Quantum Mechanics in a Nutshell

1. Quantum Mechanics in a Nutshell

2. 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)!

3. Quantum theory – Light as particles from density of states • 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, ε(ω) ~ ω2εavg = ω2(kT) • But experimental results could be recovered only if energy of a mode is an integer multiple of ħω as from equipartition theorem Classical (~ω2kT) ε(ω) The ultraviolet catastrophe experimental ω

4. 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 …

5. 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,…)

6. 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/(4pe0r2)  v = sqrt(e2/(4pe0mr)) • 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

7. Quantum theory – Electrons as waves • de Broglie (~1923): Justification: L = nħ is equivalent to nλ = 2πr (i.e., circumference is integer multiple of wavelength) ifλ = 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!

8. 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 ei(kx-ωt) as the wave function is • Schrodinger then “generalizes” his equation for a bound particle K.E. P.E. Hamiltonian operator

9. 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

10. 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

11. 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

12. 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)

13. The many-particle Schrodinger equation • The N-electron, M-nuclei Schrodinger (eigenvalue) equation: The N-electron, M-nuclei wave function The total energy that we seek 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!!!

14. 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 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

15. 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 F containing adjustable parameters, and determine the parameters that minimize <F|Helec|F> • The absolute minimum of <F|Helec|F> will occur when F = ψ0 • Note that E0 = <ψ0|Helec|ψ0> thus, strategy available to solve our problem!

16. What is Reality?