I am currently in my second undergraduate quantum course and just finished studying the addition of angular momenta. I am also in my third abstract algebra course and am now covering product groups and group actions. In my QM book (griffiths) there was a reference made to group theory. it said "...
Homework Statement
Consider f(x) = x^3-5
and its splitting field K = Q(5^{1/3}, \omega)
where \omega = e^{2 \pi i/3}
Show that B = \{1, 5^{1/3}, 5^{2/3}, \omega, \omega 5^{1/3} , \omega 5^{2/3} \}
is a vector space basis for K over Q.
The Attempt at a Solution
I am just a bit...
Homework Statement
Show that if p is prime and e^{2 \pi i/p} is constructable
then p=2^k+1 for a positive integer k
Homework Equations
e^{i \theta} = Cos \theta + iSin \theta
The Attempt at a Solution
By definition, a complex number a+bi is constructible if a and b are...
Homework Statement
This is something ive been trying to prove for a bit today. My quantum mechanics book claims that the following two definitions about hermitian operators are completely equivalent
my operator here is Q (with a hat) and we have functions f,g
\langle f \mid \hat Q f \rangle...
Homework Statement
A particle moves in one dimension in the delta function potential V= αδ(x). (where that is an 'alpha' ... not 'a')
An initial wave function is given
\Psi = A(a^2-x^2) for x between -a and a and Psi=0 anywhere else
What is the probability that an energy measurement will...
Homework Statement
If two integers a and b are relatively prime and if each divides an integer n, prove that their product ab divides n
Homework Equations
1=sa+tb for some integers s and t (thm 1.35)
gcd(a,b)=1
n=aa'=bb'
The Attempt at a Solution
I have tried many many different ways...
I have come to a bit of a misunderstanding with partial derivatives. I will try to illustrate my problem. Say we have a function f(x, y(x), y'(x)) where y'(x)=dy/dx. Now suppose that f does not explicitly depend on x. My physics book says at this point that ∂f/∂x=0, even though y(x) and y'(x)...
Hello,
I am a math and physics double major and a junior at Michigan State University. I have already taken introductory physics, the first semster of classical mechanics, a course in thermodynamics and modern physics, and linear algebra (along with the basic calculus and diff eq courses) I...
Hello,
I will be taking two semesters of quantum physics this year, and am looking for a good book to read before the fall semester to give me somewhat of a head start. Two books I am currently considering are Vol 3 of the Feynman Lectures, and Principles of Quantum Mechanics by Paul Dirac...
Homework Statement
prove that p is prime iff for any integer "a" either (a,p)=1 or p divides a
(where (a,p) denotes the gcd of a and p)
Homework Equations
(a,b)= the greatest common factor of a and b
The Attempt at a Solution
I had no trouble with proving the forward direction...
Homework Statement
Prove the following:
If 5 divides a^2 + b^2 + c^2 then 5 divides a and 5 divides b and 5 divides c.
Homework Equations
5 \mid a \implies a=5k , k \in Z
The Attempt at a Solution
My idea is to assume 5 divides a^2 + b^2 +c^2
also assume that "5 does not divide a"...
Homework Statement
Suppose that the four inspectors at a film factory are supposed to stamp the expiration date on each package of film at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 60% of...
Homework Statement
Prove by induction on n that 11^n - 4^n is divisible by 7 for all natural numbers n (not including zero).
Homework Equations
Principle of mathematical induction.
The Attempt at a Solution
Of course I start by showing that the statement holds for n=1
11^1...
Homework Statement
Let V be a vector space and \{v_1,...,v_{n+1} \} \subset V a set of linearly independent
vectors of V . Show directly: (Don't just quote a theorem!)
(a) The set \{v_1,...,v_{n} \} is linearly independent.
(b) v_{n+1} \not \in span \{v_1,...,v_{n} \}
Homework...
There are seven stacks of coins, each consisting of 100 coins.
You know that real coins weigh 10 grams, but counterfeit coins weigh 11 grams.
