Sorry, I don't know about US, since I'm from Denmark, where we don't have ph.d.-programmes as such. Here a ph.d. student is hired by a research-group to do a specific researchproject and choose their own courses, usually in the form of studygroups.
Working with a variety of different physicists I've seen many approaches, as many as I've seen physicists I suppose. Some start from curiosity over a cool experiment, others spend hours and hours discussing the actual meaning of some equation, and then there are some who seem to rely on luck and...
Yes, x is real. But since the integrand goes to zero for x\rightarrow \infty the direction of integration in the complex plane shouldn't alter the integral...
I think you would enjoy something like Lab-on-a-chip research, where you will need all your skills. But in general everything within nanoscience could be interesting for you.
Good luck.
Here's an integral that is currently giving me grey hairs:
\int_0^{\infty} \frac{1}{x} \exp(i \frac{k}{x}(a-c \cos(\theta + wx))) dx
I've tried different approaches like contour integration around x=0 and replacing the exponential by its Taylor sum to have:
\int_0^{\infty}...
Phonons are the vibrational modes of the crystal lattice. The mechanism for inelastic scattering of neutrons may be very simply described as this: When the neutron hits the nucleus of an atom in the crystal, energy may be transferred from the neutron to the nucleus and it will start oscillating...
I'm just learning this stuff myself, but I think residues are wrong, since the closed contour is the full contour and the small circle is open, and coming from above and below does not necessarily lead you to the same point, since the function may be multivalued. I believe you need to do the...
If the wavefunction is interpreted as a (time-evolving) probability distribution for the position of the particle, then immidiately after the measurement the wavefunction is restricted to the state corresponding to the measured eigenvalue. The shape of this wavefunction will depend strongly on...
Write: \int_{lower}^{upper}f(x)dx between sets of square brackets with itex and /itex respectively or between <backslash>begin{equation} and <backslash>end{equation}. The former gives
\begin{equation}
\int_{lower}^{upper}f(x)dx
\end{equation}
in-line, the latter creates a formula on a new line.
Hmm, I dindn't write that mol is defined in mass units, but from it... Anyway, we obviously agree on what 1 mol and 1 u are, so maybe we shouldn't confuse the boy further by pedantically contradicting each other.
Maybe you could be more specific about what you need it for? Someone here can probably provide you with exactly the number you need in units you are familiar with.
(But please, keep asking questions of this sort anyway!)
Ok, so for
\begin{equation}
\frac{1}{\sqrt{A}} \int_{-\sqrt{A}B}^{\infty} dx \: e^{-ix^2}
\end{equation}
with \sqrt{A}B large and positive we may extend the limit to -\infty and obtain \sqrt{\frac{\pi}{A}} e^{-i\frac{\pi}{4}}, and for \sqrt{A}B\approx 0 we get half of that, but what...
Hi! How do I approximate the integral
\begin{equation} \int_0^{\infty} dt \:e^{-iA(t-B)^2} \end{equation}
with A, B real, A > 0, and B=b \cos\theta where 0 \leq \theta < 2\pi?
I guess for B\ll 0 the lower limit may be extended to - \infty to yield a full complex gaussian integral, but what...