I've been contemplating an experiment on a 1-dimensional system which would require a single light source to be split into two separate sources which are exactly out of phase with each other. The idea is to focus the beams near each other (but not overlapping) along the 1D object. Any phonons...
I have a question about https://www.amazon.com/dp/B002RS5IKA/?tag=pfamazon01-20 product. It is an autosampler shell vial with a polyethylene plug style needle closure. Does anyone know whether the plug on this vial will be resealable? In other words, will it be able to withstand multiple...
We have a 532 nm laser that we use with our Raman tool. The only power indicator on the control is in the form of a percentage. I know that at 100%, the laser has a power of 25 mW. Does this scale linearly with power percentage? In other words, does 50% power translate to 12.5 mW?
I saw this notation in a textbook but I'm a bit confused by it. What exactly does the 3 mean? Is it shorthand for a triple integral in spherical coordinates?
\int{d^3 r}
Thanks in advance.
A colleague of mine just started a website called wikiquation.com. It's a searchable wiki-style database for equations. The database is currently quite empty, so it's not very useful as a search engine yet, but it allows anyone to add or edit an equation without creating a user account. I think...
One of my professors told me a story about a physicist and I can't remember who it was. It's driving me crazy. This is what I remember about the story.
Apparently this physicist was a new faculty member and he had a seemingly brilliant plan for some experiment. All of the older faculty members...
I'm looking for a book that gives advice on how to research as a professional scientist. I am looking for insight into the culture of industrial research, applying for grants and obtaining funding, methods and techniques to use time your time effectively, etc. I'd prefer something from a physics...
Research-related question:
What is the typical on/off ratio of a silicon transistor? This seems like basic information, but I can't find it anywhere.
Also, how high does the on/off ration need to be in order for transistors (say, in a logic circuit) to function correctly?
Thanks!
I'm graduating with a B.S. in physics and I will soon be starting a Ph.D. program in nanoscale science, which is essentially an applied physics program. I chose this program because my undergraduate research was focused in this area and I've really enjoyed it.
However, I'm wondering if my...
Homework Statement
True/False
If Ttheta is a rotation of the Euclidean plane R2 counterclockwise through an angle theta, then T can be represented by an orthogonal matrix P whose eigenvalues are lambda1 = 1 and lambda2 = -1.
Homework Equations
The Attempt at a Solution
Just checking to...
Homework Statement
Find the eigenvalues of the following matrix:
\left(
\begin{array}{ccc}
1 & 0 & -3 \\
1 & 2 & 1 \\
-3 & 0 & 1
\end{array}
\right)
Homework Equations
The Attempt at a Solution
I think I'm forgetting a basic algebra rule or something. I know there are supposed to be...
Homework Statement
Write the standard matrix representation TE for T.
Homework Equations
T\left(ax^3+bx^2+cx+d\right)=(a-b)x^2+(c-d)x+(a+b-c)
The Attempt at a Solution
[T]_E=\left(
\begin{array}{c}
a+b-c \\
c-d \\
a-b \\
0
\end{array}
\right)
I think this is right, but a...
Homework Statement
Let E={1, x, x2,x3} be the standard ordered basis for the space P3. Show that G={1+x,1-x,1-x2,1-x3} is also a basis for P3, and write the change of basis matrix S from G to E.
Homework Equations
The Attempt at a Solution
Here's what I got:
S_E^G=\left(...
Homework Statement
This is a slight variation of the last problem I posted.
Write the standard matrix representation for T1T2 and use it to find [T1T2(1,-3,0)]E.
Homework Equations
T_1\left(x_1,x_2,x_3\right)=\left(x_3,-x_1,x_3\right)...
Homework Statement
Write the standard matrix representation for T1 and use it to find [T1(1,-3,0)]E.
Homework Equations
T_1\left(x_1,x_2,x_3\right)=\left(x_3,-x_1,x_3\right)
The Attempt at a Solution
I just wanted to check to see if I am doing this right. Thanks in advance!
A=\left(...
Homework Statement
Prove that B={(1,-1,-1),(1,0,1),(0,-1,1)} spans R^3.
(Actually, the problem asks to show that B is a basis for R^3. This would require that I prove linear independence AND that it spans R^3. The first is easy, but I'm not sure about the second.)
