Quantum mechanics review

Reading for week of 1/28-2/1 – Chapters 1, 2, and 3.1,3.2 Reading for week of 2/4-2/8 – Chapter 4

Schrodinger Equation (Time-independent) where The solutions incorporate boundary conditions and are a family of eigenvalues with increasing energy and corresponding eigenvectors with an increasing number of nodes. The solutions are orthonormal.

Physical properties: Expectation values Dirac notation or bra-ket notation

Physical properties: Hermitian Operators Real Physical Properties are Associated with Hermitian Operators Hermitian operators obey the following: The value mn is also known as a matrix element, associated with solving the problem of the expectation value for A as the eigenvalues of a matrix indexed by m and n

Zero order models: Particle-in-a-box: atoms, bonds, conjugated alkenes, nano-particles Harmonic oscillator: vibrations of atoms Rigid-Rotor: molecular rotation; internal rotation of methyl groups, motion within van der waals molecules Hydrogen atom: electronic structure Hydrogenic Radial Wavefunctions

Particle-in-a-3d-Box x a V(x) V(x) =0; 0<x<a V(x) =∞; x>a; x <0 b  y ; c  z n x,y,z = 1,2,3,...

Particle-in-a-3d-Box x a V(x) V(x) =0; 0<x<a V(x) =∞; x>a; x <0 b  y ; c  z

Zero point energy/Uncertainty Principle In this case since V=0 inside the box E = K.E. If E = 0 the p = 0, which would violate the uncertainty principle.

Zero point energy/Uncertainty Principle More generally Variance or rms: If the system is an eigenfunction of then is precisely determined and there is no variance.

Zero point energy/Uncertainty Principle If the commutator is non-zero then the two properties cannot be precisely defined simultaneously. If it is zero they can be.

Harmonic Oscillator 1-d F=-k(x-x 0 ) Internal coordinates; Set x 0 =0

Hermite polynomials Harmonic Oscillator Wavefunctions V = quantum number = 0,1,2,3 H v = Hermite polynomials N v = Normalization Constant

Raising and lowering operators: Recursion relations used to define new members in a family of solutions to D.E.

Rotation: Rigid Rotor

Wavefunctions are the spherical harmonics Operators L 2 ansd L z

Degeneracy

Angular Momemtum operators the spherical harmonics Operators L 2 ansd L z

Rotation: Rigid Rotor Eigenvalues are thus: l = 0,1,2,3,…

Lots of quantum mechanical and spectroscopic problems have solutions that can be usefully expressed as sums of spherical harmonics. e.g. coupling of two or more angular momentum plane waves reciprocal distance between two points in space Also many operators can be expressed as spherical harmonics: The properties of the matrix element above are well known and are zero unless -m’+M+m = 0 l’+L+l is even Can define raising and lowering operators for these wavefunctions too.

The hydrogen atom Set up problem in spherical polar coordinates. Hamiltonian is separable into radial and angular components

n the principal quantum number, determines energy l the orbital angular momentum quantum number l= n-1, n-2, …,0 m the magnetic quantum number -l, -l+1, …, +l