths equal to a = λn / 2.
• Frequencies that can be sustained in the cavity are v = nc/2a.

Max Plancks Equation

• Max Planck solved blackbody emission by quantizing energy states depending on harmonic oscillation.
• This meant the energy per volume of radiations is p(v)dv = (8πv²/c³)kBT .
• The solution stated that the energy of a harmonic oscillator is restricted to frequencies v and a constant h approximately(h = 6.6x10−34Js) E = nhv, due to radiation emerging from collections of harmonic oscillators and energies. where the energy of the radiations from the oscillators has to be packets of hν.
• In these terms, energy can be described as (E) = (hv*e^(-hv/kT))/(1-e^(-hv/kT)). and the energy density of radiations can be evaluated as p(v)dv = (8πhv^3)/(c^3) *1/((e^(hv/kT))-1) dv.
• Max Planck unknowingly established energy quantization in systems.

Atomic Spectra

• Atomic spectroscopy identifies and quantifies elemental sample composition using distinct element spectra.
• Robert Bunsen and Gustav Kirchhoff discovered absorption lines in the solar spectra, also known as Fraunhofer lines.
• Classical atomic models could not explain discrete line emissions, but models were proposed in response stating accelerated charges radiate energy through electron orbiting and emission.
• Modern atomic models describe line spectra by referring to state transitions.

Dual Nature of Radiation

• Radiation is part of the electromagnetic spectrum, also EM waves, where conventional wave theory explains reflection, refraction, interference, diffraction and polarization of light.
• Einstein explained the photoelectric effect as a particle interaction, paving the way for wave particle duality.
• The Photo electric displayed a particle nature when the interaction is at atomic / sub atomic particles.

Compton Effect

• Arthur H Compton studied X-ray scattering.
• Arthur H Compton observed that if a an electron emitted, the beam has a different wavelength as compared to the incident radiation.
• Classical X-ray treatment failed to explain X-rays of the type that had a higher wavelength.
• Treating this problem as a particle collision with momentum of photons. This is written as P = h/ λ, with wavelength of incident x-ray is scattered alongside the transfer of energy.
• Due to considering energy and momentum after the collision, this gives an increase in the wavelength of a photon after the collision and gives .
  • λf – λᵢ = Δλ = h/mₑc (1 - cos θ)
• This value is known as the compton shift, and depends on:
• Incident of X-rays
• Scattering Material
• Angle of Xrays scattering

Dual Nature Of Matter

• Louis de Broglie proposed moving matter displays wave characteristics, a = h/mv (mv is the momentum of the particle).
• Larger particles have smaller wavelength by observation, where they are difficult to measures.
• Experiments by Davisson and Germer demonstrate wave nature using crystals, confirming matter’s dual characteristics.
• This concept has been confirmed when measuring heavier particles.
• Hitachi in the 1980s showed the electrons scattering patterns are very close to a diffraction pattern produced by a double slit experiment when showing the diffraction of electrons.

Young's Double Slit experiment

• A double slit experiment reveals electrons as particles arriving at distinct locations but forming interference patterns when many electrons pass through individually.

Wave Packets

• Matter waves require mathematical representation for moving particles, where both position and momentum can be estimated with reasonable accuracy.
• The superposition of two waves gives a resultant, as described: y = y₁ + y2 = 2Asin(ωt + kx)⋅cos ((Δωt+Δk⋅x)/2).
• Due to group of superimposed waves, the particle's momentum comes from estimating waves of a wavelet, and inferred by maximum amplitude of the amplitude.

Describing momentum and position of particles

• This process makes accurate estimations of a particle's momentum and position. Given momentum is taken from the wavelength and the position represents maximum amplitude
• Phase velocity of waves refers to the high frequency components of the wave and is given by vp= ω/к
• Group wave velocity refers to the speed of travel, expressed as Vg= dw/dк
• A relation between both velocities describes group velocity as dw/dk, where w is the angular frequency of the wave, yielding group velocity Vg in which dE = hdw.
• In a dispersive medium the group velocity is Vg = Vph + k *dvph / dλ

Schrodingers Cat

• Quantum systems are not subjected to observations, since the observations interfere with the quantum behavior of the particles being studied

Uncertainty Principle

• Position and propagation constant are intrinsically related through the uncertainty principle.
• In summary he product of the standard deviations in the estimates of the position and the propagation constant was shown to be greater than or at the most equal to. . ie., Δx. Δk ≥ 1/2
• This then translates to the standard form of the uncertainty principle when the propagation constant is transformed to the momentum through the relation p = hk

Heisenberg's Gamma ray microscope:

• Heisenberg proposed the gamma ray microscope (as a thought experiment to illustrate the uncertainty principle) where it is evident that one should use y rays to observe electrons, which in turn scatter the radiation onto the objective lens of the microscope. In that order it resolves Ax is related to momentum p with the relation : ∆x. ∆px ≈ 2h > h/4π

Electrons Inside the Nucleus

• High energy beta particles emission from nuclei challenges electron existence inside the nucleus, uncertainty principle used.
  • With related uncertainties of DeltaX≈ 10^ (-14)m the minimum momentum is given by:
    The minimum uncertainty in the momentum of the electron then can be estimated as Δp =h / 2. Δχ= 5.28 × 10^−21 kgms^-1

Wave Functions

• Moving particles are represented by packets, wave packet can be described by x, y, z in a function referred to as state functions, functions must be :
• y must be finite, continuous and single valued in the regions of interest.
• The derivatives of the wave function must be finite, continuous and single valued in the regions of interest.
• The wave function y must be normalizable

Observables:

• The physical parameters associated with the particle such as energy, momentum are observable as measured by experiment and by the wave function
• Operations can be carried out on a function f to yield results, which is why the wave function can yield results: αψ = α ψ() αψ α αψ

Linear superposition of waves from SWE

• In a quantum system when multiple states, with each being a wave equations then a separate linear combination can also be expressed as such , with 4,(x) + m.

Time dependant and independant Schrodingers Equation.

• All systems can me summarised in the following format , with some operators in the following forms : Ê= Total Energy KE Kinetic energy PE - Potential energy - Schrodinger’s form. Êψ(x,t) = ΚΕψ(x,t) + Vψ(x, t) A time indépendant system can also be defined as functions where $(t) ≠ 0: h² d²ψ(x) + Εψ(x) – Vψ(x) = 0. 2m dx2

Template for quantum equations

• (1) Define / set up the physical system (define particle nature, boundaries of potential, total energy of the particle etc)
• (2) Write the Schrodinger's wave equation and apply the known conditions
• (3) Obtain the general form of the wave function
• (4) Check/verify the wave function for it's characteristics: finiteness, discreteness and continuity of and its derivatives.
• Normalization of the wave functions
• (5) Interpret the solution, get implications on the quantum system.

Free Patrical Soluton

• Particle without external forces, implying force = 0, constant is zero, constant is 0.
• When writing 1D (one dimensional) for a free particle wave for the system : h² d²ψ(x) + Εψ(x) = 0 or 2² +E = 0

==Start of section for page 22== Some typical solved Numericals

1. A simple problem shows to Estimate the Compton's shift.