As reinforced once more, we learn that those who use Facebook and Twitter are of a different species from those of us who blog. The latest research discloses they're "much less opinionated" - to which I respond, 'DUH!" Of course they are! Facebook as well as Twitter are venues for social schmoozing and sharing slices of life, not providing political opinions that can be polarizing: upsetting your virtual or real friends, as well as family. So people using Facebook would naturally tend to keep opinions to themselves as opposed to publicizing them,
Now a Pew Research Report released Tuesday validates this. The Report found that those who use Facebook and other social media sites are much less likely to share personal opinions, even offline. These findings run counter to how many view social media - which is basically as a bunch of self-opinionated, solipsistic blabbermouths. In truth, they are more likely to blab about minor social engagements, parties they went to or special trips - while posting photos and jokes - as opposed to giving negative reviews of Obama or Obamacare or discussing Ferguson or Snowden..
In a specific case, to do with NSA whistle blower Edward Snowden, 1,800 social media users were asked their opinions. Researchers noted a "spiral of silence" to do with any opinions - meaning the FB and Twitter users were more likely to "clam up", as well as shy away from going near any hot button issues. This in total contradistinction to those of us who blog and believe our mandate is to inform about critical issues - including government mass surveillance, fracking ravages and climate change - and not keep silent.
Those of us who blog then are born with opinions and let's face it, to blog you HAVE to be opinionated! If you don't have opinions about any and every thing then you will likely be a dead end blogger running out of ideas after ten posts. Also, a boring one at that - mainly posting accounts of the last turtle race you observed on a beach or another kitty - doggie boxing match. All stuff that could easily go on FB.
The bottom line? For those who use Facebook and Twitter, there is a much greater social comfort level with "ice bucket" challenges than political banter and exchanges. Which suits those of us who blog just fine, because the person who wants some spice with his or her information will seek out our virtual spaces rather than FB or Twitter. They will then see our (often) singular opinions about one issue or another and in addition, see how we reason our justifications, say on why we don't find the "ice bucket challenge" all that cool.
Another Pew finding: The typical Facebook user - say who logs onto the site a few times a day - is half as likely to discuss Snowden's leaks as a non-FB user. Twitter users, those who cut their communications to 140 characters or less, are one fourth as likely to discuss Snowden. In a way this ought to be obvious given that any dummy can see the space allotted to discuss topics at length - in FB or Twitter - is much reduced from what a blog provides - which is why we opinionated types don't use FB or Twitter to offer our takes. Those who do use them are "hesitant to speak up for fear of starting an argument or losing a friend." And they ought to be, because it's not the right venue!
One communications professor, Keith Hampton, has this take:
"A society where people aren't able to share their opinions openly and gain from understanding alternative perspectives is a polarized society."
Maybe, prof, but there are also right and wrong venues to share those opinions, and maybe the FB and Twitter users get it. If they really want to share they can BLOG!
Saturday, August 30, 2014
Friday, August 29, 2014
Burger King's Greedy Move Betrays the U.S. and Its Customers
The news that BURGER KING - home of the 'Whopper' - could be moving to Canada, where its taxes would be paid, has many long time customers (including your truly) upset. Already, by Monday afternoon, social media exploded with disapproval with over one thousand negative comments on BK's Facebook page. According to one customer, Shawn Simpson, interviewed outside a BK in NYC:
"For them to take their headquarters and move it across the border is a negative to me."
He added that he "didn't like the idea of the company paying its taxes to another country."
Well, me neither! Since it was established in 1957 in Miami, I have been a Burger King customer - not on any frequent basis - like eating there three times a week - but more like once a month. I always preferred the taste of the Whopper to the Big Mac and the best Whopper I've ever eaten (and wifey too) was at a BK on Picadilly Circus, London in July, 1978. Some of my first malted milk shake and burger dates were at the Burger King in Miami located near the Palmetto Expressway and NW 42nd Ave.
So it is disheartening now to learn this news, almost like a betrayal. Some financial gurus have proclaimed that "it's rational" and even the great Warren Buffet has bought a bunch of new BK shares, though once proclaiming that no CEO ought to escape more taxes than his workers. Cognitive dissonance anyone?
If Buffet were serious regarding his salary and tax proclamations, he'd have dissed the move - which took no cognizance of workers and merely exploited mammoth loopholes in current tax legislation. But then, despite all his blabber, Buffet is a Wall Street investor and all that counts is share price. Let's hope with all the taxes now saved Burger King will do the right thing and at least give its workers a raise!
Here's the skinny: Burger King currently earns about $1 billion a year but its customers aren't buying like they used to. Mickey D's still leads them, but let's face some hard facts: people now have many more choices including antibiotic -free fare at Chipotle's and also there's the general trend to healthier food choices. It is thus unrealistic to expect most people to go to the King as often as even twice a week. The last Whopper I ate (actually two) was about a month ago and that was via a coupon where you got one free with the purchase. (I more often go to get breakfast, a sausage croissanwich and small black coffee).
