Quantum Mechanics in a Nutshell

Quantum theory

• Wave-particle duality of light (“wave”) and electrons (“particle”)

• Many quantities are “quantized” (e.g., energy, momentum, conductivity, magnetic moment, etc.)

• For “matter waves”: Using only three pieces of information (electronic charge, electronic mass, Planck’s constant), the properties of atoms, molecules and solids can be accurately determined (in principle)!

Quantum theory – Light as particles• Max Planck (~1900): energy of electromagnetic (EM) waves can

take on only discrete values: E = nħω– Why? To fix the “ultraviolet catastrophe”

– Classically, EM energy density, εω ~ ω2εavg = ω2(kT)

– But experimental results could be recovered only if energy of a mode is an integer multiple of ħω as

εω

ω

Classical (~ω2kT)

experimental

from density of states

from equipartition theorem

The ultraviolet catastrophe

Quantum theory – Light as particles

• Einstein (1905): photoelectric effect– No matter how intense light is, if ω < ωc no photoelectrons

– No matter how low the intensity is, if ω > ωc, photoelectrons result

– Light must come in packets (E = nħω)

• Compton scattering (1923): establishes that photons have momentum!

– Scattering of x-rays of a single frequency by electrons in a graphite target resulted in scattered x-rays

– This made sense only if the energy and the momentum were conserved, with the momentum given by p = h/λ = ħk (k = 2π/λ, with λ being the wavelength)

• By now, it is accepted that waves may display particle features …

Quantum theory – Electrons as waves

• Rutherford (~1911): Experiments indicate that atoms are composed of positively charged nuclei surrounded by a cloud of “orbiting” electrons. But,– Orbiting (or accelerating) charge radiates energy

electrons should spiral into nucleus all of matter should be unstable!

– Spectroscopy results of H (Rydberg states) indicated that energy of an electron in H could only be -13.6/n2 eV (n = 1,2,3,…)

Quantum theory – Electrons as waves

• Bohr (~1913):– Postulates “stationary states” or “orbits”, allowed only if electron’s

angular momentum L is quantized by ħ, i.e., L = nħ implies that E = -13.6/n2 eV

– Proof: • centripetal force on electron with mass m and charge e, orbiting with velocity v at

radius r is balanced by electrostatic attraction between electron and nucleus mv2/r = e2/(4πε0r2) v = sqrt(e2/(4πε0mr))

• Total energy at any radius, E = 0.5mv2 - e2/(4πε0r) = -e2/(8πε0r)

• L = nħ mvr = nħ sqrt(e2mr/(4πε0)) = nħ allowed orbit radius, r = 4πε0n2ħ2/(e2m) = a0n2 (this defines the Bohr radius a0 = 0.529 Å)

• Finally, E = -e2/(8πε0r) = -(e4m/(8ε02h2)).(1/n2) = -13.6/n2 eV

– The only non-classical concept introduced (without justification): L = nħ

Quantum theory – Electrons as waves

• de Broglie (~1923): Justification: L = nħ is equivalent to nλ = 2πr (i.e., circumference is integer multiple of wavelength) if λ = h/p (i.e., if we can “assign” a wavelength to a particle as per the Compton analysis for waves)!– Proof: nλ = 2πr n(h/(mv)) = 2πr n(h/2π) = mvr nħ = L

• It all fits, if we assume that electrons are waves!

Quantum theory – Electrons as wavesThe Schrodinger equation: the jewel of the crown

• Schrodinger (~1925-1926): writes down “wave equation” for any single particle that obeys the new quantum rules (not just an electron)

• A “proof”, while remembering: E = ħω & p = h/λ = ħk– For a free electron “wave” with a wave function Ψ(x,t) = ei(kx-ωt), energy is purely kinetic

– Thus, E = p2/(2m) ħω = ħ2k2/(2m)

– A wave equation that will give this result for the choice of e i(kx-ωt) as the wave function is

• Schrodinger then “generalizes” his equation for a bound particle

K.E. P.E.Hamiltonian operator

The Schrodinger equation• In 3-d, the time-dependent Schrodinger equation is

• Writing Ψ(x,y,z,t) = ψ(x,y,z)w(t), we get the time-independent Schrodinger equation

• Note that E is the total energy that we seek, and

Ψ(x,y,z,t) = ψ(x,y,z)e-iEt/ħ

Hamiltonian, H

The Schrodinger equation

• An eigenvalue problem– Has infinite number of solutions, with the solutions being Ei

and ψi

– The solution corresponding to the lowest Ei is the ground state

– Ei is a scalar while ψi is a vector

– The ψis are orthonormal, i.e., Int{ψi(r)ψj(r)d3r} = δij

– If H is hermitian, Ei are all real (although ψi are complex)

– Can be cast as a differential equation (Schrodinger) or a matrix equation (Heisenberg)

– |ψ|2 is interpreted as a probability density, or charge density

EH

Applications of 1-particle Schrodinger equation

• Initial applications– Hydrogen atom, Harmonic oscillator, Particle in a box

• The hydrogen atom problem

Solutions: Enlm = -13.6/n2 eV; ψnlm(r,θ,ϕ) = Rn(r)Ylm(θ,ϕ)

http://www.falstad.com/qmatom/http://panda.unm.edu/Courses/Finley/P262/Hydrogen/WaveFcns.html

Summary of quantization

• Spin (Pauli exclusion principle) not included in the Schrodinger equation & needs to be put in by hand (but fixed by the Dirac equation)

The many-particle Schrodinger equation

• The N-electron, M-nuclei Schrodinger (eigenvalue) equation:

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The total energy that we seekThe N-electron, M-nuclei wave function

The N-electron, M-nuclei Hamiltonian

Nuclear kinetic energy

Electronic kinetic energy

Nuclear-nuclear repulsion

Electron-electron repulsion

Electron-nuclear attraction

• The problem is completely parameter-free, but formidable!– Cannot be solved analytically when N > 1– Too many variables – for a 100 atom Pt cluster, the wave function is a

function of 23,000 variables!!!

The Born-Oppenheimer approximation• Electronic mass (m) is ~1/1800 times that of a nucleon mass (MI)• Hence, nuclear degrees of freedom may be factored out• For a fixed configuration of nuclei, nuclear kinetic energy is zero

and nuclear-nuclear repulsion is a constant; thus

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Electronic eigenvalue problem is still difficult to solve! Can this be done numerically though? That is, what if we chose a known functional form for ψ in terms of a set of adjustable parameters, and figure out a way of determining these parameters? In comes the variational theorem

The variational theorem• Casts the electronic eigenvalue problem into a minimization problem• Lets introduce the Dirac notation

• Note that the above eigenvalue equation has infinite solutions: E0, E1, E2, … & correspondingly ψ0, ψ1, ψ2, …

• Our goal is to find the ground state (i.e., the lowest energy state)• Variational theorem

– choose any normalized function containing adjustable parameters, and determine the parameters that minimize <|Helec|>

– The absolute minimum of <|Helec|> will occur when = ψ0

– Note that E0 = <ψ0|Helec|ψ0> thus, strategy available to solve our problem!

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What is Reality?