I am struggling with question about possible outcomes of momentum measurement and their probability. I know I can calculate it with momentum operator, but a wavefunction is of form
$$\psi (x)=3\cos\pi x+\cos3\pi x$$
and I am unsure how to deal with it, as the derivative consists of sines.
I know that $\cos{kx}=\frac{e^{ikx}+e^{−ikx}}{2}$ and $p_{x}=\hbar k$, but does it mean that the momentum is sum of eigenvalues of individual exponentials?
The letters of a message are converted to Morse code, with a space between each pair of letters and two spaces between each pair of words. Each pair of elements (for example dot-space, dash-dash, and so on) is then assigned a number from 1 to 9. A message written in this system is as follows:
512537293228231463737325742346428147
What pair of elements corresponds to each number, and what does the message say?
Hint 1:
Start by analysing 2282 .
Answer
The secret message and digit correspondences are . . .
W E L L B E G U N I S H A L F D O N E
.-- . .-.. .-.. / -... . --. ..- -. / .. ... / .... .- .-.. ..-. / -.. --- -. .
.--/./.-../.-..//-..././--./..-/-.//../...//..../.-/.-../..-.//-../---/-././
5 1 2 5 3 7 4 2 9 3 2 2 8 2 3 1 4 6 3 7 3 6 3 3 7 1 5 3 7 5 2 9 3 9 8 9 2 2
-/ ./ .. -. .- // /. -- /-
1 2 3 4 5 6 7 8 9
This puzzle is built on a superb premise. Morse code’s natural 9 dot /dash /space pairs may be recoded as digits for a perfect combination of encryption and clue-idity. Morse code’s statistical skew provides a nicely shaped search space, like slopes and valleys that concentrate hunches.
The puzzle needed minor rebuilding. Unfortunately the original encryption contained imperfections. Fortunately M Oehm figured out (!) the intended message computationally. Unfortunately that message’s encryption has an odd number of dots /dashes /spaces, so the rules need to say that the message ends with a space.
Revised encryption and rules. The 38 digits of 51253742932282314637363371537529398922 represent a combination of 76 dots, dashes and spaces. Every Morse code letter is followed by a space while an additional space separates each pair of words. For example, Morse code for LIKE THIS would be .-../../-.-/.//-/..../../.../, where each / represents a space.
Solution begins by doubling the encoded digits and profiling a statistically likely solution.
5511225533774422993322228822331144663377336633337711553377552299339988992222
-/-/-/././/../../../-./.-/-.//.../.-./-../---/---//..--/..-./.-../.-../-..-/
The -/-/-/ . . . /-..-/ profile has approximately 5 letters per word, approximately equal counts of letter lengths, and favors . over - as is statistically expected. Here is the distribution of code digits and, without elaboration, likelihoods of their counts if they were specific dot /dash /space pairs.
Pair: .. -. .- -- ./ /. /- -/ //
Expected: 5.5 4 3.5 2.5 6.5 6 4 4 1.5
Actual code digits Probabilities
3 3 3 3 3 3 3 3 3 .05 .01 .005 .0005 .09 .07 .01 .01 .00001
(last) 2 2 2 2 2 2 2 2 .09 .03 .01 .002 .13 .11 .03 .03 .00008
(first) 5 5 5 5 .16 .21 .14 .07 .11 .13 .21 .21 .05
7 7 7 7 .16 .21 .14 .07 .11 .13 .21 .21 .05
9 9 9 9 .16 .21 .14 .07 .11 .13 .21 .21 .05
1 1 1 .11 .20 .22 .22 .06 .08 .20 .20 .13
4 4 .05 .14 .18 .26 .02 .04 .14 .14 .26
6 6 .05 .14 .18 .26 .02 .04 .14 .14 .26
8 8 .05 .14 .18 .26 .02 .04 .14 .14 .26
Morse code within-word statistics. Strongly skewed in favor of dots. (Derived from Wikipedia)
Frequencies of pairs: .. 15.9% -. 10.5% /. 17.1% ./ 18.0%
.- 9.7% -- 7.4% /- 11.1% -/ 10.3%
Letters grouped by beginnings Letters grouped by endings Total by length
---------------------------------- ---------------------------------- ---------------
E /. 12.7% T /- 9.1% E ./ 12.7% T -/ 9.1% 1 21.8%
A /.- 8.2% M /-- 2.4% I ../ 7.0% A .-/ 8.2% 2 24.3%
I /.. 7.0% N /-. 6.7% N -./ 6.7% M --/ 2.4%
R /.-. 6.0% D /-.. 4.3% D -../ 4.3% K -.-/ 0.8% 3 32.0%
S /... 6.3% G /--. 2.0% G --./ 2.0% O ---/ 7.5%
U /..- 2.8% K /-.- 0.8% R .-./ 6.0% U ..-/ 2.8%
W /.-- 2.4% O /--- 7.5% S .../ 6.3% W .--/ 2.4%
F /..-. 2.2% B /-... 1.5% B -.../ 1.5% J .---/ 0.2% 4 22.0%
H /.... 6.1% C /-.-. 2.8% C -.-./ 2.8% Q --.-/ 0.1%
J /.--- 0.2% Q /--.- 0.1% F ..-./ 2.2% V ...-/ 1.0%
L /.-.. 4.0% X /-..- 0.1% H ..../ 6.1% Y -.--/ 2.0%
P /.--. 1.9% Y /-.-- 2.0% L .-../ 4.0% X -..-/ 0.1%
V /...- 1.0% Z /--.. 0.1% P .--./ 1.9%
Z --../ 0.1%
------- ------- ------- -------
/. 60.7% /- 39.3% ./ 63.6% -/ 36.4%
Solution proceeds by testing most-likely matches. The clear place to begin is with the 22 at the end. Not only must it end with a / space, but it is also so numerous that the most-likely-of-all ./ pair is the best candidate.
code
|
| substitution
| |
| | 5511225533774422993322228822331144663377336633337711553377552299339988992222
2 ./ 5511./55337744./9933././88./33114466337733663333771155337755./9933998899././
| e e
result
The next code to assign is the most-numerous 3, and the statistically favored candidate is /.
5511225533774422993322228822331144663377336633337711553377552299339988992222
2 ./ 5511./55337744./9933././88./33114466337733663333771155337755./9933998899././
3 /. 5511./55/.7744./99/.././88.//.114466/.77/.66/./.771155/.7755./99/.998899././
^^ i e ^^ // ^^^^^^
998899 means that 88 or 99 must contain a / space.
/i/e/88.// prevents anything meaningful if 88 contains a / space.
/99/i/e prevents 99 from being // or -/
So the next assignment to try is 9 = /- ( ./ and /. are already taken)
5511225533774422993322228822331144663377336633337711553377552299339988992222
2 ./ 5511./55337744./9933././88./33114466337733663333771155337755./9933998899././
3 /. 5511./55/.7744./99/.././88.//.114466/.77/.66/./.771155/.7755./99/.998899././
9 /- 5511./55/.7744.//-/.././88.//.114466/.77/.66/./.771155/.7755.//-/./-88/-././
//t e n e
//t/-88/n/e cannot be a word. ^^^^^^^^^^^^^^
So much for 3 being /.. The next most likely candidate for 3 is ...
5511225533774422993322228822331144663377336633337711553377552299339988992222
2 ./ 5511./55337744./9933././88./33114466337733663333771155337755./9933998899././
3 .. 5511./55..7744./99..././88./..114466..77..66....771155..7755./99..998899././
^^^^^^^ ^^^ e
/99..99 means that 99 must contain a / space.
99./e/ makes /- the most likely candidate for 99. ("ne" vs. "ee" or "ie")
5511225533774422993322228822331144663377336633337711553377552299339988992222
2 ./ 5511./55337744./9933././88./33114466337733663333771155337755./9933998899././
3 .. 5511./55..7744./99..././88./..114466..77..66....771155..7755./99..998899././
9 /- 5511./55..7744.//-..././88./..114466..77..66....771155..7755.//-../-88/-././
// b e h // d ^^^ n e
From here the solution unravels neatly enough without further narrative.
8 -- 5511./55..7744.//-..././--./..114466..77..66....771155..7755.//-../---/-././
// b e g ^^ h ^^ ^^ // d o n e
7 /. 5511./55../.44.//-..././--./..114466../...66..../.1155../.55.//-../---/-././
// b e g ^^ h // d o n e
6 // 5511./55../.44.//-..././--./..1144//../...//..../.1155../.55.//-../---/-././
// b e g // i s // h ^^^^ ^^ // d o n e
1 -/ 55-/./55../.44.//-..././--./..-/44//../...//..../.-/55../.55.//-../---/-././
e // b e g u ^^// i s // h a // d o n e
4 -. 55-/./55../.-..//-..././--./..-/-.//../...//..../.-/55../.55.//-../---/-././
5 .- .--/./.-../.-..//-..././--./..-/-.//../...//..../.-/.-../..-.//-../---/-././
w e l l // b e g u n // i s // h a l f // d o n e
So how did the actual statistics compare to predictions? Very well! Only two digits, representing .. and -., varied much from expectation and created just one false lead.
