For the first posting I choose the concept of generalized potential
energy from physics.
The first posting includes a snapshot from celebrated [2] textbook [1]
defining the concept of generalized potential energy in Lagrangian
mechanics.
I was forced to include a snapshot because a group of insane guys
began to say (see note below) I had misread the book! Yes it looks
surreal!
It includes one link to an old paper in the Journal of Physics A,
Mathematical, Nuclear, and General.
It also includes links to two small online chapters in (velocity-
dependent) generalized potential energies and the derivation of the
relativistic Lorentz force, and to standard acronym for generalized
potential energy function.
I wait it can help.
[1] Udwadia; R. E. Kalaba. Analytical dynamics: a new approach.
Cambridge University Press. 1996.
[2] http://www.amazon.com/exec/obidos/ASIN/0521482178
##### CONTACT AND FEEDBACK #####
My newsgroup e-mail account is disabilited because of odd personal
messages I received.
I also disabilited the comment section in the blog for avoiding to
insane guys to post more and more nonsense ad infinitum. See below.
##### NOTE #####
The choice for the first posting was motivated by an odd discussion in
sci.physics.relativity with four _supposed_ physicists (*) confused
about special relativity, potential energies, Lagrangian and
Hamiltonian formulations, AAAD, invariances, electromagnetism,
dimensional analysis, variational principles, forces, and dozens of
other stuff.
I do not cite their names. If they decide to reveal their identity
they are insane enough to do it.
I was forced to include a snapshot to page 160 of [1] because they
acussed to me (as usual)
> He is either misunderstanding or lying. Wouldn't be the first time,
> and most definitely won't be the last.
Now you could think after posting a snapshot they would recognize
their mistake and would accept the existence of potential energies in
Lagrangians, true? Well, that is not possible because they are insane.
Now, one of them is promoting another new misunderstanding. He now
confoundes energies and forces:
> nobody uses "generalized potential energy". The
> proper term for the quantity you are describing is "generalized
> forces".
Notice in the blog (see also [1]) potential energies are being denoted
by V and forces are by Q. Anyone can see V and Q are two different
symbols. But he will ignore that.
He, a self-proclaimed physics' student, also ignores something called
dimensional analysis (the units for V and Q are different as
corresponding to different physical quantities), but you would not be
surprised because can be more bizarre. Another of them, a supposed PhD
on physics, did the same mistake before and assure us that V was not
an energy.
If anyone were to try to correct that new mistake now (V is an energy
Q is a force), either they will insult or will submit another new
misunderstanding and a new cycle of replies would begin once more. It
is an unending bizarre process.
Reason which in what follows I will NOT reply to nonsenses NOT to
insane guys and liars in newsgroups. Also I have disabilited the
comment section in the blog.
This decision will impede to me to receive further feedback and
suggestions for improvement but I wait you can understand that
otherwise they would disturb any sane discussion.
(*) After some pressure now two of them present themselves like
students, one of physics and the other of math.
##### ACKNOWLEDGMENT #####
I thanks people who send me personal and public messages for defense
from irrational attacks and insults I was receiving.
That sounds like the act of a coward, afraid of contrary positions.
That is difficult to believe.
I am using full name but part of the people who decide to submit
insults and nonsensical posts, however, hidde their identities with
nicks.
I already apologized by disabiliting the comment section. I have
explained with real data why i did.
I do not see the educative reason which i would let one of them (E.
G.) to write in the comment section of the blog dozens of nonsenses.
The latter E. G. wrote before i had decided to not reply more to him
was
> nobody uses "generalized potential energy". The
> proper term for the quantity you are describing is "generalized
> forces".
A simple look to the snapshot (i needed to include a snapshot!!!!) and
you can see that V and Q are two different things in physics. And if
you try to correct him he will lie you with some other new
misunderstanding and the whole process is unending during months.
Anyone disagreeing with the content of the blog can provide their
corrections or views here. Therefore, I fail to see your point.
One of the people i am alluding (K), was loosing the debate and
decided to vote against me simulating different users, whereas he
voted up by himself. He did that hidding his true identity.
If i open the comment section to anyone, they would fill the blog with
dozens of messages containing racist or sexual insults (nazi or
shithead are common) and completely useless comments violating even
most basic stuff as dimensional analysis. In short, I would waste more
time moderating than posting.
But as remarked before anyone disagreeing with the content of the blog
can write here. I wait you can understand me.
