* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Quantum Theory of Atoms
Quantum entanglement wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Schrödinger equation wikipedia , lookup
Quantum fiction wikipedia , lookup
Tight binding wikipedia , lookup
Bell's theorem wikipedia , lookup
Erwin Schrödinger wikipedia , lookup
Quantum field theory wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Quantum computing wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Quantum machine learning wikipedia , lookup
Path integral formulation wikipedia , lookup
Quantum teleportation wikipedia , lookup
Density matrix wikipedia , lookup
Dirac equation wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Double-slit experiment wikipedia , lookup
Coherent states wikipedia , lookup
Scalar field theory wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum group wikipedia , lookup
Renormalization wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Renormalization group wikipedia , lookup
Electron configuration wikipedia , lookup
Canonical quantization wikipedia , lookup
Particle in a box wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Wave function wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Matter wave wikipedia , lookup
EPR paradox wikipedia , lookup
Atomic orbital wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Quantum state wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Wave–particle duality wikipedia , lookup
History of quantum field theory wikipedia , lookup
Hidden variable theory wikipedia , lookup
Probability amplitude wikipedia , lookup
Atomic theory wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum Theory of Atoms • The Bohr theory of Hydrogen(1913) cannot be extended to other atoms with more than one electron • we have to solve the Schrödinger equation(1925) • since the Coulomb force only depends on r, we should use spherical coordinates (r,,) instead of Cartesian coordinates (x,y,z) Quantum Theory of Atoms • For a particle in a cubic or rectangular box, Cartesian coordinates are more appropriate and there are three quantum numbers (n1,n2,n3) needed to label a quantum state • similarly in spherical coordinates, there are also three quantum numbers needed • n = 1, 2, 3, ….. • l= 0, 1, 2, …,n-1 => n values of l for a given value of n • m= -l,(-l+1),…0,1,2,…,l => 2l+1 values of m for a given l • n is the principal quantum number and is associated with the distance r of an electron from the nucleus • l is the orbital quantum number and the angular momentum of the electron is given by L=[l(l+1)]1/2 ħ • m is the magnetic quantum number and the component of the angular momentum along the z-axis is Lz = m ħ Quantum Theory of Atoms • The fact that both l and m are restricted to certain values is due to boundary conditions • in the figure, l=2 is shown • hence L=(2(2+1))1/2 ħ = ħ(6)1/2 • and m= -2, -1, 0, 1, 2 • the Schrödinger equation can be solved exactly for hydrogen • the energies are the same as in the Bohr theory En = -Z2 (13.6 eV)/n2 • they do not depend on the value of l and m • this is a special property of an inverse-square law force • The lowest energy has n=1 => l=0 and m=0 • the second lowest energy has n=2 => l=0, m=0 l=1, m=-1,0,1 • hence 4 states! • Notation: • • • • l=0 l=1 l=2 l=3 l=4 S-state P-state D-state F-state G-state • When a photon is emitted or absorbed we must have m= 0 or ±1 l = ±1 • conservation of angular momentum and conservation of energy Wave Functions • We denote the solutions of the SE as nlm (r,,) • the probability of finding the electron at some position (r,,) is P (r,,)= (nlm)2 dV where dV is the volume element • in spherical coordinates dV=r2drsindd R 2 volume of a sphere dV 0 0 0 R 2 0 0 R 2 2 2 sin d d 0 r dr sin d d 0 r dr 0 0 R3 4 2 2 R 3 3 3 Wave Functions • the ground state wave function is Cexp(-Zr/a0) where C is determined from normalization 2 2 dV 1 2 2 Zr / a 2 2 2 2 r dr sin d d C e r dr sin d d 0 0 0 0 0 0 0 C 2 4 (a0 / 4Z 3 ) 1 3 hence 100 1 Z a0 3/ 2 e Zr / a0 • Hence for the ground state, the probability density 1002 is independent of and and is maximum at the origin Wave Functions • What is the probability of finding the electron between r and r+dr? • in other words, in a spherical shell of thickness dr • volume of shell is 4r2 dr (area x thickness) • P(r)dr = (4r2 2)dr P(r) is the radial probability density Wave Functions • Maximum is at r = a0/Z = a0 for Hydrogen (first Bohr orbit) • not a well defined orbit but rather a cloud Probability density 200 C200 (2 Zr / a0 )e Zr / 2 a 0 210 C210 Zr Zr / 2 a0 e cos a0 211 C211 Zr Zr / a0 e sin e i a0