Pair Predicted Actual
Digits Morse
./ 17% 8 21% 15 20%
/. 16% 4 11% 12 16%
.. 15% 9 24% 14 19%
/- 11% 4 11% 6 8%
-. 11% 2 5% 9 12%
-/ 11% 3 8% 4 5%
.- 9% 4 11% 7 9%
-- 7% 2 5% 4 5%
// 4% 2 5% 4 5%
This may seem trivial, but I'm having some trouble deriving the finite difference form of the heat equation with a thermal conductivity function $a(x)$ depending on $x$:
$$\frac{\partial u(x, t)}{\partial t} = \frac{\partial }{\partial x}[a(x)\frac{\partial u(x, t)}{\partial x}]$$
I find some trouble deriving the right hand side. Let $h$ be the space step, and $x_{i−1}=x_i−h$ and $x_{i+1}=x_i+h$. I want to evaluate the right hand side on node $i$. If I take centered differences I get:
$$\begin{align*}\left\{\frac{\partial}{\partial x}\left[ a(x)\frac{\partial \, u(x, t)}{\partial x}\right]\right\}_i &= \frac{\left[a(x)\frac{\partial\, u(x, t)}{\partial x}\right]_{i+1/2} - \left[a(x)\frac{\partial \, u(x, t)}{\partial x}\right]_{i-1/2}}{h} \\ &= \frac{a_{i+1/2}\left[\frac{u_{i+1}-u_{i}}{h}\right] - a_{i-1/2}\left[\frac{u_{i}-u_{i-1}}{h}\right]}{h} \end{align*}$$
Is this ok? The conductivities should be in between the nodes I'm calculating? (intuitively it seems this is reasonable). Is there a way, for example by using forward or backward differences, to use conductivity nodes that overlap with the nodes I'm calculating (I calculate only the interger indexed nodes)? Can I mix a centered difference approach for one derivative and a backward or forward approach for the other one?
I would really appreciate any hint.
Thanks in advance,
Federico
Answer
At first glance this looks ok. It is also reasonable to calculate your conductivity at cell nodes.
This is a diffusion equation so the only scheme of any real practical use here is central differencing as by the very nature of diffusion the 'flow' of information is isotropic. For heat transfer through fluids (moving fluids) then upwind differencing can come into play to describe the fluid advection, but the difference scheme used for the diffusion terms in these equations would still be central differencing.
For a comprehensive derivation of this scheme and others like it for thermal heat transfer through solid and fluid media, see:
Patankar, Suhas V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere.
Tannehill, John C., et al., (1997), Computational Fluid mechanics and Heat Transfer, 2nd Ed., Taylor and Francis.
Versteeg, H. K., Malalasekera, W., (2007), An Introduction to Computational Fluid Dynamics, The Finite Volume Method. Second Edition.
I hope this was of some use.
According to the neutrino flavour oscillation formula $$ P_{\alpha\beta}=4\sum\limits_{i From the Solar neutrino experiment, can/does one measure all probabilities $P_{e\mu},P_{\mu\tau},P_{e\tau}$? It appears to me that there are various unknowns such as 3 mass squared differences, all 3 mixing angles $\theta_{12},\theta_{23},\theta_{13}$, CP phase. Therefore, how does one measure the solar neutrino mixing angle $\theta_{12}$? Is it possible to eliminate all this unknowns except $\theta_{12}$ in favour of various oscillation probabilities $P_{\alpha\beta}$, $L$, and $E$? Historically, solar mixing angle was measured first. Am I correct? Without any information of other mixing angles. Therefore, one must have eliminated other mixing angles and phases in favour of $P_{\alpha\beta}$, $L$, and $E$. Is that right? I'm not interested in experimental subtleties (such as neutrinos from Sun can't be strictly monochromatic and there must be an energy spread etc).
Answer
The neutrinos of interest to SNO were all quite low energy (a few MeV at most). This has implication to how they are allowed to interact.
In general the interaction allowed to solar neutrinos in a heavy water detector come in three types \begin{align*} \nu_e + d &\longrightarrow 2p + e \tag{Charged-current} \\ \nu_x + d &\longrightarrow p + n + \nu_x \tag{Neutral-current} \\ \nu_x + e &\longrightarrow \nu_x + e \tag{Elastic electron scattering} \;. \end{align*} Crucially the charged-current reaction only occurs for electron neutrinos while the other reactions are possible with all flavors.
SNO was not sensitive to the difference between fluxuations to mu-type or tau-type neutrinos. Just how many are not detected as electron type, because there is insufficient energy for for reactions like $$\nu_l + d \to 2p + l \;,$$ for $l$ a heavy lepton ($\mu$ or $\tau$).
This means SNO measured both the flux of electron neutrinos and the total flux of solar neutrinos at one time. This gives us $P_{e\mu} + P_{e\tau}$ or equivalently $P_{ee}$.
The data has been analyzed for mixing parameters, but SNOs core accomplishmnet was unambiguously resolving the solar neutrino deficit issue.
We know that level-k Abelian 2+1D Chern-Simons theory on the $T^2$ spatial torus gives ground state degeneracy($GSD$): $$GSD=k$$
How about $GSD$ on $T^2$ spatial torus of:
SU(N)$_k$ level-k Chern-Simons theory?
SO(N)$_k$ level-k Chern-Simons theory?
Sp(N)$_k$ level-k Chern-Simons theory?
What are the available methods to compute them? such (i) algebraic geometry; (ii) Lie algebra; (iii) topological theory or (iv) quantum hall fluids parton construction?
an example of SU(2)$_k$ using this approach shows $GSD$ on $T^2$ is $$GSD=(k+1)$$
SU(3)$_k$ using this approach shows $GSD$ on $T^2$ is $$GSD=(k+1)(k+2)/2$$.
If there are examples of $G_2,F_4,E_6,E_7,E_8$ (level-k?) Chern-Simons theory and its $GSD$ on $T^2$, it will be even nicer. Reference are welcome.
Answer
After stating the solution, I'll try to give some physical insights to the best of my knowledge and some more references.
The dimension of the required state space is given by the Verlinde formula, having the following form for a general compact semisimple Lie group $G$ on a Riemann surface with genus $g$ corresponding to the level $k$:
$$ \mathrm{dim} V_{g,k} =(C (k+h)^r)^{g-1} \sum_{\lambda \in \Lambda_k}\prod_{\alpha \in \Delta}(1-e^{i\frac{\alpha .(\lambda+\rho)}{k+h}})^{(1-g)}$$
(Please see Blau and Thompson equation 1.2.). Here, $C$ is the order of the center, $h$ is dual Coxeter invariant, $\rho$ is half the sum of the positive roots, and $r$ is the rank of $G$. $g$ is the genus, $\Delta$ is the set of roots and $\Lambda_k$ is the set of integrable highest weights of the Kac-Moody algebra $G_k$.
For the torus ($g=1$), this formula simplifies to:
$$ \mathrm{dim} V_{\mathrm{Torus},k} = \# \Lambda_k $$
i.e., the dimension is equal to the number of integrable highest weights of the Kac-Moody algebra $G_k$.
The integrable highest weights of a level-$k$ Kac-Moody algebra are given by the following constraints:
$$ \lambda - \mathrm{dominant}, 0 \leq \sum_{i=1}^r \frac{2 \lambda. \alpha^{(i)}}{ \alpha^{(i)}. \alpha^{(i)}}\leq k$$
Where $ \alpha^{(i)}$ are the simple roots, please see, for example, the following review by Fuchs on Kac-Moody algebras.
(My favorite reference for the representation theory of Kac-Moody algebras is the Goddard and Olive review which seems not available on line)
For example for $SU(3)_k$ whose dominant weights are $2$-tuples of nonnegative numbers $(n_1, n_2)$, the above condition reduces to:
$$\mathrm{dim} V^{SU(3)}_{\mathrm{Torus},k} = \# (n_1\geq 0, n_2\geq 0, 0\leq n_1 + n_2 \leq k )= \frac{(k+1)(k+2)}{2}$$
To perform the computations for the more general cases, one can use the seminal review by Slansky.
The Verlinde formula was discovered before the Chern-Simons theory came into the world. Originally it is the dimension of the space of conformal blocks for the WZW model. This formula has been derived in a large variety of ways, please, see footnote 26 in the Fuchs review. It is still an active research topic, please see for example a new derivation in this recent article by Gukov.
The Chern-Simons theory may be the most sophisticated example in which the Dirac quantization postulates can be carried out in spirit. (More precisely their generalization in geometric quantization). I mean starting from a phase space and utilizing a specified set of rules to associate a Hilbert space to it. In the case of the Chern-Simons theory, the phase space is the set of solutions of the classical equations of motion. The classical equations of motion require the field strength to vanish, in other words the connection to be flat.
This phase space (the moduli space of flat connections) is finite dimensional, it has a Kähler structure and it can geometrically quantized as a Kähler manifold, just like the case of the harmonic oscillator. Thus the problem can be reduced in principle to a problem in quantum mechanics.
The case of the torus is the easiest because everything can be carried out explicitly in the Abelian and the non-Abelian case, please see the following explicit construction by Bos and Nair, (a more concise treatment appears in Dunne's review).