If you have some comment or suggestion please let me know.
And confirmed your status as a total idiot.
Rest of usual junk snipped and not even read.
Bill
What is this about?
> I do not cite their names. If they decide to reveal their identity
> they are insane enough to do it.
I bet that most of them are on my list of shit-throwing chimpanzees:
Eric Gisse
Androcles
Bilge
Bill Hobba
Dirk Van de moortel
YBM
Dono a.k.a. karandash2...@yahoo.com
Sam Wormley
Poor little Shooby can't even defend his crap in USENET without
revealing his massive persecution complex.
Which you copy directly from a textbook....fascinating.
>
> The first posting includes a snapshot from celebrated [2] textbook [1]
> defining the concept of generalized potential energy in Lagrangian
> mechanics.
>
> I was forced to include a snapshot because a group of insane guys
> began to say (see note below) I had misread the book! Yes it looks
> surreal!
You made it two whole paragraphs before you started insulting us in
your typical passive-aggressive manner! I'm impressed by your
restraint.
[...]
> My newsgroup e-mail account is disabilited because of odd personal
> messages I received.
Awwwww. Poor baby.
What is it like to have an ego that is so fragile that it cannot
handle even the most light criticism? Do you feel the sun's
disapproving gaze when you wander outside?
>
> I also disabilited the comment section in the blog for avoiding to
> insane guys to post more and more nonsense ad infinitum. See below.
God forbid you have to actually, say, do some work.
>
> ##### NOTE #####
>
> The choice for the first posting was motivated by an odd discussion in
> sci.physics.relativity with four _supposed_ physicists (*) confused
> about special relativity, potential energies, Lagrangian and
> Hamiltonian formulations, AAAD, invariances, electromagnetism,
> dimensional analysis, variational principles, forces, and dozens of
> other stuff.
>
> I do not cite their names. If they decide to reveal their identity
> they are insane enough to do it.
You do not say who you are talking about because you are a coward who
is frightful enough of confrontation that you disable your personal
email account and disallow commenting on your blog. But apparently not
scared enough to consider _not_ posting shit to USENET on a regular
basis.
I wonder if your constant displays of pitiful passive-aggressive
behaviors are a character trait that you use in person.
>
> I was forced to include a snapshot to page 160 of [1] because they
> acussed to me (as usual)
>
> > He is either misunderstanding or lying. Wouldn't be the first time,
> > and most definitely won't be the last.
>
> Now you could think after posting a snapshot they would recognize
> their mistake and would accept the existence of potential energies in
> Lagrangians, true? Well, that is not possible because they are insane.
Of course you are the perfect judge of mental health.
Where did you say you obtained your PhD in Psychology? Or is it a case
of recognizing one's own?
>
> Now, one of them is promoting another new misunderstanding. He now
> confoundes energies and forces:
This is cute. You found another irrelevant and stupid topic to whine
about in the most inane manner manageable.
>
> > nobody uses "generalized potential energy". The
> > proper term for the quantity you are describing is "generalized
> > forces".
>
> Notice in the blog (see also [1]) potential energies are being denoted
> by V and forces are by Q. Anyone can see V and Q are two different
> symbols. But he will ignore that.
Are you so unfamiliar with the popular literature that you are
_confused_ by what I said? I find it odd that someone who makes so
much noise about classical mechanics would be unfamiliar with the term
"generalized forces".
>
> He, a self-proclaimed physics' student, also ignores something called
> dimensional analysis (the units for V and Q are different as
> corresponding to different physical quantities), but you would not be
> surprised because can be more bizarre. Another of them, a supposed PhD
> on physics, did the same mistake before and assure us that V was not
> an energy.
Ah, your misunderstandings never cease to amuse. Don't ever change.
1) I never said that generalized forces had units of force. Those are
words you put in my mouth because you are completely unfamiliar with
the subject or the terminology in which it is taught.
2) Bilge never said what you claim he says. I can't tell whether your
inability to read for comprehension is because your grasp of English
is often tenuous, or because you are stupid. Then again this could be
me being narrow - it could easily be both.
>
> If anyone were to try to correct that new mistake now (V is an energy
> Q is a force), either they will insult or will submit another new
> misunderstanding and a new cycle of replies would begin once more. It
> is an unending bizarre process.
>
> Reason which in what follows I will NOT reply to nonsenses NOT to
> insane guys and liars in newsgroups. Also I have disabilited the
> comment section in the blog.