Burger King also pays $51 million a year in corporate taxes and that's where they want to try to make a tax cut. They believe that by anchoring operations in Canada (via buying out the Canadian coffee and doughnut chain Tim Horton's ) they can cut taxes nearly 30% while also creating the 3rd largest fast food restaurant. But is bigger always better? Actually no! Ask MacDonald's, which has seen steady losses over the past couple years mostly due to competition from the likes of Chipotle's and Panera Bread but also from the greater health consciousness of consumers who now often eat at home.
Thus, merely jumping to a Canadian tax base via a takeover is not likely to help BK long term.
Meanwhile, Burger King spokesman, Miguel Piedra, claims that while the new headquarters will be in Canada, Burger King would still continue to be run out of Miami. That's some consolation, but not much.
Here's the nasty thing about this tax inversion impetus, dozens of companies are now doing it (they say to cut costs) and are moving billions to offshore accounts.. And yeah, though the lazy corporate media report the U.S. has the highest corporate tax rate in the world, at 36% - the facts are substantially different. While the on-paper tax rate is 36%, the actual amount paid by the typical corporation is 12 1/2 percent. Which effective rate is really among the lowest in the world.
Part of the reason for this aberration is that Uncle Sam himself "has allowed companies the right to write off 100% of the purchase price of new equipment in 2010 and 2011". (According to one report from TIME< in Feb. 2012) This was done to theoretically "spur investment and create jobs". One wonders then how come these companies are still sitting on almost $1.5 trillion and are doing little to create more jobs. They'd rather use the $$$ to buy back shares. (Let's keep in mind here the population replacement rate for jobs is nearly 180k a month, i.e. the number required to provide steady, full time work to new monthly labor force entrees - and we still have some 4.5 million jobs to make up that were lost after the 2008 financial crisis.)
Whatever the reason, the engineered corporate tax avoidance has had a dunning effect on our society along with the top 1% escaping their own fair tax assessments. According to TIME (ibid.), corporations account for only 9% of the nation's tax bill compared to 40% in 1943. That means citizens are now left to make up that difference, and the ones who are happen to be in the lower quintiles of income - even as the upper 5% have seen their wealth soar.
This means that more companies using the ruse of tax inversion will create even more tax burden on the rest of us by virtue of abdicating their own responsibilities. The most misplaced aspect is how the corps that do this now have over $150b sitting offshore in accounts while the U.S. scans the offshore accounts of private citizens - which are usually pitiful by comparison. (Citizens with offshore accounts must report each having more than $10,000 in it, on a FATCA form.)
So, basically, the corporations are allowed to get away with tax avoidance 'murder' while very strict oversight laws now apply to private citizens. No surprise then that President Obama and Congress have criticized tax inversions because they means a loss of tax revenue for the U.S. government. So why not DO something about it then? Where is the political will to act if neither likes it?
Most likely because they are terrified of pissing off Wall Street. The proof? Burger King's stock surged $5.78 a share or up 21 percent on Monday to $32. 89. Once more, Wall Street rejoices like it does when higher productivity gains come at the expense of having fewer workers do the jobs of every two laid off. And as they do by awarding higher share prices to airlines that employ "densification" - stuffing ever more people into aircraft cabins to maximize profit! Our nation, sadly, is still hostage to Wall Street.
See also:
http://www.smirkingchimp.com/thread/dave-johnson/57890/now-burger-king-renouncing-u-s-citizenship-lets-eat-somewhere-else
"For them to take their headquarters and move it across the border is a negative to me."
He added that he "didn't like the idea of the company paying its taxes to another country."
Well, me neither! Since it was established in 1957 in Miami, I have been a Burger King customer - not on any frequent basis - like eating there three times a week - but more like once a month. I always preferred the taste of the Whopper to the Big Mac and the best Whopper I've ever eaten (and wifey too) was at a BK on Picadilly Circus, London in July, 1978. Some of my first malted milk shake and burger dates were at the Burger King in Miami located near the Palmetto Expressway and NW 42nd Ave.
So it is disheartening now to learn this news, almost like a betrayal. Some financial gurus have proclaimed that "it's rational" and even the great Warren Buffet has bought a bunch of new BK shares, though once proclaiming that no CEO ought to escape more taxes than his workers. Cognitive dissonance anyone?
If Buffet were serious regarding his salary and tax proclamations, he'd have dissed the move - which took no cognizance of workers and merely exploited mammoth loopholes in current tax legislation. But then, despite all his blabber, Buffet is a Wall Street investor and all that counts is share price. Let's hope with all the taxes now saved Burger King will do the right thing and at least give its workers a raise!
Here's the skinny: Burger King currently earns about $1 billion a year but its customers aren't buying like they used to. Mickey D's still leads them, but let's face some hard facts: people now have many more choices including antibiotic -free fare at Chipotle's and also there's the general trend to healthier food choices. It is thus unrealistic to expect most people to go to the King as often as even twice a week. The last Whopper I ate (actually two) was about a month ago and that was via a coupon where you got one free with the purchase. (I more often go to get breakfast, a sausage croissanwich and small black coffee).
Burger King also pays $51 million a year in corporate taxes and that's where they want to try to make a tax cut. They believe that by anchoring operations in Canada (via buying out the Canadian coffee and doughnut chain Tim Horton's ) they can cut taxes nearly 30% while also creating the 3rd largest fast food restaurant. But is bigger always better? Actually no! Ask MacDonald's, which has seen steady losses over the past couple years mostly due to competition from the likes of Chipotle's and Panera Bread but also from the greater health consciousness of consumers who now often eat at home.