In the case of the torus, the moduli space of flat connections in the Abelian case is also a torus and in the non-Abelian case it is:
$$\mathcal{M} = \frac{T \times T}{W}$$
where $T$ is the maximal torus of $G$. Basically, a Fock quantization can be carried away, but there is a further restriction on the admissible wave functions coming from the invariance requirement under the large gauge transformations (please see for example, the Dunne's review ). The invariant wave functions are called non-Abelian theta functions and they are just in a one to one correspondence with the Kac-Moody algebra integrable highest weights. (In the Abelian case, the wave functions are the Jacobi theta functions).
In the higher genus case, although the quantization program leading to the Verlinde formula can be carried out in principle, few explicit results are known, please see the following article by Lisa Jeffrey (and also the following lecture notes). The dimension of these moduli spaces is known. In addition. Witten in an ingenious work computed their symplectic volumes and their cohomology ring in some cases.
Witten's idea is that as in the case of a simple spin, the dimension of the Hilbert space in the semiclassical limit ($k \rightarrow \infty$) becomes proportional to the volume and the leading exponent of $k$ is the complex dimension of the moduli space (please observe for example, that in the case of $SU(3)$ on the torus, the leading exponent is $2$ which is the rank of $SU(3)$ which is the dimension of the maximal torus $T$).
I have been trying to convince myself that it is consistent to replace basis vectors $\hat{e}_\mu$ with partial derivatives $\partial_\mu$. After some thought, I came to the conclusion that the basis vectors $\hat{e}_\mu$ were ultimately just symbols which represent what we think of as arrows, so it is not a problem to use a different symbol. The only requirement is that one can manipulate the $\partial_\mu$ in the same way as the $\hat{e}_\mu$.
However, raising/lowering indices seems to create an inconsistency. In switching our representation of the basis vectors, we make the substitutions:
$$\hat{e}_\mu \rightarrow \partial_\mu$$
$$\hat{e}^\mu \rightarrow dx^\mu$$
However, while we previously could write $\hat{e}^\mu=g^{\mu \nu}\hat{e}_\nu$, we fail to be able to write the same relationship in the new representation:
$$dx^\mu \neq \partial^\mu =g^{\mu \nu} \partial_{\nu}$$
My questions are:
Answer
Raising and lowering indices in a vector is not a valid operation. Basis vectors are no exception. While $x_\mu=g_{\mu\nu}x^\nu$ is a valid operation, $\hat e^\mu=g^{\mu\nu}\hat e_\nu$ is not. The reason is that in the first case you are dealing with the components of a vector, and in the second case you are dealing with a vector itself.
Let me elaborate. Given a vector $\hat X$ $$ \hat X=x^\mu\hat e_\mu $$ you can lower the index $\mu$ in $x^\mu$ through $$ x_\mu\equiv g_{\mu\nu}x^\nu $$
That is: raising and lowering indices is an operation that is defined for the components of a vector (or covector).
The index $\mu$ in $\hat e_\mu$ is not a vector index; it just labels the different basis vectors. You cannot raise/lower this index, because $\hat e_\mu$ does not denote the components of any vector. The operation $$ \phantom{\color{red}{\text{NO!}}}\qquad\hat e^\mu\equiv g^{\mu\nu}\hat e_\nu\qquad\color{red}{\text{NO!}} $$ is a meaningless operation.
The same thing can be said about covectors. Given a covector $\tilde X$ $$ \tilde X=x_\mu\tilde e^\mu $$ you can raise the index in $x_\mu$. But you cannot lower the index in $\tilde e^\mu$, because that index does not denote the components of a covector; it just labels the different basis covectors.
Most importantly, while $\hat e_\mu$ is a basis of the space of vectors, and $\tilde e^\mu$ is a basis for the space of covectors, these objects are not related through $$ \phantom{\color{red}{\text{NO!}}}\qquad\hat e^\mu= g^{\mu\nu}\tilde e_\nu\qquad\color{red}{\text{NO!}} $$ or any similar relation.
In short: you can raise/lower indices when those indices denote the components of an object - either a vector or a covector - but you cannot raise/lower the indices of the bases of vectors/covectors, because those indices do not denote the components of anything. They are just labels.
However, see Musical isomorphism.
I hope that at this point, you are still with me. Given an arbitrary vector $\hat v$ (like $\hat X$ or $\hat e_\mu$), and a certain function $f$, we can define the action of $\hat v$ on $f$ as follows: we define $$ \hat e_\mu[f]\equiv \frac{\partial f}{\partial x^\mu}\in\mathbb R $$ and we extend this through linearity: if $\hat v=v^\mu \hat e_\mu$, then $$ \hat v[f]=v^\mu\frac{\partial f}{\partial x^\mu}\in\mathbb R $$
I'm not going to discuss why this new operation is useful. But let me stress that this operation is something new, something that you might have never seen before: now vectors can act on functions! In any case, useful or not, this new operation motivates us to consider the following convenient notation: we will write $\hat \partial_\mu$ instead of $\hat e_\mu$: $$ \hat \partial_\mu\equiv \hat e_\mu $$
With this, our equation from before now becomes $$ \hat\partial_\mu[f]=\frac{\partial f}{\partial x^\mu} $$
Note that we are using the same symbol, $\partial$, with two different meanings: on the one hand, it denotes a basis vector, and on the other hand, it denotes a partial derivative. The usual thing we do is to drop the distinction: we just write $\partial_\mu$ for both, and let context decide what the symbol means.
In the same vein, we usually use the symbol $\mathrm dx^\mu\equiv\tilde e^\mu$. That is, we denote the basis of covectors by the symbol $\mathrm dx^\mu$. It's just notation.
Let us now move on to the gradient. We define the covector $\mathrm d f$ as the covector that has $\frac{\partial f}{\partial x^\mu}$ as components: $$ \mathrm d f=\frac{\partial f}{\partial x^\mu}\tilde e^\mu $$ or, using our new notation, $$ \mathrm d f=\partial_\mu f\,\mathrm dx^\mu $$
You can raise and lower the $\mu$ index in $\frac{\partial f}{\partial x^\mu}$, because this index denotes the components of a covector. In this sense, you could say that you can raise/lower the $\mu$ index in $\partial_\mu$, whenever this symbol denotes a derivative. But you cannot raise/lower the $\mu$ index in $\hat \partial_\mu$, whenever this symbol denotes a basis vector (for the same reason you cannot raise/lower the $\mu$ index in $\hat e_\mu$).
In short: the objects $\partial_\mu$ and $\mathrm dx^\mu$ replace the old notation $\hat e_\mu$ and $\tilde e^\mu$, but they denote the exact same object: they are a basis for the space of vectors and covectors. This means that you cannot raise/lower their indices. On the other hand, the object $\partial_\mu f$ denotes the components of the covector $\mathrm df$, and as such, you can raise/lower its index.
We know there are general solutions for 1-body problem and 2-body problem and also we know in some "special cases" there are some possible solutions for $n$-body problem for $n \geqslant 3$ and there might not be a general solution to this problem. But I heard it could be possible to prove there is no general solution in terms of elementary functions for $n$-body problem in general.
Now my question is: What is this proof? Can anybody explain it? Do we have to use the method of "Proof by contradiction"?
And also another question: What are the approximate methods used to solve this problem? Numerical analytical methods? Perturbation theory?
Hamilton's principle states that the actual path a particle follows from points $p_1$ and $p_2$ in the configuration space between times $t_1$ and $t_2$ is such that the integral
$$S = \int_{t_1}^{t_2}L(q(t),\dot{q}(t),t)dt$$
is stationary. And then we have that $L = T - U$ is the lagrangian. Now, how this was found? I mean, how could someone find that picking the quantity $T-U$, considering the integral and extremizing it would give us the actual path on the configuration space?
I know that it works, and the books show this very well. But historically how physicists found that this would give the path? How they found the quantity $L$ and thought on studying it's integral?
The claim is often made that the discovery of the Higgs boson will give us information about the origin of mass. However, the bare masses of the up and down quarks are only around 5 MeV, quite a bit smaller than their "constituent" or "dynamical" mass of around 300 MeV. (Remember that a neutron, for example, is one up and two down quarks and has a total mass of 939 MeV.) What then, is the reasoning behind the claim that the Higgs will address the origin of mass when by far the majority of the mass of the neutron (and proton) is related instead to the dynamical breaking of chiral symmetry?
If astronauts could deliver a large quantity of breathable air to somewhere with lower gravity, such as Earth's moon, would the air form an atmosphere, or would it float away and disappear? Is there a minimum amount of gravity necessary to trap a breathable atmosphere on a planet?
Answer
Gravity is a major factor in planets retaining atmospheres over the eons. But there are other factors that must be taken into consideration to consider the volatility of an atmosphere.