Of course you won't reply. You are too much of a coward.
This is your preferred way of replying to critics - through the most
passive-aggressive manner possible. You are unable to withstand direct
criticism, and it shows. Badly.
If you had the stones to simply _ask_ about the choice of terminology,
I would have explained it to you. But instead, you have to be a little
bitch and dedicate a few paragraphs in your latest whine to insulting
me. Perhaps if you would crack open a textbook, say Goldstein, and
read it instead of having it as a coffee table decoration you would be
at least on speaking terms with modern formalisms of classical
mechanics.
>
> This decision will impede to me to receive further feedback and
> suggestions for improvement but I wait you can understand that
> otherwise they would disturb any sane discussion.
How's that false sense of moral superiority working out for you,
hypocrite?
I'm just more upfront about my feelings than you - I simply call you a
fucking moron and move on. You, on the other hand, compose post after
post whining about your critics in an underhanded fashion because you
are simply incapable of responding rationally to what is said.
>
> (*) After some pressure now two of them present themselves like
> students, one of physics and the other of math.
Naturally the same cannot be said of you, because you refuse to
discuss your formal education in either physics or mathematics.
>
> ##### ACKNOWLEDGMENT #####
>
> I thanks people who send me personal and public messages for defense
> from irrational attacks and insults I was receiving.
I thank you too, Juan R. Henri Wilson won't argue with me anymore
since he realized I won't forget that he presented me with a forged
diploma. Androcles won't even grace me with a "Fuck off, cunt". Shooby
simply realized he can't answer my questions. Which leaves you and
guskz.
Sure, you pepper these newsgroups with posts whining about me but at
least you have the courtesy to not use my name in the subject. Plus
the 5 seconds of thinking required to find holes in your thinking is
about 5 seconds longer than it takes to make guskz look stupid. Don't
ever change - what will I do when all the entertaining cranks stop
responding to me?
It is easy to understand. In another thread i wrote
> > > > The term "generalized" in generalized potential energy [1] is usually
> > > > droped
Where [1] is the same reference to Uwadia textbook on dinamics i cite
in the blog. It is a celebrated textbook, see for instance
http://www.amazon.com/exec/obidos/ASIN/0521482178
Then Bill said that generalized potential energy would not exist
according to him.
> > > Care to give a reference to its definition? I own quite a few physics
> > > textbooks that define gerneralised momenta, forces etc - but not generalised
> > > pectineal energy. Or is this another of your advanced concepts the basic
> > > texts I have don't cover?
And Eric complainted with him whereas acussing me
> > He is either misunderstanding or lying. Wouldn't be the first time,
> > and most definitely won't be the last.
Fortunately i had included a snapshot, therefore anyone can decide by
him or herself if i misunderstood "generalized potential energy" from
the textbook or if effectively the page 160 of that book defines V
like the "generalized potential energy".
Then Eric, instead recognizing he accused me without proof, add in
defense of Bill
> > He is confused because nobody uses "generalized potential energy".
Fortunately, I also linked in the blog to standard acronym GPEF as
example that the concept "generalized potential energy" is so used in
science that a standard acronym was defined.
Eric even wrote after!!!!!
> > The proper term for the quantity you are describing is "generalized
> > forces".
You can decide from yourself if V and Q could be confunded by someone
sane or if they are lying again.
Now Bill only can insult me. This confirm my decision to block the
comment section in the blog.
But i want to finish this odd debate about those insane guys. I prefer
talk about physics.
Well, two of them already revealed their identity, you can see.
> Eric Gisse
> Androcles
> Bilge
This is a supposed physicist who called me moron when i said that V =
e*phy - e*A was an energy. And shout that V was not an energy.
Look to the online course i link in the blog. Exactly to
http://www.phys.uri.edu/~gerhard/PHY520/mln86.pdf
and search above for "Velocity-dependent potential energy"
Well, Bilge, a supposed PhD for physics do not know dimensional
analisys which is though in the first course of the University.
> Bill Hobba
> Dirk Van de moortel
> YBM
> Dono a.k.a. karandash2...@yahoo.com
This is interesting, because Dono also attacked me in another thread
with insults (and zero physics).
karandash2...@yahoo.com was discovered the other day by voting by
himself in more than 20 times, simulating different users supporting
him. He also voted against me simulating different users. Dono was
probably another of the accounts he used, but he may be using many
more.