Thus, merely jumping to a Canadian tax base via a takeover is not likely to help BK long term.
Meanwhile, Burger King spokesman, Miguel Piedra, claims that while the new headquarters will be in Canada, Burger King would still continue to be run out of Miami. That's some consolation, but not much.
Here's the nasty thing about this tax inversion impetus, dozens of companies are now doing it (they say to cut costs) and are moving billions to offshore accounts.. And yeah, though the lazy corporate media report the U.S. has the highest corporate tax rate in the world, at 36% - the facts are substantially different. While the on-paper tax rate is 36%, the actual amount paid by the typical corporation is 12 1/2 percent. Which effective rate is really among the lowest in the world.
Part of the reason for this aberration is that Uncle Sam himself "has allowed companies the right to write off 100% of the purchase price of new equipment in 2010 and 2011". (According to one report from TIME< in Feb. 2012) This was done to theoretically "spur investment and create jobs". One wonders then how come these companies are still sitting on almost $1.5 trillion and are doing little to create more jobs. They'd rather use the $$$ to buy back shares. (Let's keep in mind here the population replacement rate for jobs is nearly 180k a month, i.e. the number required to provide steady, full time work to new monthly labor force entrees - and we still have some 4.5 million jobs to make up that were lost after the 2008 financial crisis.)
Whatever the reason, the engineered corporate tax avoidance has had a dunning effect on our society along with the top 1% escaping their own fair tax assessments. According to TIME (ibid.), corporations account for only 9% of the nation's tax bill compared to 40% in 1943. That means citizens are now left to make up that difference, and the ones who are happen to be in the lower quintiles of income - even as the upper 5% have seen their wealth soar.
This means that more companies using the ruse of tax inversion will create even more tax burden on the rest of us by virtue of abdicating their own responsibilities. The most misplaced aspect is how the corps that do this now have over $150b sitting offshore in accounts while the U.S. scans the offshore accounts of private citizens - which are usually pitiful by comparison. (Citizens with offshore accounts must report each having more than $10,000 in it, on a FATCA form.)
So, basically, the corporations are allowed to get away with tax avoidance 'murder' while very strict oversight laws now apply to private citizens. No surprise then that President Obama and Congress have criticized tax inversions because they means a loss of tax revenue for the U.S. government. So why not DO something about it then? Where is the political will to act if neither likes it?
Most likely because they are terrified of pissing off Wall Street. The proof? Burger King's stock surged $5.78 a share or up 21 percent on Monday to $32. 89. Once more, Wall Street rejoices like it does when higher productivity gains come at the expense of having fewer workers do the jobs of every two laid off. And as they do by awarding higher share prices to airlines that employ "densification" - stuffing ever more people into aircraft cabins to maximize profit! Our nation, sadly, is still hostage to Wall Street.
See also:
http://www.smirkingchimp.com/thread/dave-johnson/57890/now-burger-king-renouncing-u-s-citizenship-lets-eat-somewhere-else
Solutions to Quantum Mechanics Problems (2)
1)
We see that from the diagram below that if L = 3 we get one value for ℓ. and three for j :
So
ℓ 1 = 1 and
ℓ 2 = 2
therefore: ℓ 1 + ℓ 2 = 1 + 2 = 3.
Meanwhile, S can be defined by only one value of s, or s = ½
The possible j-values are:
j = ℓ + s = ℓ 1 + s = 1 + ½ = 3/2
j = ℓ2 +s = 2 + ½ = 5/2
Meanwhile, S can be defined by only one value of s, or s = ½
The possible j-values are:
j = ℓ + s = ℓ 1 + s = 1 + ½ = 3/2
j = ℓ2 +s = 2 + ½ = 5/2
j
= ℓ - s = ℓ 1 - s = 1 - ½ = ½
j = ℓ 2 - s = 2 - ½ = 3/2
So in total: j = ½, 3/2 and 5/2
Note that, conforming to j-selection rules, all the j's differ by an integral amount, though they are half-integral (e.g. 3/2, 5/2) themselves.
To obtain any J (total angular momentum) we need an L-S coupling vector that yields J = 5/2. Two possible L-S couplings are available: [L + S] and [L + S – 1] and it is the last that yields the appropriate result: [3 + ½ - 1] = 5/2
This means we need values such that ℓ 1 = 1, ℓ 2 = 2 and s = ½ to make this work.
j = ℓ 2 - s = 2 - ½ = 3/2
So in total: j = ½, 3/2 and 5/2
Note that, conforming to j-selection rules, all the j's differ by an integral amount, though they are half-integral (e.g. 3/2, 5/2) themselves.
To obtain any J (total angular momentum) we need an L-S coupling vector that yields J = 5/2. Two possible L-S couplings are available: [L + S] and [L + S – 1] and it is the last that yields the appropriate result: [3 + ½ - 1] = 5/2
This means we need values such that ℓ 1 = 1, ℓ 2 = 2 and s = ½ to make this work.
