Gaussian integrals
Calvin W. Johnson
January 22, 2014
1
The basic Gaussian and its normalization
The Gaussian function or the normal distribution,
exp x2 ,
(1)
is a widely used function in physics and mathematical physics, including in
quantum mechanics.
Complex numbers and Eulers formula
Calvin W. Johnson
January 22, 2014
1
Complex arithmetic
I wont start from the very beginning, but assume some familiarity with this
topic.
Complex numbers are very useful in mathematics and physics. They are of
the form
Physics 610A
HW #4 due March10, 2014
Shankar problems 1.10.3, 1.10.4, 4.2.1
plus
Johnson 4.1: Expand the eigenvectors of Sy, |Sy = +1/2 > and |Sy = -1/2 >, in terms
of the eigenvectors of Sx, |Sx = +1/2 > and |Sx = -1/2 >. (The matrices Sx , Sy can be
fou
Physics 610A
Feb 12, 2014
HW #3 due Feb 26, 2014
Shankar problems 1.8.10, 1.9.2, 1.10.1, 1.10.2 plus
0 1
Johnson 3.1 Consider the matrix H =
1 0 . Find its eigenvalues and
eigenvectors. (There will be only two.) Show explicitly that the eigenvectors a
Physics 610
February 5, 2014
HW #2 due Feb 17, 2014
Shankar problems 1.6.2, 1.6.3, 1.6.6, 1.7.1, 1.7.2 (you may assume and use the general property
that det AB = det A det B), 1.8.2, 1.8.3, 1.8.4, 1.8.5, 1.8.6, 1.8.7
Dimensional analysis, an introduction
Calvin W. Johnson
January 22, 2014
1
Dimensional analysis
Dimensional analysis is a powerful and important tool in calculations. In fact,
you should make it a habit to automatically use dimensional analysis
to check n
Dimensional analysis in checking integrals
Calvin W. Johnson
January 22, 2014
Previously Ive written about the importance of paying attentions to the
dimensionality of your answer. Less obvious is that one can also use dimensionality as a partial check on
Orthonormality, completeness, and projection
Calvin W. Johnson
January 22, 2014
Unitary matrices, that is, matrices which satisfy
U1 = U ,
(1)
are ubiquitous and important in quantum mechanics, in particular because they
have some useful properties well d
A quick tour of common household matrices and
their uses
Calvin W. Johnson
January 22, 2014
The following denitions are very important to know by heart. This material
is also covered in the Appendix to Griths.
Matrices are two-dimension arrays of numbers:
The Dirac -function
Calvin W. Johnson
January 22, 2014
You already know the Kronecker delta for discrete indices:
1 if m = n,
.
0 if m = n
mn =
(1)
The analog for continous variables such as position x is the Dirac -function,
(x). (Technically, it is not
Dierentiating under the integral, an alternative
to integration by parts
Calvin W. Johnson
January 22, 2014
1
The basic idea
When we learn denite integrals, one of the tools is integration by parts. Here I
introduce an alternative which can be applied to
Physics 610
Spring 2014
January 29, 2014
HW #1 due Feb 5, 2014
Shankar problems 1.1.3, 1.1.4, 1.1.5, + the following
1 + i
1
1
Johnson 1.1: Consider the following vectors: V1 = 2 , V2 = 2i , and V1 = 2
3 i
3
3
(a) What is the length o