Some Practice with Commutators
Here are some exercises to practice using commutators. (These are not required, and are not for creditdont turn in answers, though you can ask if
you have questions.)
General advice: if you want to caldulate a commutator of
Some Practice with Indices
Here are some exercises to practice using indices, the summation convention,
and the and tensors. (These are not required, and are not for credit
dont turn in answers, though you can ask if you have questions.)
1. Show that im v
Some Useful Facts for the Physics 115B Final Exam
Schrdinger Equation: i = H
o
h t
Some operators: pi =
h
i ri
Hamiltonian: H =
p2
2m
(ri and xi mean the same thing)
2
h
+ V (x) = 2m
2
+ V (x)
Li =
[i , pj ] = i ij
r
h
ijk rj pk
Adjoint: |A = A | ; A
A little bit of vector and tensor math:
Indices: cfw_i, j, k, l, . . . take values 1, 2, 3,
with x1 = x, x2 = y, x3 = z
Indices label the components of a vector v: v1 = vx , v2 = vy , v3 = vz
Einstein summation convention:
when an index appears twice in
Some Useful Facts for the Physics 115B Midterm
Schrdinger Equation: i = H
o
h t
Some operators: pi =
h
i ri
Hamiltonian: H =
(ri and xi mean the same thing)
p2
2m
2
h
+ V (x) = 2m
2
+ V (x)
Li =
[i , pj ] = i ij
r
h
ijk rj pk
Adjoint: |A = A | ; A is
Detection of Fluorescence from Single Molecules
8
LA B O VE RV IE W
The goal of this lab is to align and use an optical detection system that excites fluorescent
molecules with laser light and detects the resulting emission with single fluorophore resolut