2)
The vector solutions (which the reader can do) following the same tack as the
previous probme, will show:
j = ℓ + s = 3 + ½ = 7/2
and:
m J = -7/2, -5/2, -3/2, -½, ½, 3/2, 5/2 and 7/2
meanwhile:
j = ℓ - s = 3 - ½ = 5/2
So:
m J = -5/2, -3/2, -½, ½, 3/2, 5/2
Note that these last m J quantum numbers would also be the ones applicable to the original problem for which: L = 3 , S = ½, and J = 5/2.
j = ℓ + s = 3 + ½ = 7/2
and:
m J = -7/2, -5/2, -3/2, -½, ½, 3/2, 5/2 and 7/2
meanwhile:
j = ℓ - s = 3 - ½ = 5/2
So:
m J = -5/2, -3/2, -½, ½, 3/2, 5/2
Note that these last m J quantum numbers would also be the ones applicable to the original problem for which: L = 3 , S = ½, and J = 5/2.
3) a) The numerical value of the total angular
momentum is given by:
L
= [ℓ (ℓ + 1)]1/2 (ħ)
Where
ℓ = 3,
then:
L
= [3 (3 + 1)]1/2 (ħ) = [3 (4)]1/2 (ħ) = 3Ö2 ( ħ)
b)
The z-component of the orbital angular momentum is given by:
L(z)
= m ℓ
ħ
For
this election, m ℓ = 3, so that:
L(z)
= 3 ħ
(4)
We have ℓ1 =3 and ℓ2 = 2, then:
Therefore,
the possible values of L will be found
from letting ℓ1 =3 and adding each next descending value of m ℓ from 2, to 1, to 0, to -1, to -2:
(3) + 1 = 4
(3) + 0 = 3
(3) + (-1) = 2
(3) + (-2) = 1
So the total angular momentum L can have the values:
5, 4, 3, 2 and 1.
The
f
electron has ℓ =3 so that the total angular momentum quantum number
possibilities are:
j = ℓ + ½,
ℓ - ½
Then: j = 7/2,
5/2
(5) For
4s 3d
we have:
ℓ 1 = 2, s1 = 1/2, ℓ 1 = 0 and s1 = 1/2. Then for the maximum value:
ℓ 1 = 2, s1 = 1/2, ℓ 1 = 0 and s1 = 1/2. Then for the maximum value:
ℓ
1 + ℓ 2 = 2
+ 0 = 2
and: s= s1 + s2 = 1/2 + 1/2 = 1
The lowest energy level is then:
and: s= s1 + s2 = 1/2 + 1/2 = 1
The lowest energy level is then:
4s
3d (3D1)
Since
2s' + 1 = 3, leading to minimum:
j' = [s' - ℓ '] = 1.
Using the assorted combinations, for ℓ '= 0 and ℓ ' = 2, to get the respective j' values (in combination with s' = 0) and then further for s' = 1, we arrive at the energy configuration diagram shown below:
Using the assorted combinations, for ℓ '= 0 and ℓ ' = 2, to get the respective j' values (in combination with s' = 0) and then further for s' = 1, we arrive at the energy configuration diagram shown below:
Thursday, August 28, 2014
An Introduction to Quantum Mechanics (2)
2. Electron Spin and the Stern-Gerlach Experiment
So far, we've seen three different quantum numbers: n (principal), ℓ (angular momentum), and m ℓ (magnetic). We now examine how the fourth quantum number (ms) or electron spin came about.
By the early 1920s, quantum mechanics had developed to the point that theorists realized an electron magnetic moment ought to exist. In any external magnetic field B, the magnetic moment ought to experience a force similar to what a magnetic compass needle experiences in Earth's magnetic field. According to quantum mechanics, the assumed values should be quantized, so the magnetic moment can assume only certain values, given by:
2 ℓ + 1
Thus, the orientation number will depend on the second or azimuthal quantum number, ℓ. The total angular momentum (L) would therefore be:
L = ħ [ℓ (ℓ +1)]1/2
In 1922, the German physicists Otto Stern and Walther Gerlach passed a beam of electrons through a non-uniform magnetic field B as shown in the top sketch of the diagram (C denotes collimator and D, detector plate). A non-uniform field meant the field was stronger on one side of the beam than the other. As predicted from theory, the force on the magnetic moment of the electrons is such that the field ought to deflect the beam according to the orientation of its moment.
The field should therefore split the beam into 2 ℓ + 1 parts (according to 2 ℓ +1 orientations). Stern and Gerlach found their beam of hot silver atoms split into two parts. At first this appeared surprising and at odds with theory but later work showed the conflict could be resolved if the electrons going into the opposing (up and down) beams, had their own spins, or intrinsic angular momenta.
Fig. 7.
Illustrating the Stern-Gerlach Experiment.
In
this case, the upper beam would feature electrons with "spin up"
angular momentum of (+1/2) while the lower beam would feature electrons with
"spin down"
angular momentum of (-1/2) . These are shown at the lower section of the
diagram. Note that in each case, spin up or down, the orientation accords with
the direction of rotation: clockwise or anti-clockwise.
The refining experiment was actually done by Phipps and Taylor using a beam of heated H-atoms. They began by defining the magnetic dipole moment such that:
m ℓ z = - g(ℓ) u B m ℓ
where, as before, m ℓ = - ℓ, - ℓ +1, 0, ....+ ℓ
and u B is the Bohr Magneton: u B = eh/4p m
where m is the mass of the H-electron.