> Sam Wormley
Someone just added above:
> 1) I never said that generalized forces had units of force.
I did not remark this in the blog because i considered that was
evident that a generalized force has units of force, but i see now it
is not evident for someone.
The units of Q (generalized forces) are units of force, as follows
from the equation (5.115) on
http://canonicalscience.blogspot.com/
The next micro-thought is about "Relativistic Lagrangian and
limitations of field theory". For the next other i choose dimensional
analysis as topic.
I forgot to say that i accept suggestions for topics for next micro-
thoughts.
I prefer to write about general topics rather than very specific for a
small comunity, and i prefer to write about topics are usually
misunderstood or not well thougth in traditional courses of science
and engineering rather than repeating is said in many textbooks.
Other topics i have in mind are differences between infinitesimals and
differentials (math), trouble with paradoxes of p orbitals
(chemistry), the second law of thermodynamics and living organisms
(biology and physics).
Now bye, i am on vacations.
He called you a moron for saying it is a potential energy. Which is
not the same thing.
[...]
> karandash2...@yahoo.com was discovered the other day by voting by
> himself in more than 20 times, simulating different users supporting
> him. He also voted against me simulating different users. Dono was
> probably another of the accounts he used, but he may be using many
> more.
I've noticed that about Dono a.k.a. karandash2...@yahoo.com also but
one doesn't need multiple accounts to do it. There's a bug in google
groups that allows just one person to vote as many times as they like.
Dono a.k.a. karandash2...@yahoo.com has given me the lowest possible
score on just a few posts literally hundreds of times within an hour.
He has done the same to you in this thread. What else do you expect
from a shit-throwing chimpanzee that can't argue physics?
Notice that your opening post has a vote of 122 users. Dono a.k.a.
karandash2...@yahoo.com voted about a hundred times against you with
the lowest score. To balance that out a little, I voted for you about
20 times with the highest score. It takes too much time to play that
game and I don't want to waste my time like the chimpanzee Dono. I
rather spend my time writing intelligent posts.
Shubee
> ##### CONTACT AND FEEDBACK #####
>
> My newsgroup e-mail account is disabilited because of odd personal
> messages I received.
>
> I also disabilited the comment section in the blog for avoiding to
> insane guys to post more and more nonsense ad infinitum. See below.
>
>
> ##### NOTE #####
>
> The choice for the first posting was motivated by an odd discussion in
> sci.physics.relativity with four _supposed_ physicists (*) confused
> about special relativity, potential energies, Lagrangian and
> Hamiltonian formulations, AAAD, invariances, electromagnetism,
> dimensional analysis, variational principles, forces, and dozens of
> other stuff.
The internet is a jungle. Welcome to the jungle! :-))
So, you must protect yourself against all kinds of virusses and madmen, but
at the same time try to filter out most of the useful stuff. It's well known
to be a constant struggle. Also, you definitely risk to filter out some most
useful postings that attempt to correct some true errors. Possibly (because
of less emotions and more thoughts) the more polite disagreeing messages
have a higher chance to be correct, but I'm not sure if this has been
demonstrated - there may have been studies about that.
Good luck!
Harald
PS: it's not "disabilited" but "disabled"
How can the two of you be sure that it was him? Apparently this regularly
happens in this group, as I noticed statistically impossible numbers of
votes for particular postings and it certainly looks as if he is behind
that. If it can be proved that he is such a dishonest jerk, then I won't do
him a favor again.
> Notice that your opening post has a vote of 122 users. Dono a.k.a.
> karandash2...@yahoo.com voted about a hundred times against you with
> the lowest score. To balance that out a little, I voted for you about
> 20 times with the highest score. It takes too much time to play that
> game and I don't want to waste my time like the chimpanzee Dono. I
> rather spend my time writing intelligent posts.
Yes, that's a rather silly game to play. :-)
Harald
There are several reasons why I strongly suspect Dono. One clue is
that he has posted a really asinine comment attacking me yesterday at
http://groups.google.com/group/sci.physics.relativity/browse_frm/thread/8c81dcd7e7d30e93
supposedly meriting 5 stars 54 times. When I pointed out how truly
demented he was to vote for himself 54 times, he didn't deny it or
respond as any sane person would. For that protest I received 70 1-
star votes and he lowered his estimate of himself to only 13 5-star
votes. Then to throw off suspicions from himself and his dishonesty,
he voted for Eric Gisse in a post critical of me 20 times at 5 stars.