Solar wind is the main factor of erosion on any atmosphere. But a healthy magnetic field can deflect most of the solar radiation and decrease the erosion. It has been a matter of debate recently if exo-moons of jovian planets in habitable zones of their host stars would be able to sustain atmospheres: such moons are most likely tidally-locked, so their magnetic fields are not expected to be high, but their host planets will likely have strong radiation belts. But is not clear at the moment if the radiation belts will protect or erode the atmosphere. Saturn has a benign level of radiation, so we have Titan, which has an atmosphere that is thicker than earth's
I am stuck varying an action, trying to get an equation of motion. (Going from eq. 91 to eq. 92 in the image.) This is the action
$$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$
And this is the solution,
$$\ddot{h} + 2 \frac{\dot{a}}{a}\dot{h} - \nabla^{2}h~=~0. $$
This is what I get
$$\partial_{0}(a^{2}\partial_{0}h)-\partial_{0}(a^{2}\nabla h)-\nabla(a^{2}\partial_{0}h)+\nabla^{2}(ha^{2})~=~0.$$
I don't really see my mistake, perhaps I am missing something. (dot represents $\partial_{0}$)
It is this problem (see Lectures on the Theory of Cosmological Perturbations, by Brandenburger):
Let us begin at the beginning
With all my mates and endless spinning
I'm not so hot and not so cold
I'm the sea and the sun, and quite as old
When shove comes to push
In your hand, I'm worth more than a bush
Day in, day out, hidden talents
Up and down, I keep the balance
I give birth to the flow
Downhill my children go
Through me you can see clearly
Now name me, riddler, dearly
Answer
The answer is
spring
Great thanks to jafe and Brandon_J
Let us begin at the beginning
spring --> beginning of the year
With all my mates and endless spinning
other seasons, rotating every year
I'm not so hot and not so cold
spring is a season where the temperature is moderate
I'm the sea and the sun, and quite as old
nice one! sea+sun sounds like season, where spring is one of them
When shove comes to push
when force is applied
In your hand, I'm worth more than a bush Day in, day out, hidden talents Up and down, I keep the balance
spring balance
I give birth to the flow
spring as in water flow
Downhill my children go
a stream of water usually flows downhill
Through me you can see clearly
we can see through water clearly
Now name me, riddler, dearly
spring
Why posting this riddle in this time of the year?
Spring Equinox?
Why does the formalism of QM represent reversible changes (eg the time evolution operator, quantum gates, etc) with unitary operators?
To put it another way, can it be shown that unitary transformations preserve entropy?
Answer
The fact that evolutions of quantum mechanics are unitary after finite periods of time can be proven from the Schrödinger equation, and hinges on the characterization of unitary operators as those linear operators which are norm-preserving.
Recall the Schrödinger equation: $$ \frac{\mathrm d}{\mathrm d t} |\psi\rangle \;=\; -i H |\psi\rangle \;,$$ where $H$ may or may not be time-dependent, but in any case has real eigenvalues, so that $H = H^\dagger$. As a result, the way in which $|\psi\rangle$ changes instantaneously with time is in such a way that its magnitude, as a vector in Hilbert space, does not increase. We can see this by simply computing: $$ \frac{\mathrm d}{\mathrm dt} \langle\psi|\psi\rangle \;=\; \left[\frac{\mathrm d}{\mathrm dt} \langle\psi|\right]|\psi\rangle + \langle\psi|\left[\frac{\mathrm d}{\mathrm dt}|\psi\rangle\right] \;=\; i\langle\psi|H^\dagger|\psi\rangle - i\langle\psi|H|\psi\rangle \;=\; 0.$$ because $H = H^\dagger$. Then at all times the state vector remains the same length. That is to say, the norm is preserved under finite-time evolution.
Because unitary operators are exactly those ones which preserve the norm, it follows that the finite-time evolution will be unitary.
In the Schrödinger equation, at any given time $t$ we should jointly add another sub equation, like $$||\psi_t(x)|| = 1$$ where $\psi_t(x) = \Psi(x,t)$, and then try to solve the two equations simultaneously. Why not? I know it does not yield, but I am always baffled, who is doing the normalization of the wave function? Observers, the system, the measuring process, God?
Anyone can tell me which are the non zero components of the Riemann tensor of the Schwarzschild metric? I'm searching for this components about 2 weeks, and I've found a few sites, but the problem is that each one of them show differents components, in number and form. I´ve calculated a few components but I don't know if they are correct. I'm using the form of the metric:
$$ds^2 = (1-2m/r)dt^2 + (1-2m/r)^{-1} dr^2 + r^2 d\theta^2 + r^2\sin^2\theta \, d\phi^2$$
I know other people asked the same question time before, but I read a few posts and I didn't find a satisfactory answer to the question, probably because it is a foundational problem of quantum mechanics.
I'm talking about the Hilbert space Separability Axiom of quantum mechanics. I'd like to understand why it was assumed this condition in the set of postulates of QFT. Is there a physical motivation of this, or was it only a way to simplify computation?
Mathematically speaking such an assumption is understandable. I read the argument about superselection sectors, where, even in presence of a non separable Hilbert space in QFT, every sector can be assumed to be separable and one can work inside this one, agreeing in this way with the said axiom. But the trouble remain unsolved, why this sector has to be separable?
If you know some old post or some book where I can find this answer and I didn't see please notify me.
Why does an object with smaller mass hits the ground at same time compared to object with greater mass? I understand the acceleration due to gravity of earth will be same but won't the object with greater mass will fall faster?
Answer
That is an excellent example for a nice quote I read on the internet: "Common sense may be common, but it certainly isn't sense" :-)
As it is hard to lift heavy objects, we assume that it must be easier for them to drop.
Now, Newton's laws point out that light and heavy objects will fall with the same velocity. But is there an intuitive reason? Yes!
The mass of an object contributes to two different phenomena: Gravity and inertia.
When people think that heavy objects should fall faster, they only think of the first point. But in reality, the first and second point cancel out each other: Yes, the earth pulls stronger on a heavy object, but the heavy object is more reluctant to get moving.
I've always assumed/been told that Newton's 2nd law is an empirical law — it must be discovered by experiment. If this is the case, what experiments did Newton do to discover this? Is it related to his studies of the motion of the moon and earth? Was he able to analyze this data to see that the masses were inversely related to the acceleration, if we assume that the force the moon on the earth is equal to the force the earth exerts on the moon?
According to Wikipedia, the Principia reads:
Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.
Translated as:
Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.
My question is how did Newton come to this conclusion? I get that he knew from Galileo Galilei the idea of inertia, but this doesn't instantly tell us that the change in momentum must be proportional to the net force. Did Newton just assume this, or was there some experiment he performed to tell him this?
Answer
First of all, it would be preposterous to think that there was a simple recipe that Newton followed and that anyone else can use to deduce the laws of a similar caliber. Newton was a genius, and arguably the greatest genius in the history of science.
Second of all, Newton was inspired by the falling apple - or, more generally, by the gravity observed on the Earth. Kepler understood the elliptical orbits of the planets. One of Kepler's laws, deduced by a careful testing of simple hypotheses against the accurate data accumulated by Tycho Brahe, said that the area drawn in a unit time remains constant.
Newton realized that this is equivalent to the fact that the first derivative of the velocity i.e. the second derivative of the position - something that he already understood intuitively - has to be directed radially. In modern terms, the constant-area law is known as the conservation of the angular momentum. That's how he knew the direction of the acceleration. He also calculated the dependence on the distance - by seeing that the acceleration of the apple is 3,600 times bigger than that of the Moon.
So he systematically thought about the second derivatives of the position - the acceleration - in various contexts he has encountered - both celestial and terrestrial bodies. And he was able to determine that the second derivative could have been computed from the coordinates of the objects. He surely conjectured very quickly that all Kepler's laws can be derived from the laws for the second derivatives - and because it was true, it was straightforward to prove him this conjecture.
Obviously, he had to discover the whole theory - both $F=ma$ (or, historically more accurately, $F=dp/dt$) as well as a detailed prescription for the force - e.g. $F=Gm_1m_2/r^2$ - at the same moment because a subset of these laws is useless without the rest.
The appearance of the numerical constant in $F=ma$ or $p=mv$ is a trivial issue. The nontrivial part was of course to invent the mathematical notion of a derivative - especially because the most important one was the second derivative - and to see from the observations that the second derivative has the direction it has (from Kepler's law) and the dependence on the distance it has (from comparing the acceleration of the Moon and the apple falling from the tree).
It wasn't a straightforward task that could have been solved by anyone but it was manifestly simple enough to be solved by Newton. So he had to invent the differential calculus, $F=ma$, as well as the formula for the gravitational force at the same moment to really appreciate what any component is good for in physics.
I have several questions. Perhaps it would be better to separate them into different posts. However, given their relative closeness to each other, I think putting it all in one place would be better. On suggestion, I will modify this post.
I am reading Penrose's paper on the conformal treatment of infinity (find here). The basic concept that underlies this is the fact that in order to study asymptotic behaviour of a space-time with metric ${\tilde g}_{\mu\nu}$, we may instead study a space-time that is conformally related to it by defining an unphysical metric on a compact manifold $g_{\mu\nu} = \Omega^2 {\tilde g}_{\mu\nu}$. He then goes on to say that the asymptotic properties of fields can then be investigated by studying local behaviour of fields at infinity on this unphysical manifold provided that the relevant concepts can be put into a conformally invariant form
What are the relevant conformally invariance concepts that one can study? What I can think of is the causal structure of space-time, gravitational waves (since they are described by a conformally invariant Weyl tensor). What else is there?