Meanwhile, g(ℓ) is known as the "orbital g-factor".
From their low oven temperature, Phipps-Taylor recognized that ℓ = 0 for the electrons coming off and entering the magnetic field. Therefore, m ℓ = 0. Since this was so, then:
m ℓ z = 0
Phipps -Taylor therefore assumed the beam would be unaffected or not split at all. Yet, it split into two symmetric components, e.g. deflected symmetrically, as shown in the graphic below. Given the earlier Stern-Gerlach experiment, plus their own, they therefore had to expect the electron had its own magnetic dipole moment..
They assumed this to be a spin magnetic moment m s, due to the electron having an intrinsic angular momentum S analogous to L (angular momentum) so that:
S = [s(s + 1)]1/2 ħ and S z = m s (ħ)
The refining experiment was actually done by Phipps and Taylor using a beam of heated H-atoms. They began by defining the magnetic dipole moment such that:
m ℓ z = - g(ℓ) u B m ℓ
where, as before, m ℓ = - ℓ, - ℓ +1, 0, ....+ ℓ
and u B is the Bohr Magneton: u B = eh/4p m
where m is the mass of the H-electron.
Meanwhile, g(ℓ) is known as the "orbital g-factor".
From their low oven temperature, Phipps-Taylor recognized that ℓ = 0 for the electrons coming off and entering the magnetic field. Therefore, m ℓ = 0. Since this was so, then:
m ℓ z = 0
Phipps -Taylor therefore assumed the beam would be unaffected or not split at all. Yet, it split into two symmetric components, e.g. deflected symmetrically, as shown in the graphic below. Given the earlier Stern-Gerlach experiment, plus their own, they therefore had to expect the electron had its own magnetic dipole moment..
They assumed this to be a spin magnetic moment m s, due to the electron having an intrinsic angular momentum S analogous to L (angular momentum) so that:
S = [s(s + 1)]1/2 ħ and S z = m s (ħ)
Then m s is the electron spin or spin quantum number, 1/2 or - 1/2,
Then the electron magnetic dipole moment would be:
m s = - g(s) m B S / ħ
To nail down the basis quantitatively, Phipps-Taylor knew that the net force felt by the electrons traversing the field would be:
F z = - (dB z /dz) m B g(s) m s
where m s is the putative spin quantum number.
Since they knew the Bohr magneton: m B = 9.27 x 10-24 J/T
and dB/dz could be measured (e.g. the difference in the B-field over the collimation width dz)
then
:
g(s) m s = F z / - (dB z /dz) u B = + 1
as measured.
Since g(s) = 2, then by deduction:
m s = + 1/2
Thus, Phipps and Taylor discovered that the splitting effectively showed two possible values for the orientational potential energy, or:
g(s) m s = F z / - (dB z /dz) u B = + 1
as measured.
Since g(s) = 2, then by deduction:
m s = + 1/2
Thus, Phipps and Taylor discovered that the splitting effectively showed two possible values for the orientational potential energy, or:
D(E) = - m s B = +g(s) u B
B/2.
As
an example, let's compute S for the value m s = 1/2.
S = [s(s+ 1)]1/2 ħ = [1/2(1/2 + 1) 1/2 ħ =
[1/2 (3/2)] 1/2 ħ = [3/4] 1/2 ħ = (Ö3/2) ħ
S = [s(s+ 1)]1/2 ħ = [1/2(1/2 + 1) 1/2 ħ =
[1/2 (3/2)] 1/2 ħ = [3/4] 1/2 ħ = (Ö3/2) ħ
This
result can be shown in diagrammatic form on a spin diagram, e.g.
Example
Problem:
(See the Solution for Problem (2) from Previous set posted yesterday)
3. Space quantization and
L-S Coupling:
For now we just want to focus on basic principles of angular momentum for classical physics. In terms of an orbiting planet, for example, we can specify the orbital angular momentum as:
L = m v r
That is, assuming a circular orbit of radius r, and a body of mass m, with velocity v, the orbital angular momentum will be L.
Now, what about at the atomic level? Bohr, in his Bohr model, visualized electrons behaving like tiny planets and orbiting a central nucleus. For the simplest atom, hydrogen, this meant one electron circling a single central proton. Bohr specified the rule for orbital angular momentum - using the principal quantum number, n, as:
mvr = n (ħ)
where again, h is Planck's constant (6.6254 x 10 -34 J-s) and ħ the reduced Planck constant.
However, this model failed to correctly predict the orbital angular momentum for the hydrogen electron, yielding a value of 1 unit, which is wrong. In addition, if L = 0 one would find the electrons oscillating in a straight line through the proton nucleus - which is impossible! Thus, it had to be modified. The modifications were possible once one removed the "planetary model" (which is classical and deterministic) and turned to the wave model. In this case, the orbital angular momentum assumes certain specific values such that:
L = [ℓ (ℓ + 1)]1/2 (ħ)
(ℓ = 0, 1, 2, ......., n-1)
In the above case, when ℓ = 0 we find L = h/2p = ħ = 1. 054 x 10 -34 J-s
The fact L can be zero and is acceptable discloses why classical mechanical models fail at the quantum level.
