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1. Splitting a single light source into two out-of-phase sources?

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...
2. HPLC Question

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...
3. Optics question - laser power sources

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?
4. Integral Notation Question

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.
5. A wiki for equations

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...
6. Physics History

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...
7. Any good books on how to do research?

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...
8. On/Off Ratio

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!
9. Research: Applied vs. Fundamental

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...
10. Help with Eigenvalues problem

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...
11. Finding eigenvalues of matrix

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...
12. Linear Algebra (Similar Matrices)

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...
13. Linear Algebra (Change of Basis)

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(...
14. Another Linear Transformation

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)...
15. Linear Transformation

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(...
16. Linear Algebra (Span)

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...
17. Basic Linear Algebra Proof

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...
18. Linear Algebra (Projection)

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...
19. Complex Inner-Product Space

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...
20. Orthonormal Basis

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...
21. Inner Product

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...
22. Cross Product

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...
23. Linear Transform

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...
24. Linear Algebra (Meaning of Rank)

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...
25. Linear Algebra (Transform a Basis Set)

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...
26. Linear Algebra (Subspaces)

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...
27. Linear Algebra (Linear Transformation)

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...
28. Linear Algebra (Linearly Independent vs. Linearly Dependent)

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...
29. Optics - I have no clue where to start

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...
30. Time-Dependent Perturbation Theory - Two-level System

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}...