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 = nħ  –Why? To fix the “ultraviolet catastrophe” –Classically, EM energy density,  ~  2  avg =  2 (kT) –But experimental results could be recovered only if energy of a mode is an integer multiple of ħ  as   Classical (~  2 kT) 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  <  c  no photoelectrons –No matter how low the intensity is, if  >  c, photoelectrons result –Light must come in packets (E = nħ  ) 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/  = ħk (k = 2  / , with  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/n 2 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 = /n 2 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  mv 2 /r = e 2 /(4  0 r 2 )  v = sqrt(e 2 /(4  0 mr)) Total energy at any radius, E = 0.5mv 2 - e 2 /(4  0 r) = -e 2 /(8  0 r) L = nħ  mvr = nħ  sqrt(e 2 mr/(4  0 )) = nħ  allowed orbit radius, r = 4  0 n 2 ħ 2 /(e 2 m) = a 0 n 2 (this defines the Bohr radius a 0 = Å) Finally, E = -e 2 /(8  0 r) = -(e 4 m/(8  0 2 h 2 )).(1/n 2 ) = -13.6/n 2 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 n  = 2  r (i.e., circumference is integer multiple of wavelength) if  = h/p (i.e., if we can “assign” a wavelength to a particle as per the Compton analysis for waves)! –Proof: n  = 2  r  n(h/(mv)) = 2  r  n(h/2  ) = mvr  nħ = L It all fits, if we assume that electrons are waves!

Quantum theory – Electrons as waves The Schrodinger equation: the jewel of the crown Schrodinger (~ ): writes down “wave equation” for any single particle that obeys the new quantum rules (not just an electron) A “proof”, while remembering: E = ħ  & p = h/  = ħk –For a free electron “wave” with a wave function Ψ(x,t) = e i(kx-  t), energy is purely kinetic –Thus, E = p 2 /(2m)  ħ  = ħ 2 k 2 /(2m) –A wave equation that will give this result for the choice of e i(kx-  t) 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 E i and ψ i –The solution corresponding to the lowest E i is the ground state –E i is a scalar while ψ i is a vector –The ψ i s are orthonormal, i.e., Int{ψ i (r)ψ j (r)d 3 r} =  ij –If H is hermitian, E i are all real (although ψ i 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

Applications of 1-particle Schrodinger equation Initial applications –Hydrogen atom, Harmonic oscillator, Particle in a box The hydrogen atom problem Solutions: E nlm = -13.6/n 2 eV; ψ nlm (r,θ, ϕ ) = R n (r)Y lm (θ, ϕ )

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: The total energy that we seek The 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 (M I ) 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

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: E 0, E 1, E 2, … & correspondingly ψ 0, ψ 1, ψ 2, … 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 –The absolute minimum of will occur when  = ψ 0 –Note that E 0 =  thus, strategy available to solve our problem!

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