Now, what if instead we have the angular momentum quantum number ℓ = 2?
Then: L = [ℓ (ℓ + 1)] 1/2 (ħ) = [2(2 +1)] 1/2 (ħ) = Ö6 (ħ)
and a set of allowed projections for L are obtained as shown in the diagram (a) as well as orientations.
Fig. 9. Space
quantization and the quantum number, L
In general L z is specified according to:
L(z) = m ℓ ħ
where m ℓ is the magnetic quantum number.
Recall that m ℓ can range from - ℓ to ℓ, so for ℓ = 2, we have:
m ℓ = +2, +1, 0, -1, -2
When we multiply each of these by (ħ) we obtain the quantization of L z as depicted in diagram (a) of Fig. 9.
It should also be obvious to anyone who's taken trigonometry that one can obtain the angle between the vertical projection L z of the vector L, and L. ( See diagram (b) of Figure 9).
The angle (q) can indeed be found using the cosine relation (adjacent over the hypoteneuse) which yields:
cos (q) = L z / L = m ℓ / [ℓ (ℓ +1)]1/2
Of course, it ought to be self-evident that we are obtaining allowed values for the angle, since obviously m ℓ is going to range from -2 to +2)
Thus, for the problem we've considered (with ℓ = 2) we have the allowed cosines and angles:
m ℓ = 0 so cos(q) = 0 so t = 90 deg
m ℓ = 1 so cos (q) = 1/ Ö6 so q = 65.9 deg
m ℓ = -1 so cos(q) = -1/ Ö6 so q = 114.1 deg
m ℓ = 2 so cos(q) = 2/Ö6 so q = 35. 2 deg
m ℓ = -2 so cos(q) = -2/Ö6 so q= 144.7 deg
Question: Say an electron in an atom (e.g. hydrogen) has zero orbital angular momentum (ℓ = 0) does that mean it has zero total angular momentum?
NO, because in quantum mechanics we find that for every electron in a given atom we have to process two kinds of angular momentum: orbital (L) and spin (S). Even if the electron experiences no precession or torque it must still exhibit a total angular momentum.
Earlier, we saw for the total orbital angular momentum:
L = [ℓ (ℓ + 1)] 1/2 ħ
We also saw that each electron has a spin angular momentum:
S = [s(s + 1)] 1/2 ħ
for which s assumes one or other of the electron spin quantum numbers, m s = +1/2 or -1/2.
It can be shown that S is always: [3/4] 1/2 ħ
Now, we reckon in what we call the total angular momentum or J, such that:
J = [j(j + 1)] 1/2 ħ
and j = ℓ + s (note the common letters apply to different quantities than the capital ones!)
Thus, for an electron with zero orbital angular momentum (l=0) we have:
j = ℓ + s = 0 + 1/2 or ℓ + s = 0 - 1/2
so: j = +1/2 or -1/2
Then we have:
J = [j(j + 1)] 1/2 ħ = [3/4] 1/2 ħ
for either j (which readers can also verify)
We can also find (as we did with L for Lz, the projection of the total angular momentum quantum number on the z-axis (Jz ) which will be:
J(z) = m J ħ
where m J = -j, -j+1, ......+j
Readers with an intuitive grasp of vectors, or if they've worked with vectors - say in high school or college physics, will quickly see that the name of the game is to obtain a vector sum such that:
V = V(1) + V(2)
In this case, L plays the role of V(1), and S plays the role of V(2)
Then we obtain for J:
J = L + S
As any physics student knows, the way to obtain the vector sum is via the law of cosines and this is demonstrated in the diagram below along with the computations. This is for the case:
L = 3
S = 1/2
J = 5/2
and the angle (q) can also be obtained as shown in the diagram below:
Note that in a weak magnetic in which the atom is situated, the L-S coupled system depends on j, in other words this very angle between the vectors L and S. Meanwhile, the orientation of the atom on the whole, depends on m J.
Another useful activity is to draw energy level diagrams using
the given electron configurations for a particular atom. For example, consider
the schematic
energy diagram for the 2p 3s configuration for the carbon atom (12 C 6) with each
level labeled with spectroscopic
notation:
For example, for the 4s 3d level
we have:
ℓ 1 = 2, s1 = 1/2, ℓ 1 =
0 and s1 = 1/2. Then for the maximum value:
ℓ 1 + ℓ 2 = 2 + 0 = 2
and: s= s1 + s2 = 1/2 + 1/2 = 1
The lowest energy level is then:
4s
3d (3D1)
Since 2s' + 1 = 3, leading to minimum: j' = [s' - ℓ '] = 1.
Problems:
1) State which values of ℓ, s, and j would apply to the case in the diagram, such that the assorted angular momentum vectors have the values identified.
2) Enumerate
the possible values of j and m J for the states in which
ℓ = 1 and s
= 1/2 and draw the associated vector diagrams.
3)
Consider an electron for which n = 4, ℓ = 3, and m ℓ = 3. Calculate:
a) the numerical value of L, the total orbital angular momentum
b) the z-component of the orbital angular momentum.
a) the numerical value of L, the total orbital angular momentum
b) the z-component of the orbital angular momentum.