I have also often heard that massless fields satisfy conformally invariant equations in curved spacetimes. (see this question). Indeed Penrose claims that this can be done if ""interpreted suitably". What does he mean by this? Further, since massless particles can only reach $\mathscr{I}^\pm$, does his formalism only apply to null infinity?
What about massive particles? Surely, the equations for such particles are not going to be conformally invariant. Further, these particles would reach $i^\pm$. One can't apply the above formalism to massive particles right? Is there an alternative way of constructing the asymptotic structures of $i^\pm$?
I have trouble imagining how two point-particles can have different properties.
And how can finite mass, and finite information (ie spin, electric charge etc.) be stored in 0 volume?
Not only that, but it can also detect all fields without having any structure. Maybe it can check curvature of spacetime to account for gravity, but how can a point contain the information of what the other fields-vectors are? This seems to mean that also the information/volume in space is infinite.
Mathematically, a point cant have any intrinsic structure, so how does physics which is a mathematical theory explain this?
Answer
When one says that an elementary particle is point-like, one is referring to the fact that theoretically, there's no limit to how small a region a detector can localize a particle to. For the sake of argument, let's imagine two wrong things (a) that such an ideal detector is possible and (b) complications arising from Planck scale physics don't change anything conceptually.
Even if you allow for that, your worry that information is being stored in a zero volume region is still unfounded. It would be a legitimate worry in pre-relativistic-QFT physics. But we know particles aren't pellets that move around carrying information. They can disappear and be spontaneously created out of the vacuum. What this is hinting at (though some might prefer a different picture) is that particles aren't fundamental - fields are.
The quantum fields for various particles are defined everywhere in space. Once you specify what kind of structure the quantum field is (a scalar, vector, spinor, etc.) and what its other properties are (say, the symmetry group under which it has local gauge invariance), you have specified what spin, charge, mass etc. its particle excitations will carry. Since the field is defined everywhere in space, there's plenty of room for all that information. So in a certain sense, the information that a detector detects is encoded everywhere in space (because the field is everywhere) - and the specific structure of the detector just picks out the right information you asked for.
Finally, two point-particles (say an electron and a muon) have different properties because they are excitations of two different fields defined everywhere in space - and the detector you build specifically for the electron will pick out the "signal" from the electron.
Poiseuille's law states that the rate of flow of water is proportionate to $r^4$ where $r$ is the radius of the pipe. I don't see why.
Intuitively I would expect rate of flow of fluid to vary with $r^2$ as the volume of a cylinder varies with $r^2$ for a constant length. The volume that flows past a point is equal to rate of flow fluid, and thus it should vary with $r^2$.
I cannot understand the mathematical proof completely. Is there an intuitive explanation for this?
Answer
The flow rate is the average velocity times the area.
If the velocity was constant, you would get a flow rate that scaled with $r^2$ (the area). But the velocity goes up for larger pipes - in fact, velocity scales with the square of the radius. And the product of these two squares gives us the 4th power relationship.
Let's break this into a few steps:
You may know that the velocity profile is a parabola - that is, if the velocity at the center is $v_0$, then the velocity at a distance $r$ from the center of a pipe with radius $R$ is given by
$$v(r) = v_0\left(1-\left(\frac{r}{R}\right)^2\right)$$
The mean velocity is exactly half the maximum velocity - you can see the proof for this in my earlier answer. So we just need to figure out how the maximum velocity scales with $r^2$. Once we know the velocity profile is quadratic, this is easy - because if we make the pipe a little bit bigger, the profile continues to follow the same parabolic shape with the same curvature.
It remains to prove for ourselves that the parabolic velocity profile is correct. This follows from the fact that the shear stress in a fluid is proportional to the viscosity times the velocity gradient. Looking at an annulus of liquid at a distance $r$ from the center, if there is a velocity gradient $\frac{dv}{dr}$ we know that the total force on the liquid inside the annulus is the pressure times the area, or $F = P\cdot A = P \cdot \pi r^2$. We also know this must equal the force due to the shear, which is the force per unit length of the annulus multiplied by the length of the circumference, so
$$\pi r^2 P = 2\pi r \frac{dv}{dr}\eta$$
It's easy to see this is a differential equation in $v$, with a parabolic solution.
$$\frac{P}{2\eta} r dr = dv\\ v = \frac{Pr^2}{4\eta} + C $$
For the boundary condition $v=0$ at $r=R$, we find that $C=-\frac{PR^2}{4\eta}$. With simple manipulation we find the same expression we had before with
$$v_0 = \frac{PR^2}{4\eta}$$.
So there you have it. Mean velocity is half the peak velocity, and peak velocity goes as the square of the radius; multiplied by the area we have the $R^4$ relationship.
I have expressed my self in a detailed way. I've labeled the points to which I want an answer.
I have doubt in the Einstein's rail experiments which he mentioned in his book.
In this experiment we preassume events A and B are simultaneous with respect to the embankment. The observer $M$ observes the events $A$ and $B$ simultaneously.
The point of view of $M$ for the observer $M^{'}$ is:
"According to me $M^{'}$ will observe event $A$ sooner and event $B$ later,that is "dissimultaneously" due to motion towards right with respect to me(embankment)."
$3.$ What is the point of view of $M^{'}$ for himself,that is would the events $A$ and $B$ as seen(observed) by himself be "dissimultaneous?
My guess is that there exist oly one "reallity" of observer $M^{'}$. We know for sure this reallity as calculated by the observer $M$. If there is only one reallity of $M^{'}$ then the point of view of $M^{'}$ for himself will be:
"Today I was in atrain, I don't know if it was moving or not. I saw two flashes of light namely A and B. They were not simultaneous. Infact A occured earlier than B.
I ask why the reallity of $M^{'}$ should be the same?
One can answer: If the reallity of an observer is different for different observers then this will be a paradox. To explain this paradox consider this thought experiment:
One's experiment
A metallic ball $b_0$ is placed on a wedge inside the train as shown:
Train is moving towards right w.r.t outside observer. Two sisters "Sheela" and "Munni" are standing near the wedge. Naturally when the centre of the wedge coincide with the outside standing observer say $O$ two fashes of light $P$ and $Q$ occurs simultaneously in train's frame. After some time when train stops $O$ goes to the train and meets "Sheela" and "Munni". He finds both of them alive.
He thinks "were $P$ and $Q$ simultaneous in my frame?" But if so then in my frame train is moving so one of the light rays would hit the ball earlier than the other and one of the two sisters will die with the heavy ball fallen. This can't be true because in reality there can't be paradoxes, either the girl should alive or not so reality for my frame and the train's frame should be same. And i've seen them alive so $P$ and $Q$ are not simultaneous in my frame.
As the train is moving towards right in my frame so $P$ should occur later than $Q$ so that both the rays of light would hit the ball simultaneously.
We see that the reality(fate) of "Sheela" and "Munni" is that they both are alive so we should accept the consequence of "STR"(i.e. the breaking of simultaneity for different observers).
We also see that the demand for the reality to be same is that $P$ and $Q$ must not be simultaneous for $O$.
I argue to one's experiment:
"The reality for observer $M^{'}$ can be different from what the point of view of $M$ is for the reality of $M^{'}$. In other words if reality is same then $M$ s thinking is that $M^{'}$ has faced both A and B dissimultaneously so when $M$ will meet $M^{'}$ after stopping of train $M^{'}$ should tell him that A and B occurred dissimultaneisly but it can also be the other way round.
When $M$ will meet $M^{'}$ the reply of $M^{'}$ can be:" NO! both A and B were simultaneous".
If reality of $M^{'}$ is different from $M$s point of view we will be in a paradox. This paradox exists. A similar can be realized in practice as explained below:
Anupam's experment
Consider an apparatus managed by $M^{'}$ inside the train:
A long stick is placed on a wedge. Two metallic balls are placed at its two ends. Another wedge is placed at the mid of the stick upon which an egg is balanced. Two observers are there $R$ and $R^{'}$. $R$ is sitting inside the train near the wedge. $R^{'}$ is standing outside. Train is moving towards right w.r.t. $R^{'}$.. Naturally the wedge and $R^{'}$ coincides at this very moment two light rays are sent from the wedge in the opposite directions as shown towards $b_1$ and $b_2$.
The reality of $R$ for himself is:" The two rays hit the balls $b_1$ and $b_2$ simultaneously. The balls falls simultaneously. The stick remains balanced. The egg is alive!."
The reality of $R$ from the point of view of $R^{'}$ is:
"Since the train is moving towards right so the ball $b_1$ will be hit earlier than $b_2$. $b_1$ and $b_2$ do not fall simultaneously. The stick has lost its balance. The egg has fallen. The egg is dead!."
The paradox has appeared in reality.
The reality(fate) of $R$ is different from his point of view than $R^{'}$'s.
$4$ What will happen to the egg when $R^{'}$ will meet $R$ ?.
Since there can be two different realities as I shown so "one's assumption" is wrong.
The point of view of $M^{'}$ for himself in Einstien's experiment can be different from the point of view of $M$ for $M^{'}$ i.e. A and B can be simultaneous in the train's frame.
