Quantum Mechanics and Atomic Physics
Notable facts about fn
1. If a particle is in a stationary state which is taken to be real, <p> = 0 2. If the potential is even then fn is either even of odd 3.E>Vmin has to be true for the function to be normalizable
Representations of functions in a given basis
<x|f> = f(x) representation of f in x basis <g|f> = inner product of the functions f and g
Hermitian operator
A=A^t All their eigenvalues are real All of their eigenfunctions are orthogonal
3D Harmonic Oscillator
E = (n1 + n2 + n3 +3/2)hbarw
Energies of hydrogen like atoms
E = Z^2 (13.6eV)/n^2
Hydrogen atom energy
En = -hbar^2/2ua^2n^2 = ue^4/2k^2hbar^2n^2 GROUND STATE = -13.6 eV, rest scale as u/n^2 (u varies for hydrogen like atoms)
Virial Theorum
For a harmonic oscillator, the kinetic and potential energy are 1/2 total energy.
Harmonic oscillator Hamiltonian in terms of raising and lowering operators
H = (a^t*a - 1/2)hbar w
The Hydrogen Atom Hamiltonian
H = -hbar/2u(del)^2 - ke^2/r Potential between an electron and a proton. u is reduced mass of electron. k on all points here is 1/4piepsilon
Angular momentum operators
L = rxp still, break it up into its components [Li,Lj] = ihbarLk*permutator(ijk)
Eigenvalues of Lz, L^2
L^2|l,m> = l(l+1)hbar^2|l,m> Lz|l,m> = mhbar|l,m>
Delta function
Only admits one bound state with Psi = sqrt(mA)/hbar e^-(mA|x|/hbar^2); E = -mA^2/2hbar^2
Free particle
Psi = e^+-kx; E = hbar^2k^2/2m = hbar*w note w is quadratic in k, where for a classical wave w = ck Just E = P^2/2m with de Broglie formula
Hydrogen atom wavefunction
Psi ~ e^-r/a for the unnormalized ground state wavefunction GROUND STATE ENERGY IS -13.6 eV
Solutions to time dependent Schrodinger equation will be linear combinations of
Psi(x,t) = e^-iEnt/hbar * fn(x)
Infinite square well
Psin = sqrt(2/a) sin(npix/a); En = n^2pi^2hbar^2/2ma^2 START COUNTING FROM n=1
Transmission and Scattering
R = |B|^2/|A|^2; T = |C|^2/|A|^2 R + T = 1 Reflection and transmission probabilities are the same for a delta well and a delta barrier
Hydrogen atom facts
Wavefunction is 0 at the origin for functions with l = 0, but the probability of being found at the origin increases as l increases Fine-structure constant alpha ~ 1/137, characterizes the quantum mechanical strength of electromagnetic interactions
Commutator relation
[AB,C] = A[B,C] + [A,C]B
Bohr Radius
a = hbar^2/uke^2, scales as 1/u To find the Bohr radius for Hydrogen-like atoms, adjust the reduced mass u. Positronium has u = me/2, Hydrogen as u ~ me so the Bohr radius of positronium is about twice that of Hydrogen
Raising and lowering normalization factors
a^t|n> = sqrt(n + 1)|n + 1> a|n> = sqrt(n)|n-1>
Measurements
cn = int(fn(x)*Psi(x)) dx where fn are basis functions of Psi Expectation value of a measurement: <A> = summ(lambda_k |c_k|^2) |c_K|^2 is probability
Rydberg formula for frequency of light emitted from hydrogen atom
f ~ 1/nf^2 - 1/ni^2 derived from E ~ f where ~ means proportional
Schrodinger equation
ihbar d/dt(Psi) = H(Psi) = E*Psi where H = p^2/2m + V(x) Eigenvalues of H are all possible values of energy and the eigenfunctions are fn
de Broglie formula
p = hbark = h/lambda
Generalized uncertainty principle
sigmaA^2*sigmaB^2 >= (1/2i<[A,B]>)^2 Recall [x,p] = ihbar => dxdp ~ dEdt ~ hbar
Position and momentum operators
x = x; p = -ihbar(d/dx) in x space p = p; x = ihbar(d/dp)