4) A two-electron atom for which the
orbital angular momentum quantum numbers are ℓ1 =3 and ℓ2 = 2 can have what values
for the total orbital angular momentum number L? Determine the possible values of the total angular momentum
quantum number of single f electron.
5)
For 4s 3d configuration we have:
ℓ 1 = 2, s1 = 1/2, ℓ 1 = 0 and s1 = 1/2.
ℓ 1 = 2, s1 = 1/2, ℓ 1 = 0 and s1 = 1/2.
Using
the assorted combinations, for ℓ '= 0 and ℓ ' = 2, to get the respective j' values (in
combination with s' = 0), and then further for s' = 1, sketch the energy configuration
diagram
How the Frackers Prevailed in Colorado
Thanks to an extended report, we now know how Gov. John Hickenlooper, his Neoliberal Dem cronies and the fracker brigade managed to prevail in Colorado. By "prevail" I mean that they managed to finagle a closed doors, or back door deal - presented as a compromise - that resulted in Dem rep Jared Polis removing his two local fracking initiatives from the November Ballot. As the Post put it in its report ('Hickenlooper's Deal', p. 1A, Aug. 24):
"Nearly every person interviewed by the Denver Post doubted that anyone but Hickenlooper, a former geologist who has allies in the energy community and environmental communities, could have pulled off a compromise."
But the problem is that it's no real compromise, only a puny token sham that managed to scrap actual potential for action. According to the Denver Post report back on Aug. 8:
"A 20 - person commission was announced Monday as part of a compromise to keep two oil and gas oriented initiatives from the fall ballot."
And what, pray tell, would we see from this commission?
"The commission will have 6 months to make recommendations to the legislature which will then consider further legislation."
Oh, terrific! Six months to make "recommendations" and then the (bought out) legislature will have its say in making new laws. All you need to know is that the two largest fracking companies - from Texas- cheered wildly at the news. Can you figure it out now? Because never mind the legislature, the 6 month span will allow the drillers to put down 5,000 more frack wells, threatening our air, soil and ground water, see e.g.
http://brane-space.blogspot.com/2014/05/now-fracking-pollutes-soil-as-well-as.html
Anyway, Hick and his compadres were propelled to act to get a deal - any deal - after learning from Brian Lawrence (who founded Yorktown Partners - a company that invests in energy) that while the firm had planned to invest $1 billion in Colorado energy companies they were unable to proceed while any fracking control items were on the ballot. Thus the impetus for Hick to come up with any kind of bogus deal he could.
The breakthrough came when U.S. Rep Ed Perlmutter organized a dinner at a downtown Denver steakhouse to discuss the ongoing fracking debate and try to come to a resolution. No members of the fracktivists were allowed to attend, only the inside pols and their lackeys. Oh ...and the oil and gas representatives. According to the Post (p21 A, ibid.):
"To Chris Castilian, director of government affairs for Andarko Petroleum and deputy chief of staff for former Republican governor Bill Owens, the dinner was a watershed moment."
Oh, I am sure it was as the weasels - in their secluded meetings- found concordance on how to craft a deal to screw the public out of an opportunity to vote for local control of frack operations. Now, they're at the mercy of "commissions" and an industry-driven "task force".
To make a long story short, Hickenlooper according to the Post, "practiced shuttle diplomacy" until he got all the participants on board as well as PR firms - to be able to sell this sham to the public. After all, following all the hoopla of the ballot initiatives you had to toss them a rag when the measures were removed. (It helped of course that the state Dept. Of Natural Resources houses the Colorado Oil and Gas Conservation Commission - which is the industry 'pet' group designed to run interference on fracktivists )
As Hickenlooper beheld the "hangman's noose", i.e. the date of Aug. 4 approaching with all the signed petitions that would enable the two local fracking control initiatives, he and his minions pushed into overdrive to secure the bogus deal. The break point came when Polis caved under the pressure and dropped all additional demands concerning the composition of the task force designated to oversee the issues via the commission set up for the compromise. The road was then free and clear to bamboozle the public. (Polis was told if he really wanted to have local fracking control he could run for a special state office.)
So we can then plausibly conclude that Polis has likely been playing us all, or at least only exploited the initiative drive to get more press for himself.. Sadly, with the initiatives pulled, it means all those 100,000+ who've signed them will likely not be tricked into doing so again - for another sham cover by a lying politico only seeking to protect his own hide. Meanwhile, the Neoliberal Dem Governor Hickenlooper has again resorted to his usual codswallop, once describing the push for the local control initiatives and the action by five municipalities to place bans on moratoriums for drilling as "a rebellion". Obviously showing he has no clue what a REAL one looks like, making me wish I could transport him back in time to Yorktown or Lexington in 1777, oh, in a Redcoat uniform.
And in his most recent Post remarks:
"There are still a bunch of critics who say we're kicking the can down the road. But if those critics had their way we would be in the middle of a ballot war right now. "
You have that one right, sonny. And we'd have had a full throated expression of the people's will which you and your imps have now prevented by a back room deal.
Some things never change, but the people of Colorado won't forget. You can make book on it!
Preventing Another Ferguson Incident: The Body Camera!