Answer
Let me address Anupam's experiment, because it seems to me this is the real thrust of your question. It seems to you that different things happen depending on the frame. In one frame the egg survives while in the other it's broken. To explain why this doesn't happen let me modify your experiment slightly:
This starts the same as Anupam's experiment. In step 1. We send out a signal from the egg towards the spheres. We'll take the moment the signal is emitted as time $t = 0$, and the position of the egg as $x = 0$, so the signal is emitted at the spacetime point $(0, 0)$ - I'm giving the coordinates as $(t, x)$. The signal travels out to the spheres so it hits the left sphere at $(d/c, -d)$ and it hit the right sphere at $(d/c, d)$.
In your step 2. the spheres fall off and the beam will tilt or not tilt depending on whether the spheres fall off at the same time. In my modification the spheres send back some signal travelling at speed $v$ (which may or may not be equal to $c$). The signal nudges the egg, so if both signals arrive at the same time the egg stays where it is, while if one signal arrives first it will push the egg over.
So in the rest frame the egg doesn't fall. But in your thought experiment above the egg does fall. Well let's see what happens in my experiment.
In the rest frame $R$ the signals are emitted at $(0, 0)$, call this point $A$, and the two responses arrive back at the egg at $(d/c + d/v, 0)$, call this point $B$. To find out what happens in the moving frame $R'$ we have to Lorentz transform $A$ and $B$ to get their locations in the moving frame $A'$ and $B'$. Actually we'll choose our coordinates so that $A = A' = (0, 0)$, so we only have to find $B'$.
Suppose we do this, then we'll find the signals are emitted from the egg at $(0, 0)$ and they arrive back at the egg at $B'$. But hang on, in our moving frame the two signals arrive back at the egg at the egg at the same point $B'$, that is they arrive simultaneously just like in the rest frame. So the egg stays balanced. No matter how fast the moving frame is travelling the egg stays balanced.
So how come in Anupam's experiment the egg stays balanced in the rest frame but falls over in the moving frame, but this doesn't happen in my experiment. It's because in Anupam's experiment you have assumed that the beam starts rotating instantly when one sphere falls off. But this can't happen because when one sphere falls off the other sphere can't react in a time shorter than $2d/c$ (assuming the beam length is $2d$) because the change in the force on the beam can't propagate faster than $c$. This is the point of my modification to your experiment - in my experiment the change in the force on the beam propagates at the speed $v$ that I put into my experiment. In the moving frame this speed $v$ is different for the two spheres and this difference exactly balances out the original difference in the light signal arrival time.
It would be pretty complicated to work out exactly how the torque on the beam evolves in the moving frame, but the point of my experiment is that you don't have to because no matter what happens in between you can be guaranteed that the effects of changes in the spheres will arrive back at the egg simultaneously in both frames. There is no paradox - the egg stays balanced.
How many $fb^{-1}$ of integrated luminosity at the 7 Tev LHC do physicists expect are needed, to make a $5\sigma$ discovery of the most likey 115 Gev Higgs, if it exists?
Answer
your question is answered by this graph:
It was taken from Tommaso Dorigo's blog.
The 115 GeV Higgs is the very left beginning of all the lines in the graph. The full lines are the 2x 3.5 TeV beams; the dotted lines describe the 2x 4 TeV beams that were planned as a small upgrade of the energy for a while but the plans were abolished and 7 TeV was kept.
You see that with 5 inverse femtobarns at 7 TeV, you only reach about 2.7 sigma (the left beginning of the full blue line). Because the significance in standard deviations only goes as $\sqrt{N}$ where $N$ is the number of collisions, to achieve 5 sigma, you need $(5/2.7)^2=3.43$ times higher amount of data than 5/fb which is about 17 inverse femtobarns. Because of statistical flukes, the 5-sigma discovery could of course occur earlier or later.
The 115 GeV Higgs is the most challenging one to be seen by the LHC. But the run to be closed by the end of 2012 - before the 1-year upgrade - was ultimately chosen in such a way that the LHC should safely accumulate even those 17 inverse femtobarns by the end of 2012 - and maybe even sooner - so that it will discover the Higgs boson even at 115 GeV. One assumes that the additional improvements to the luminosity will materialize. But even if they used the bunch schemes they use now, without any further improvements, they should get to 10/fb or so by the end of 2012 - enough for a 4-sigma "discovery" or so.
Of course, that assumes that the cross sections for the Higgs production are given by the Standard Model. New physics such as supersymmetry may make the Higgs both easier and harder to be detected - usually scenarios are studied where it is easier because the production rate is enhanced. Also, if there's more new physics than just the Standard Model Higgs, the LHC could find something more interesting than the Higgs.
The discovery of the 125 GeV Higgs in 2011
For a more realistic Higgs mass at 125 GeV, each detector - CMS or ATLAS - should produce about 3.5-sigma positive signals for a Higgs boson of this mass because each detector will have collected about 5/fb at 7 TeV, see the full blue curve in the chart, by the end of the year 2011.
Of course, this confidence level is just an estimate. So after a press conference sometime at the end of this year, for example on December 13th, 2011, the ATLAS may say that they have seen a 3.6-sigma evidence for a 126 GeV Higgs while CMS may say they have only seen a 2.2-sigma combined evidence for a 124 GeV Higgs. The overall significance from both experiments may be close to 4 sigma at that time. I included some realistic differences in their measured masses which may be both due to statistical fluctuations as well as systematic errors and differences between both detectors.
I was having a house-leaving party at my old house, and challenged four of my friends to guess the number of the new address, which had three digits.
The responses I got were 280, 376, 304, and 370.
I was amused, as each one of them got one digit correctly and in its right place, while the other two digits of each guess did not match in any place.
What's my house number?
Answer
274
376 and 370 must share one number and not another.
based on possibilities of first number and can't be three or the last number would be repeated twice in one of them.
seeing as 3 isn't the first number 7 must be the second number because of 376 and 370.
can't be 6 or 0 for the third number or else there would be two numbers with 2 correct guesses so that leaves 4.
(1) As we know, we have theories of second quantization for both bosons and fermions. That is, let $W_N$ be the $N$ identical particle Hilbert space of bosons or fermions, then the "many particle" Hilbert space $V$ would be $V=W_0\oplus W_1\oplus W_2\oplus W_3\oplus...$ , and further we can define creation and annihilation operators which satisfy commutation(anticommutation) relations for bonsons(fermions).
So my first question is, do we also have a "second quantization theory" for anyons like bosons and fermions?
(2) Generally speaking, anyons can only happen in 2D. Is this conclusion based on the assumption that the particles are point-like?
In Kitaev's toric code model, the quasiparticles are point-like due to the local operators in the Hamiltonian. My question is, in 3D case, whether there exists a simple model whose Hamiltonian contains local operators and spatially extended operators, so that it has both poit-like quasiparticles(say, $\mathbf{e}$) and knot-like quasiparticles(say, $\mathbf{m}$), then the $\mathbf{e}$ and $\mathbf{m}$ particles have nontrivial mutual statistics in 3D?
Answer
(1) For anyons to be created locally in a physical model they must be created in groups such that the local excitation is a boson or a fermion. However, the local excitation can fractionalize into anyonic parts which can propagate independently. In terms of second quantized operators the expectation is that the the local fermionic/bosonic degree of freedom can be written as a product of anyon creation/annihilation operators. This can be explicitly realized in exactly solvable models such as the Toric Code or the Kitaev Honeycomb model. So the answer to whether anyons have creation and annihilation operators is yes.
However, as pointed out by @delete000, we need knowledge of the exclusion statistics to characterize the Fock space of an anyon type. In exactly solvable models this is apparent if there is a direct algebraic fractionalization as just described. But, I don't think there is a complete understanding of exclusion statistics for an anyon given set of fractional quantum numbers so I cannot completely answer part (1) of your question, although there is a recent discussion for the special case of parafermions.
(2) As pointed out in the comments, there are Toric Code models whose quasiparticles are extended operators that realize non-trivial mutual statistics. One good example is the recent exactly soluble 3D models by Lin and Levin realizing realizing braiding between points and loops.
The particle-loop braiding picture is also important for the gapless phase of the 3D variants of the Kitaev Honeycomb model (although gaplessness makes it difficult to identify the anyon in the wavefunction, it exists at the operator level) where the confinement transition takes place at finite temperature on the 3D lattice because the loops need not only to exist, but be very large to lead to the cancellation between spinon paths through about around the loops [cite: 1309.1171 and 1507.01639]. This is unlike the 2D case where a point defect already does the job, making the 2D Kitaev spin liquid unstable to finite temperatures.
The cross section of top quark pair production is dominated at the LHC by gluon-gluon fusion, whereas at Tevatron, quark-antiquark annihilation is more prevalent. Why is this?
I know the fundamental difference between these colliders is that the LHC collides $pp$ pairs and the Tevatron collides $p\bar{p}$ pairs, so is it related to the parton distribution functions in some way?
Answer
In order to create a top quark pair you need at least an energy of $2 m_t$.