I also think its logical to point out that Dono's posts which attack
me demonstrate the least possible understanding of the physics that
I'm addressing and he does so with greatest blind rage than anyone
else on my list of shit-throwing chimpanzees:
Eric Gisse
Androcles
Bilge
Bill Hobba
Dirk Van de moortel
YBM
Dono a.k.a. karandash2...@yahoo.com
Sam Wormley
Shitbert,
Since I showed you your latest blunder in your latest "masterpiece"
http://www.everythingimportant.org/relativity/simultaneity.htm
you lost your bearings altogether. I know it is tough for the person
that attacks "Einstein's blunders" (Shitbert) to come to terms that
some college kids are laughing at the pompous ass you are. You know,
Shitbert, if you make basic arithmetic blunders like the one you made
in the above link, it is easy for you to become the butt of all
jokes :-)
By whining so hard about SCORES ON USENET, you are giving into his
[subtle] trolling.
I have to admit, it is pretty funny.
A supposed student of physics who think that a generalized force Q has
not units of force (see above) now said:
> He called you a moron for saying it is a potential energy. Which is
> not the same thing.
Apart from trying to liar and insulting me again, it illustrates how
insane are. I have corrected those guys before. The full quote and
Bilge reply was
> > > I wrote L = T - U with U = [e * phi] - [j * A] many, many time ago
> > > in this same thread.
> >
> > Look, moron... That expression is not a potential energy. An energy
> > is the time component of a four-vector, not the scalar product of two
> > four-vectors. jeeezzzz...
Bilge reply are two parts:
i) The first where he says that U is not a potential energy. A look to
the blog and you can see that is not true. However, for avoiding
further misunderstandings, look to
http://www.phys.uri.edu/~gerhard/PHY520/mln86.pdf
Search for "velocity dependent potential energy". It is the same U i
wrote but I used other conventions for the notation.
ii) The second part of the message is when Bilge 'defines' energy and
states that energy is not the scalar product of two four-vectors (U
can be written like that product U = j_bA^b).
I corrected this also in the past and explained the link between P^0
and U. To remark: U has units of energy.
As said before i will add a micro-thought about dimensional analysis
for people can get the units fo physical quantities. In that way, i
wait anyone would get the correct units for U , and i wait anyone
would get the correct units for Q.
Well, he (i asume not woman could be so weird) voted for himself more
than 20 times and for his friend Eric more still. And in one of last
times i see the score he had voted against me more than 350 times. Now
that falsified score was modified and i get only four one stars, which
loks right because just fit with Eric Gisse, Bill Hobba, Karandash2,
and Bilge: 1,2,3, and 4.
Also the falsfied 24 votes supporting Karandash2 reduced to realistic
3: Eric Gisse, Bill Hobba, and Bilge.
Yes, Karandash2 is playing the same dishonest game in this thread. And
i suspect he will do the same with any posting from mine with
independence of its scientific or educative content.
I am not suprised because karandash2 is known by attacking people,
journals, or entire countries without knowing they first.
> Notice that your opening post has a vote of 122 users. Dono a.k.a.
> karandash2...@yahoo.com voted about a hundred times against you with
> the lowest score. To balance that out a little, I voted for you about
> 20 times with the highest score. It takes too much time to play that
> game and I don't want to waste my time like the chimpanzee Dono.
Jenny named that name by Kanradash2 like "Chronic masturbation?"
> I rather spend my time writing intelligent posts.
Yes, please ignore it and use your time for more interesting tasks and
thanks by your attempt to balance the situation.
> Shubee
Thanks.
> So, you must protect yourself against all kinds of virusses and madmen, but
> at the same time try to filter out most of the useful stuff. It's well known
> to be a constant struggle.
Yes, Pmb also said avoid Bilge as the plage or something similar. I
agree. I avoid Bilge and Karandash2 and Bill and Eric, but the problem
is when they submit nonsense: U is not an energy, there is not
relativistic lagrangian, a generalized force has not units of force,
and so on. I can see they are writting nonsense and i ignore them "as
the plage".
I am rather sure most of people here can see that. But what about
novices? What about young students being confused by them? I see many
examples of that each day. In fact, all this began when in group
decided to attack to another post with completely wrong and personal
attacks. Then i defended the other post against they and now i am the
target.