A Denver Police officer tries out the miniature camera.
Could the upheaval and violence in Ferguson, MO all have been prevented or at least minimized? These are questions now being asked, and very often the response is 'Yes, it could!' To understand why we note the observation in a recent TIME article ('The Tragedy of Ferguson', Sept. 1, p. 22) that:
"the heart of it all lay two or three minutes just after noon on Saturday, August 9, a fatal span that brought Brown and Wilson together on a stretch of Canfield Drive."
Meaning that within this time interval assorted people reported their separate observations of what transpired and the officer his own. Alas, no settled accounts of the time sequence or actual events exist. This could all have evidently been avoided. How?
Via the use of special miniaturized digital body cameras worn on the side frames of officers' glasses or sun glasses. The tiny cameras record every last detail in any encounter with the public for any reason, and hence provide a firm and accurate document of the officer's actions and the perp's responses or crimes. Basically, it is insurance to protect police from false allegations of excessive force, and also suspects from overly aggressive police actions. The body camera ensures literal, visible accountability. According to Denver Police Chief Robert White :
"The body camera will clear up those moments of conflict"
This was revealed in an NBC News spot last night which focused on the police force in Laurel, Maryland, where officers have been authorized to use these cameras as a standard operating procedure for the past 18 months. The result? Crime has dropped 46 percent and the police force has seen accountability for all arrests enhanced by virtue of everything being on record. (In a few cases, as noted in the Denver Post, p. 5A), "people making allegations against the police have withdrawn complaints when they learned their encounters were recorded.")
One instance shown in the Laurel, MD segment involved an officer stopping a speeding vehicle. The entire transaction was recorded including when the officer asked to see identification and the perp floored her accelerator and left the scene - soon later apprehended down the highway.
The benefits from this digital camera system are self-evident. It's no longer a case of the officer's word against anyone else's because he can always provide the evidence in the form of the digital recording. Did he really fire six times into a suspect's body? The camera won't lie and will record all the action from his perspective. If he fired, that will be shown, as well as any reactions by the suspect.
Thus, had Ferguson police officer Darren Wilson been outfitted with such a miniature camera - like the police in Laurel, MD - it is doubtful things would have gotten to the stage they did in Ferguson. All the events would have been immediately accessible and there'd have been no "fatal span" of time within which witness accounts conflicted.
Clearly, these devices need to be dispensed throughout the U.S. to all police departments, and I would say as a vastly higher priority than dispensing militarized gear like Humvees.
Wednesday, August 27, 2014
Solutions to Quantum Mechanics Problems - Pt. 1
(1) In an experiment, with
protons moving through a potential difference V = 4,000 volts, the kinetic
energy gained should be equal to the work done, or:
½ m p v2
= eV
Where
m p the proton mass, is: 1.7 x 10 -27
kg
And
the proton charge = e = 1.6 x 10-19
C
The
velocity then is:
v
= Ö (2eV/mp)
lD= h/p = h/Ö (2eVm p)
For
a voltage V = 4,000 V one would find:
lD =
(6.626
x 10-34 J-s) / [(3.2 x 10-19 C) (4000V) ( 1.7 x 10-27
kg)]1/2
lD = 4.49 x 10 -13 m
(2) Write
out the electron configuration for oxygen (O16), then write out the values for
the set of quantum numbers : n, ℓ, m ℓ, m s for each of the electrons in O16
Finding the electron configuration means one must use the Pauli Exclusion Principle to make sure the electrons are distributed so that no two electrons have the same set of quantum numbers. This means we need to have 2 in the 1s shell, 2 in the 2s shell, and 4 in the 2p shell, so:
1s(2) 2s(2) 2p(4)
For the 2 electrons in the 1s shell: n = 1, ℓ = 0, = 0 and m s = ½ (and (- ½), for second)
For the 2 electrons in the 2s shell: n=2, ℓ =0, m ℓ = 0 and m s = ½ (and (- ½), for second)
For the 4 electrons in the 2p shell:
Since ℓ =1 corresponds to p: n=2, ℓ =1, m ℓ = +1, m s = + ½
Then: n=2, ℓ =1, m ℓ = -1, m s = -½
And: n=2, ℓ =1, m ℓ = 0, m s = +½
Finally: n=2, ℓ =1, m ℓ = 0, m s = -½
Finding the electron configuration means one must use the Pauli Exclusion Principle to make sure the electrons are distributed so that no two electrons have the same set of quantum numbers. This means we need to have 2 in the 1s shell, 2 in the 2s shell, and 4 in the 2p shell, so:
1s(2) 2s(2) 2p(4)
For the 2 electrons in the 1s shell: n = 1, ℓ = 0, = 0 and m s = ½ (and (- ½), for second)
For the 2 electrons in the 2s shell: n=2, ℓ =0, m ℓ = 0 and m s = ½ (and (- ½), for second)
For the 4 electrons in the 2p shell:
Since ℓ =1 corresponds to p: n=2, ℓ =1, m ℓ = +1, m s = + ½
Then: n=2, ℓ =1, m ℓ = -1, m s = -½
And: n=2, ℓ =1, m ℓ = 0, m s = +½
Finally: n=2, ℓ =1, m ℓ = 0, m s = -½
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