Homework Equations
None...
Homework Statement
Prove the following:
For every n × n complex matrices A and B, (\alpha AB)^*=\bar{\alpha }B^*A^*.
Homework Equations
None
The Attempt at a Solution
Okay, I'm just getting started on this problem. All the ideas I have come up with so far involve using two...
I apologize for the excessive use of Latex, but for this particular problem I think the notation would be extremely difficult to read otherwise. I usually try to keep my use of Latex to a minimum.
Homework Statement
\text{Let }
\mathbb{C}^3
\text{ be equipped with the standard inner product...
Homework Statement
In an arbitrary complex inner-product space V which of the following is not always true?
a. <αu + βv, w> = α<u, w> + β<v, w>
b. |<u, v>|2 ≤ <u, u> <v, w>
c. <u, αv> = α<u, v>
d. <0, u> = 0
Homework Equations
None
The Attempt at a Solution
The...
Homework Statement
True/False:
The set of vectors B={(-1,-1,1,1),(1,0,0,0),(0,1,0,0),(-1,-1,1,-1)} is an orthonormal basis for Euclidean 4-space \mathbb{R}^4.
Homework Equations
None
The Attempt at a Solution
I said false because \langle (-1,-1,1,1),(-1,-1,1,1) \rangle =2\ne1, which shows...
Homework Statement
Prove that if u and v are given non-zero vectors in the arbitrary inner-product space V, and are such that <u,v>=0, then {u,v} is a linearly independent subset of V.
Homework Equations
The Attempt at a Solution
I have no idea where to start. It's hard to prove because the...
Homework Statement
Which of the following functions is not an inner product in \mathbb{R}^2, where u=(a, b), v=(c,d)?
a. <u,v> = ac+bd
b. <u,v> = 4ad+7bc
c. <u,v> = ac/2+bd/3
d. <u,v> = 2ac+9bd
Homework Equations
None
The Attempt at a Solution
u x v = det(u v) = ad-bc
I...
Homework Statement
Prove the following:
The action of a linear transformation T:U\rightarrow V is completely determined by its action on a basis B=\left\{u_1,u_2,\text{...},u_n\right\} for the domain U.
Homework Equations
None
The Attempt at a Solution
Okay, I feel like my solution is...
Homework Statement
True or False:
If A is an n x n matrix, then the rank of A equals the number of linearly independent row vectors in A.
Homework Equations
None
The Attempt at a Solution
Okay, I know this is a ridiculously easy question, but I'm wondering if there is a catch...
Homework Statement
If T:U\rightarrow V is any linear transformation from U to V and B=\left\{u_1,u_2,\text{...},u_n\right\} is a basis for U, then the set T(B)=\left\{T\left(u_1\right),T\left(u_2\right),\text{...} T\left(u_n\right)\right\}
a. spans V.
b. spans U.
c. is a basis for V...
Homework Statement
Okay, this is the last True/False question I will post.
True or False:
\text{The set of all solutions to the }m\times n\text{ homogeneous system of equations }Ax=0\text{ is a subspace of }\mathbb{R}^m.
Homework Equations
None
The Attempt at a Solution
I...
Homework Statement
True or False:
T(x,y)=(2x+5y,-x+2)\text{ is a linear transformation from }\mathbb{R}^2\text{ to }\mathbb{R}^2.
Homework Equations
None
The Attempt at a Solution
I thought the answer was true, but the correct answer is false. Here is my reasoning for true...
Homework Statement
If S = {v1, v2, v3} where v1, v2 and v3 are in \mathbb{R}^2, then the vectors v1, v2 and v3 are linearly independent.
Homework Equations
None
The Attempt at a Solution
I thought the answer was true, but I know the correct answer is false and I'm not sure why. Here was my...
Homework Statement
See attached.
Homework Equations
See attached.
The Attempt at a Solution
I have no idea where to start. I've been banging my head against this problem for 2 hours now. I just don't see how on earth I can find theta with respect to x when it appears nowhere in the...
Homework Statement
See attached. The problem is labeled "Peatross 1". Don't worry, it's short. I just didn't feel like retyping it.
Homework Equations
Included in attempt.
The Attempt at a Solution
I'm not sure if I am doing this correctly, but here it goes.
I'll just do it for H'_{10}...