At Tevatron, the energy available is about 1 TeV per beam, where beam1 is a proton beam and beam2 an anti-proton beam. For simplicity, let's pick-up a parton with an energy $m_t = 173$ GeV from a proton and another parton with $E=m_t$ from the anti-proton. The ratio of the energy of the partons with respect to the (anti-)proton will be of the order of $x=0.17$ ($x$ is the Bjorken variable). Looking at the parton distribution function (a.k.a. PDF), you'll see that such relatively large $x$ is dominated by valence quarks and thus $u,u,d$ in proton and $\bar{u},\bar{u},\bar{d}$ in anti-proton. (An example of PDF is given in figure 19.4 of this link: http://pdg.lbl.gov/2014/reviews/rpp2014-rev-structure-functions.pdf ). Hence, the production of $t\bar{t}$ from quark-anti-quark is likely to happen at Tevatron.
At LHC, the situation is different because the energy of the beams is about 6.5 times larger. Hence, $x$ value is decreased by this factor. Typically, for $x$ below 0.1, you'll see that the probability to pick-up a gluon is much larger than a quarks (or anti-quarks). Notice, that the PDF of the gluon is divided by a factor 10 in the plot! So LHC collisions are largely dominated by gluon fusion production.
This is an ultra-soft question about relatively recent history. While reading some of Mandelstam's papers, I noticed that he cites David John Candlin consistenly whenever he does anything with Grassman path-integral. Everyone else cites Berezin.
So I read Candlin's 1956 paper, and I was stunned to find a complete and correct description of anticommuting variables, presented more lucidly than anywhere else, with a clear definition of Grassman integration, and a proof that it reproduces the Fermionic quantum field. This is clearly the original source of all the Grassman methods. I was stunned that the inventor of this method is quietly buried away.
I wrote the Wikipedia page on the guy, but I couldn't find out anything beyond the sketchy stuff I found on an old Princeton staff listing. The fellow doesn't google very well at all.
Here are the questions:
I mean, this is one of the central tools of modern physics, it is used every day by every theorist, and the inventor is never mentioned. It's 50% of the path integral. Why the silence?
I know that the rest mass of a photon is zero. but the photon can be bent by gravity (which can also be explained by the curvature of space-time due to the effect of mass), this implies that it must have some effective mass, while in motion, therefore does it also bend space-time? can the mass of the photon be defined in common(SI) units, how?
Answer
The concept of relativistic mass is obsolete. We do not need to ascribe mass to a photon in order to see that it distorts space:
As an excitation of the electromagnetic field, a photon contributes to the stress-energy tensor $T_{\mu\nu}$, which, through the Einstein field equations, will distort the metric on spacetime, and thus exert gravity.
Spontaneous symmetry breaking can be defined as follows (Teresa & Antonio, 1996; pg89):
A physical system has a symmetry that is spontaneously broken if the interactions governing the dynamics of the system possess such a symmetry but the ground state of this system does not.
I am slightly confused about the meaning of 'ground state' in this expression in the context of condensed matter. In the case of the Heisenberg model the ground state can be take as: $$\rho\propto \sum_n \left|GS_n\right>\left
Does this system therefore strictly speaking have spontanous symmetry breaking? and what is meant by the term 'ground state' in the above definition does it refer to individual $\left| GS_n\right>$ or to the density matrix $\rho$? Does it refer to before or after symmetry breaking fields are applied?
Answer
I would like to offer a contrasting answer to that of @tparker. I want to emphasize the fact that it is actually not necessary to introduce any symmetry-breaking field, provided you use the proper setup. (Of course, you can use symmetry-breaking fields, or suitable boundary conditions to achieve that, but I believe that it is conceptually interesting that you don't have to.)
I'll only discuss the case of classical systems, as I am much more familiar with the relevant mathematical framework.
The goal is to define a Gibbs measure for an infinite system. This is necessary, as only truly infinite systems undergo genuine phase transitions (of course, large finite systems can display approximate, smoothed-out "phase transitions"). The usual definition relying on the Boltzmann weight $e^{-\beta H}$ is useless in this case, as the energy of an infinite system is usually undefined.
The most efficient framework to solve this problem, the so-called Dobrushin-Lanford-Ruelle theory, works as follows. I discuss the case of the Ising model on $\mathbb{Z}^2$ for simplicity, but the approach is completely general. One says that a probability measure $\mu$ on the set of infinite configurations $\{-1,1\}^{\mathbb{Z}^2}$ is an infinite-volume Gibbs measure if, for any finite set $\Lambda\subset\mathbb{Z}^2$ and any configuration $\eta\in\{-1,1\}^{\mathbb{Z}^2\setminus\Lambda}$, the conditional probability of seeing a configuration $\sigma$ inside $\Lambda$, given that the configuration outside $\Lambda$ is given by $\eta$, is given by the finite-volume Gibbs measure in $\Lambda$ with boundary condition $\eta$. The latter is well-defined (through the usual Boltzmann weight), since $\Lambda$ is finite.
For models with compact spins (such as the Ising model or the Heisenberg model), existence of infinite-volume Gibbs measures is guaranteed. Uniqueness, however, does not hold in general. For the Ising model on $\mathbb{Z}^2$, for example, there exists a critical value $\beta_{\rm c}\in(0,\infty)$ of the inverse temperature such that there is a unique infinite-volume measure when $\beta\leq \beta_{\rm c}$, while there are infinitely-many infinite-volume Gibbs measures when $\beta>\beta_{\rm c}$. It turns out that any infinite-volume Gibbs measure $\mu$ can be expressed as a convex combination of two of them: $\mu = \alpha \mu^+_\beta + (1-\alpha)\mu^-_\beta$, for some $0\leq\alpha\leq 1$. These two measures $\mu^+_\beta$ and $\mu^-_\beta$ thus contain all the relevant physics, and there are good reasons to consider them as the physically truly relevant ones. It turns out that the average magnetization under $\mu^+_\beta$ is equal to $m^*(\beta)>0$, the spontaneous magnetization (that is the value of the magnetization that you would get, had you first added a magnetic field $h>0$ and then let $h$ decrease to $0$). Under $\mu^-_\beta$, on the other hand, the average magnetization is $-m^*(\beta)<0$. In this precise sense, there is spontaneous symmetry breaking, even though the procedure described above does not explicitly break the symmetry at any step.
Let me now briefly link the above approach (still for the Ising model on $\mathbb{Z}^2$) with the one using an explicit symmetry breaking. A standard way to proceed in this case is to consider an increasing sequence of finite subsets $\Lambda_n\subset\mathbb{Z}^2$. We then consider the Gibbs measure $\mu^+_{\Lambda_n;\beta}$ associated to the Ising model in $\Lambda_n$, with $+$ boundary condition. One can then consider the (weak) limit of the probability measures $\mu^+_{\Lambda_N;\beta}$ as $\Lambda_n$ grows to cover $\mathbb{Z}^2$. It turns out that the limiting measure coincides with the infinite-volume Gibbs measure $\mu_\beta^+$ derived above. Of course, one recovers the measure $\mu^-_\beta$ by using a sequence with $-$ boundary condition.
So the two approached yield the same result, but I insist that the former does not require explicit symmetry breaking. (Moreover, it provides a much more powerful framework.)
How much energy would be necessary to slow down Earth rotation such that a year was 360 days long?
In the same spirit: how much energy would be necessary to make the Earth rotate faster around the Sun such that a year was 360 days long?
(Just for fun: how would you do it practically?)
Answer
The Earth is currently rotating at one revolution per sidereal day. Converting to radians, this is an angular velocity of $\omega_0 = \frac {2\pi}{\text{sidereal day}}$. You want to have 360 solar days per year, or 361 sidereal days per year. That means a rotation rate of one revolution per 1/361 tropical year, or an angular velocity of $\omega_1 = \frac {722\pi}{\text{tropical year}}$.
The rotational kinetic energy of a rotating object is given by $E=I\omega^2$, where $I$ is the moment of inertia about the rotation axis and $\omega$ is the angular velocity. The minimum amount of energy needed to accomplish the desired change is thus $I_\text{earth}({\omega_0}^2 - {\omega_1}^2)$. Using a rather old value for the Earth's moment of inertia about the Earth's rotation axis of $8.034 \times 10^{37}$ kg m2, the minimum amount of energy needed to accomplish this change is $9.8\times10^{27}$ joules (calculation at Wolfram Alpha).
The minimum energy needed to change the Earth's orbit so that a tropical year is 360 days long is to perform a Hohmann transfer. To accomplish this, we need to decrease the Earth's orbit velocity about the Sun by 72.04 m/s (calculation) and then half a year later, decrease the Earth's orbit velocity once again by 72.22 m/s (calculation). Assuming some magic that allows us to convert energy to momentum, the minimum energy needed to accomplish these changes is $3.1\times10^{28}$ joules (calculation).
Either way, that's a lot of energy. Given that all of humanity consumed $5\times10^{20}$ joules during 2010, if we were to do this without any outside help, we'd have to stop consuming energy for about 60 million years. By then we'll have stored enough energy needed to perform either of these tasks, and surely in 60 million years we'll have developed the magic needed to convert energy into momentum.
There are other ways to accomplish the second task. We could use gravity assists to transfer momentum to Jupiter (Korycansky 2001). Capture an asteroid and outfit it with thrusters. Set it into a flyby of Jupiter that drops it to a flyby of Earth. Repeat this, many, many times over. We'll need a bit of fuel to readjust the asteroid's orbit, but the vast majority of the needed energy will come from the Jovian flybys. Eventually (after hundreds of millions of years), the Earth will be in the desired orbit.
