Hi. I'm a math instructor and this problem is given to 1. year economics students.
1. The problem statement
In an episode of the TV show "All in the Family", Mike claimed that he could identify different brands of cola by taste alone. He was challenged and presented with three glasses, one...
I've been staring at this for hours. Any hints?
Let the vector Y = (Y_1,Y_2,\dots,Y_k) have a multinomial distribution with parameters n and \pi = (\pi_1,\pi_2,\dots,\pi_k):
\sum_{i=1}^{k}Y_i = n, \quad \sum_{i=1}^{k}\pi_i = 1
Show that the conditional distribution of Y_1 given...
Hi. I'm asked the following question:
What isomers does the complex [\text{Fe}(\text{H}_2\text{O})_6] give rise to? Is it optically active?
My answer to the first question would be 'none': The 6 identical H2O molecules are arranged in an octahedral fashion around the central Fe-atom, so...
Hi. I need help with (part of) this exercise:
c) By considering the differences in bonding determine which of the combustion reactions (in part a) I have written combustion reactions for the following hydrides: B2H6, CH4, NH3 and H2S) must be expected to have the lowest energy of activation...
Hi. I'm preparing for a chemistry exam and I have trouble with this former exam problem:
The solubility of I_2\text{(s)} in water at 25ºC is 0.0013 M.
Calculate the solubility of I_2\text{(s)} in a 0.1 M solution of KI(aq) by considering this equilibrium:
I_2\text{(aq)}+I^-\text{(aq)}...
Hi.
How come that Cs reacts more violently with water than Li, when the enthalpy of hydration for Li is -520 kJ/mol while it is only -276 kJ/mol for Cs. More energy is released in the Li reaction, but still it reacts more quietly?
(EDIT: Oops, maybe this should be in the homework section?)
Hi,
I was just reading up on some QM, and I was wondering this: under what circumstances can you treat a bunch of electrons as occupying "single particle states", and when do you have to use one wavefunction depending on all the coordinates, psi(r1, r2, r3, ...)? Hope you know what I mean...
Hi.
I'm reading a simple introduction to groups. A group is said to be a set satisfying the following axioms (called the 'group axioms'):
1) Associativity.
2) There is a neutral element.
3) Every element has an inverse element.
4) Closure.
My questions is simply: why are they...
Hi.
Can anyone explain to me what spin-up effects are? The context is a waterwheel that consists of an ordinary wheel with papercups with holes in the bottom suspended along the rim. It is stated that for this wheel there are two sources of damping: ordinary frictional damping and "inertial"...
Hi.
How can I "experimentally" (by way of computer simulation) calculate an approximate value for the dimension of a fractal object? The object in question is the Lorenz strange attractor, which has a dimension between 2 and 3.
Also, I know there is a number of different ways to define...
Hi.
I'm starting a project on chaos very soon and I was just wondering...
One of the distinguishing features of a "chaotic system" is the sensitive dependence on initial conditions. It is stated that if we knew the initial conditions with infinite precision we would also be able to...
Hi.
Can anyone explain the meaning of this formula from statistical physics to me:
S = -k\sum_r{p_r\ln p_r}
Ok, I know that S is the entropy, the p's are probabilities of some sort - but somehow this is not satisfactory :-)
Hi.
I'm reading a book on deterministic chaos, and an important distinction seems to be the one between dissipative and conservative dynamical systems. A dissipative system is defined as a system whose "phase space volumes shrink", whereas in a conservative system phase space volumes are...
Hi,
A very hot gas is enclosed in an oven with a small window. The gas molecules emits radiation at a characteristic wavelength. I assume that because of the thermal motion of the molecules the emitted wavelengths will form a spectrum of some kind (Doppler broadening.) I am trying to derive...
Hi.
I'm having trouble with this statistical physics thing again. I am given this exercise:
Problem 9 – A spin model
In a solid at temperature T the atoms have spin 1 so that the m quantum number takes on the values m = 0, ±1. Due to an interaction with the electrostatic field in the...