You have an analytical scale that is accurate to .1 gram.
Each stack consists of either all real coins or all counterfeit coins, none of the stacks...
Describe a simple way to force a particle to move along two different circular trajectories at the same time. (Two circles that are not concentric) Hint: The trajectories do not have to be measured from the same frame of reference.
It need not be an elementary particle (could be even a small ball)
I just had a thought that seems somewhat significant, so I thought I'd share it.
The structure of most "arguments" that make up the flimsy backbone of pseudoscience very much remind me of the type of arguments that I, and all other children in my school, were encouraged to make in english...
Homework Statement
An accelerometer is made of a piece of wire with a bead on it that can slide on the wire with no friction. The wire is formed as a parabola y = kx2, as shown in the drawing. The bead rests at the lowest point of the parabola when it is at rest. When accelerated parallel to...
In my modern physics course we learned that electrons, when shot through two close together slits, will produce an interference pattern just light light. However, when you set up an experiment to observe which slit the electron goes through, there is no longer an interference pattern. To my...
I was wondering if anybody knows about the way in which science was studied in the past. For example, did the great scientists that we read about (Kepler, Galileo, Newton, Euler, Maxwell, Einstein, Dirac... etc) study Math and Physics in similar ways as we? Did they learn by solving example...
Homework Statement
I am having trouble with part B of this problem
N people, each of mass mp, stand on a railway flatcar of mass mc. They
jump off of one end of the flatcar with velocity u relative to the car. The car rolls in
the opposite direction without friction.
(a) What is the final...
Homework Statement
I have attached a copy of the homework as there is also a picture of the pump. This problem is number 6.
A centrifugal pump processes j = 18m3/min of water, with inlet and outlet
velocities of v = 15m/s. The impeller rotates in the clockwise direction around the...
Homework Statement
An object is ejected straight up into the air at an initial velocity v0.
(a) Determine the time for reaching the maximal elevation when the object is subject
to gravity alone.
(b) Determine the time for reaching the maximal elevation when the object is subject
to gravity...
Homework Statement
A sphere of radius 0.500 m, temperature 27.0°C, and emissivity 0.850 is located in an
environment of temperature 77.0°C. At what rate does the sphere (a) emit and (b) absorb
thermal radiation? (c) What is the sphere's net rate of energy exchange?
Homework Equations...
Homework Statement
Expand sin x about the point x = pi/4. Hint: Represent the function as
sin x = cos (y + pi/4) and assume y to be small
Homework Equations
Taylor Series Expansion
f(x)= f(a) + f'(a) (x-a)+ (1/2!) f''(a) (x-a)^2+ (1/3!) f"'(a) (x-a)^3+.....+ 1/n! f(n)(a) (x-a)^n
The...
Preparation for "Thermodynamics & Modrn Physics"
Hello,
I will be transferring to Michigan State University next semester (from Washtenaw Community College) and will be taking a course called "PHY 215: Thermodynamics & Modern Physics", which is actually in the Chemistry Building.
I have...
Homework Statement
I was looking through my notes, and I saw that my professor solved a propellor blade problem in a way that I don't understand. I took this class two semesters ago, so I cannot ask the professor himself. The problem that I am having is with his setup of a differential element...
Here's a little puzzle that I thought of earlier. I gave it to my roommate and he wasn't able to solve it without a pretty good hint, but I'm sure somebody here will get it.
|A|+|B| = 2
|A+B| = √(2)
What are A and B?
(The bars are absolute value signs, and the " √(2) " is the square...
Hello,
After this semester I will have completed Math up to Elementary Diff Eq, and both freshman level Calculus based Physics courses. I will be transferring to a university from community college next semester. I am currently taking 20 credit hours: Physics 2, Bio 101, Chem 1, Geography and...
Hello Everybody
I am a Physics Major transferring to Michigan State University next semester. I have completed Calculus based Physics 1 and 2, as well as Calculus 1,2,3, Linear Algebra, and Differential Equations. I am wondering which kind of math class some of you may recommend to take next...