What they can say about me, the hundred of negative votes the insults
and the irrationality i ... [well i will remain polite]. But i see
some other people get problems with they.
I wait the blog can be useful in that way. I wait i can post about
aspects where standard literature can be confusing and about aspects
where standard textbooks fail to do enough emphasis.
For example I see that the Goldstein can be confusing when defines
"generalized potential energy" but however drops the term energy,
using only "generalized potential", which can be confusing for
novices.
The textbook i cited however avoid that confusion by writting
"generalized potential energy" in full. The pdf i cited avoid any
confusion by naming to U potential energy also.
Since they cannot insult me and cannot submit nonsense to the blog it
can remain sane and clean.
My mistakes can be discussed here. This newsgroup would still contain
a lof of noise and insults, but i cannot do nothing for that.
> Also, you definitely risk to filter out some most
> useful postings that attempt to correct some true errors. Possibly (because
> of less emotions and more thoughts) the more polite disagreeing messages
> have a higher chance to be correct, but I'm not sure if this has been
> demonstrated - there may have been studies about that.
>
> Good luck!
> Harald
>
> PS: it's not "disabilited" but "disabled"
Yes, my mistake.
I seems Google administrators analized Karandash2 account (in Google
groups you may be signed before rating)
See recent posts about Karandash2 silly behavior by Jenny
After being discovered by Google, I also noticed that from time on
time Karandash2 try again to falsify the score and add to me from
dozens to a hundred of one-star votes.
Some of his friends would call to hospital. It is urgent.
It does have units of force. Why did I say that it didn't?
The Q's are the forces that cannot be derived from a potential energy
function. The forces that can are stuffed into the Lagrangian.
>
> > He called you a moron for saying it is a potential energy. Which is
> > not the same thing.
>
> Apart from trying to liar and insulting me again, it illustrates how
> insane are. I have corrected those guys before. The full quote and
> Bilge reply was
I'm sorry, you forgot to explain where you obtained your degree in
physics. Where was it that taught you that energy and potential energy
are the same thing in both classical and relativistic systems?
>
> > > > I wrote L = T - U with U = [e * phi] - [j * A] many, many time ago
> > > > in this same thread.
>
> > > Look, moron... That expression is not a potential energy. An energy
> > > is the time component of a four-vector, not the scalar product of two
> > > four-vectors. jeeezzzz...
>
> Bilge reply are two parts:
>
> i) The first where he says that U is not a potential energy. A look to
> the blog and you can see that is not true. However, for avoiding
> further misunderstandings, look to
Welp, this is the entire fucking point.
1) [e * phi] - [j * A] is not a potential energy.
2) L = T - U is not valid for relativistic systems.
3) He said that it is not a potential energy. What he never, EVER even
once claimed is that it _does not_ have units of energy. This is a
mistake you have been repeating for weeks.
>
> http://www.phys.uri.edu/~gerhard/PHY520/mln86.pdf
>
> Search for "velocity dependent potential energy". It is the same U i
> wrote but I used other conventions for the notation.
Who cares?
Find a few more irrelevant PDFs, please.
>
> ii) The second part of the message is when Bilge 'defines' energy and
> states that energy is not the scalar product of two four-vectors (U
> can be written like that product U = j_bA^b).
Um, no, it can't.
Energy in relativistic systems is the time [0'th] component of the
momentum four-vector.
>
> I corrected this also in the past and explained the link between P^0
> and U. To remark: U has units of energy.
Nobody ever disputed that it did have units of energy, moron.
>
> As said before i will add a micro-thought about dimensional analysis
> for people can get the units fo physical quantities. In that way, i
> wait anyone would get the correct units for U , and i wait anyone
> would get the correct units for Q.
Um, force?
What "paradoxes" of p orbitals????
>
> Now bye, i am on vacations.- Hide quoted text -
An apparent paradox with any orbital with nodal planes, which is very
common in chemical literature.
p orbitals have two lobes of high probability density separated by a
nodal plane of zero density. The paradox is usually formulated like
"how does the electron cross the plane?"
But as said above it is just apparent and follows from a misconception
about quantum mechanics.
Funny how all of your problems end up being misconceptions.
So you can so do something for that.
i -> I
they -> them
What "misconception"? The p orbital has a node. s orbitals don't.
So "how does the electron cross the plane"?
Or "Why did the electron cross the nodal plane?"
"To get to the other lobe!"