Hi. I've just started a course on statistical physics and the first assignment is this:
A system possesses 3 energy levels, E_1 = \epsilon, E_2 = 2\epsilon and E_3 = 3\epsilon. The degeneracy of the levels are g(E1) = g(E3) = 1, g(E2) = 2. Find the heat capacity of the system.
I've...
Hi,
I was just reading up on some astrophysics and I saw the line element (general relativity stuff) written in spherical coordinates as:
ds^2 = dr^2 + r^2(d\theta^2 + \sin\theta\d\phi)
I don't get this. dr is the distance from origo to the given point, so why isn't ds^2 = dr^2 without...
Hi,
I have yet another problem with this diffraction thing. I have little clue on these questions:
e) The radius of the O^2- ion is assumed to be 0.126 nm. By x-ray diffraction experiments the dimensions of the unit cells for MgO, CaO, SrO and BaO has been determined to be 0.4213 nm...
Hi,
In the solid form FeO, CoO and NiO all has the NaCl-structure (simple cubic). In a series of diffraction experiments with x-rays (\lambda = 0.15406~\text{nm}) one found reflexes from the (111), (200) and (220)-planes with the following \theta-values (\theta is the angle in Bragg's law...
Hi,
How do one calculate the pH-value of a solution that does not contain water, pH being equal to -log([H3O+])?
I need to calculate the pH-value of a solution where BaCO3 and SrCO3 is dissolved in acetic acid (4 M). The only information I am given is, that half of the acetic acid is...
Hi,
I'm doing this chemistry report, and I have absolutely no experience in chemistry.
We have this solution containg (among some other stuff) Ba^2+, Sr^2+ and Ca^2+ ions along with SO4^2-ions in excess. Now we add some Na2CO3 to transform the sulphates to carbonates. They ask me these...
Hi,
How can you infer from these equations,
a = b_{max}(b_{max}+\hbar) \quad \text{and} \quad a = b_{min}(b_{min}-\hbar),
that b_{max} = -b_{min}? It is used in the derivation of the angular momentum eigenvalues...
I'm reading in Sakurai's 1st chapter that this follows from the "associative axiom":
\langle\beta|\cdot\left(X|\alpha\rangle\right) = \left(\langle\beta|X\right)\cdot|\alpha\rangle
so we might as well write \langle\beta|X|\alpha\rangle. I know this is basic stuff, but I thought this...
Hi.
Can anybody help me with this exercise in Introductory Nuclear Physics:
One of the magic numbers for nuclei is 28. How many stable isotopes exist with N = 28. (N being the number of neutrons)
I have no idea on how to solve this.
Thanks.
Hi
I have to draw a Feynman diagram for the following reaction:
K^{*+} \rightarrow K^0 + \pi^+
The K^{*+}-meson is composed of an u and an anti-s quark, the K^0-meson is composed of a d and an anti-s quark, and the \pi^+-meson is composed of an u and an anti-d quark. I have drawn...
Okay, I've read and re-read the section on tensor operators and the Wigner-Eckart theorem in Sakurais book, but I'm still confused. Could anyone explain to me how to think about vector and tensor operators and the significance of the Wigner-Eckart theorem? :confused:
Thanks.
We have a particle in a harmonic oscillator potential. The eigenstates are denoted {|0>,|1>,...,|n>,...}. Initially the particle is in the state |s> = exp(-ipa)|0>, where p is the momentum operator.
I need to find <x> as a function of time using the Heisenberg picture. The problem is, how do...
I have some basic questions concerning operators. What is actually meant by the following:
1) The dimensionality of an operator? E.g., what does it mean to say that the operator K has the dimension of 1/length (an example from Sakurai's book)? Operators act on abstract mathematical states to...
Okay, I have this function defined as an infinite series:
f(x) = \sum_{n=1}^{\infty}\frac{\sin(nx)}{x+n^4}
which is converges uniformly and absolutely for x > 0. I have shown that f is continous and has a derivative for x > 0. Now I have to show that f(x) \rightarrow 0 as x \rightarrow...
I really need help with this exercise. Consider the power series
\sum_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{2n+1}.
for z\in\mathbb{C}.
I need to answer the following questions:
a) Is the series convergent for z = 1?
This is easy; just plug in z = 1 and observe that the alternating...