Another approach is to build a big (a very, very big) solar array that is somehow maintained in static equilibrium relative to the Earth (McInnes 2002). The center of mass of the sail-Earth system will slowly accelerate (or decelerate, depending on sail orientation), once again eventually moving the Earth.
Note that both of the cited papers address the problem of moving the Earth away from the Sun. We'll need to do that eventually because G-class stars generate ever more energy as they age. The Earth will be completely uninhabitable in less than a billion years if we don't do something. But there's nothing technically wrong with using Korycansky's flybys or McInnes's big solar sail to move the Earth closer to the Sun.
Korycansky, D. G., Laughlin, G., & Adams, F. C. (2001). Astronomical engineering: a strategy for modifying planetary orbits. Astrophysics and Space Science, 275(4), 349-366.
McInnes, C. R. (2002). Astronomical engineering revisited: planetary orbit modification using solar radiation pressure. Astrophysics and Space Science, 282(4), 765-772.
I found this puzzle online:
On the top floor of a castle lives a princess. The floor has 17 bedrooms arranged in a row. Each bedroom has doors connecting to the adjoining bedrooms as well as to the outside corridor. The princess sleeps in a different bedroom each night by opening the door to an adjoining bedroom and spending the night and the next day in that room.
One day a prince arrives at the castle and is desirous of marrying the princess. The guardian angel at the castle tells him of the princess' sleeping patterns and informs him that each morning he may knock on one of the outside doors. If the princess happens to be behind that door, she will open it and consent to marry him. The prince also has a return ticket to his kingdom in 30 days, so he can make at most 30 attempts. Can the prince win the hand of the princess, and if so, what is his strategy?
An unstated assumption here is that the princess can move to any adjacent room (she's not restricted to moving in one direction). So a possible sequence could be 3, 2, 3, 4, 5, 4, 3, 2, 1, 2...
Further down the thread at the link, this solution is given:
The Prince should knock on the second door from one of the ends of the corridor (call it door #2), and knock on the next adjacent door each successive day until he reaches the second door from the opposite end of the corridor (door #16). The day after that, he should begin the same process in reverse order (meaning he will knock on the 16th door two days in a row). By the time he reaches his starting point (door #2 on the 30th day), he will have found the princess.
Number the doors 1 thru 17.
If the princess occupies an even numbered room on the day the prince first knocks on a door (#2), then she will either occupy the same room he knocks on or she will be an even number of rooms away. Since both move to an adjacent room each day, this will hold true so she will never be in a room adjacent to the one he knocks on, and thus never be in a position to move past him the next day. She will have nowhere else to go by the time he reaches the 16th door, and will have by then been located. If she, on the other hand, was in an odd numbered room when he began, then she will be in an even numbered room when the prince starts the process again on day 16 (when he knocks on door #16 the second time).
I've been trying to follow the solution, but I don't quite get it. The main assumption the logic in the solution seems to hinge on is this: "she will never be in a room adjacent to the one he knocks on, and thus never be in a position to move past him the next day." I can't figure out what evidence from the puzzle supports that assumption. If we begin knocking at door #2 and the princess is in room #3, our next move (following this solution) will be to knock on door #3. But at this point the princess could have moved to room #2, and we will keep moving to higher numbered rooms, and the princess could easily remain in the lower-numbered rooms.
So I'm having trouble understanding why this is the solution. Can someone explain it more clearly/in a different manner? Why will this process ensure the princess is found?
Answer
Ilmari's answer covers the logic of the answer perfectly, but in case anyone's still confused, here's a version of the diagram I find more intuitive (I made it myself while solving the puzzle).
The Pink squares are rooms where the princess could be on the given day, the blue squares are where the prince knocks that day, and the black squares are rooms in which she logically cannot be. On day 30 all rooms but room 2 have been eliminated, meaning that if the prince has not found the princess already, he will find her there on day 30.
So I've been doing a lot of reading about the twin paradox and have encountered several different explanations that strive to resolve it. First off let me start by saying general relativity is not an adequate explanation and in fact has nothing to do with resolving the paradox. (That much has been made clear to me from what I have read, as it has been pointed out that believing general resolves the paradox is a common misconception) To drive that point home let me propose a slight variation on the twin paradox that removes acceleration all together.
Some ancient race of aliens long ago set up an experiment for us without our knowledge to help us understand space time. The experiment contains two space craft with clocks on board separated by a very large distance. The first clock, clock A, was accelerated to .866 speed of light millions of years ago (Thus time runs at half speed) and is set on a trajectory to fly past earth. As it flies past earth it resets its clock to 0 and continues on its way. (The aliens also left behind a clock on Earth, clock C, that starts ticking the moment Clock A passes earth and resets itself to 0) The other clock, clock B, was also accelerated to .866 the speed of light long ago and is on the same trajectory as clock A but in the opposite direction so that it heads towards Earth. The two ships and their respective clocks pass each other at a distance of four light years away from earth at which point clock A transfers its time reading to Clock B. Clock B flies past earth and relays its time measurement so as to be compared to Clock C. The time reads half the time elapsed by Clock C, but how could this be possible if time dilation is always symmetrical?
An affinely connected spacetime with a metric compatible connection can, in principle, have a non-vanishing anti-symmetric part; where the definition of the connection is given by defining parallel transport by
$$dA^{\mu} = - dx^{\sigma}\Gamma_{{\sigma}{\rho}}^{\mu}A^{\rho}$$
Now, in the light of Equivalence Principle, one can assert that at a point $\mathcal{P}$, one can always find a local inertial coordinate system with the coordinates of the point $\mathcal{P}$ being {$\xi^{\mu}$} and in this system, the parallel transported version of a vector $\vec{A}$ (whose coordinates at the point $\mathcal{P}$ are $\{A^\rho\}$) from $\mathcal{P}$ to a nearby point situated at coordinates $\{$$\xi^\mu+d\xi^\mu$$\}$ will also be $\{A^\rho\}$. Therefore, the components of the parallel transported version of $\vec{A}$ in a generic coordinate system with coordinates $\{x'^\mu\}$ will be
$\dfrac{\partial x'^{\mu}}{\partial \xi^{\rho}}\bigg|_{ \xi+d\xi}A^\rho$ and the components of the original vector $\vec{A}$ will be $\dfrac{\partial x'^{\mu}}{\partial \xi^{\rho}}\bigg|_{ \xi}A^\rho$. It is important to keep the distinction between coordinate transformation matrices of two different points because they will not generically be equal. Therefore, the difference in the coordinates of the parallel transported and the original vector in the generic coordinate system will become
$$ \begin{aligned} dA'^{\mu} &= \bigg(\dfrac{\partial x'^{\mu}}{\partial \xi^{\rho}}\bigg|_{ \xi+d\xi} - \dfrac{\partial x'^{\mu}}{\partial \xi^{\rho}}\bigg|_{ \xi}\bigg) A^{\rho}\\ &= \dfrac{\partial^2x'^{\mu}}{\partial \xi^{\sigma}\partial\xi^{\rho}}d\xi^{\sigma}A^{\rho} = \dfrac{\partial^2x'^{\mu}}{\partial \xi^{\sigma}\partial\xi^{\rho}}\dfrac{\partial \xi^{\sigma}}{\partial x'^{\kappa}} dx'^{\kappa}A^{\rho} = \dfrac{\partial^2x'^{\mu}}{\partial \xi^{\sigma}\partial\xi^{\rho}}\dfrac{\partial \xi^{\sigma}}{\partial x'^{\kappa}} \dfrac{\partial \xi^{\rho}}{\partial x'^{\nu}}dx'^{\kappa}A'^{\nu} \end{aligned} $$
Thus, under the light of Equivalence Principle, one can conclude that
$$\Gamma_{{\kappa}{\nu}}^{{'}{\mu}} = - \dfrac{\partial^2x'^{\mu}}{\partial \xi^{\sigma}\partial\xi^{\rho}}\dfrac{\partial \xi^{\sigma}}{\partial x'^{\kappa}} \dfrac{\partial \xi^{\rho}}{\partial x'^{\nu}}$$
The expression obtained is manifestly symmetric in the lower two indices of the connection and thus, the anti-symmetric part of the connection is zero. Doesn't this argument suffice to conclude that although there is no apriori mathematical reason to believe that torsion ($T_{{\mu}{\nu}}^{\lambda}:=-2\Gamma_{[{\mu}{\nu}]}^{\lambda}$) must be zero, under the light of Equivalence Principle, it is proven that the torsion must always vanish? I know that in General Relativity, torsion is indeed zero but I have read at many places that torsion can be incorporated into a theory of gravity and it is just an assumption made in General Relativity that torsion vanishes. I find this inappropriate as the Equivalence Principle dictates the vanishing torsion and one doesn't need to assume that. Also, I wonder how a theory allowing non-vanishing torsion can possibly accommodate Equivalence Principle - without which I think the considered theory should be of no merit to a Physicist.
A thought passed by me and I wondered if projecting black light was possible.
We can make lights with many other colors, but I'm not so sure about black.
So, is it possible? Has it been done?
I need a good book about elementary particles. I am a high school student and don't want anything to technical. I read a brief history of time and the universe in a nutshell but I want something that centers more around elementary particles.
