| Topic: |
Science > Physics |
| User: |
"Gregory L. Hansen" |
| Date: |
07 Feb 2004 06:49:14 PM |
| Object: |
The meaning of a^dag. |
I'm starting to pay a lot of attention to the harmonic oscillator. It
seems to be key to many things. So I thought, why not try the ladder
operators in classical mechanics?
a = sqrt(mw/2) x + i sqrt(1/2mw) p
a^dag = sqrt(mw/2) x - i sqrt(1/2mw) p
The Hamiltonian for the simple harmonic oscillator is then
H = p^2/2m + 1/2 m w^2 x^2 = w a^dag a
No 1/2 term because they commute in classical mechanics. The equations of
motion are, as usual, given by the Poisson brackets with the Hamiltonian.
da/dt = {a,H} = -i w a
da^dag/dt = {a^dag,H} = i w a^dag
With the solutions
a(t) = a(0) exp(-iwt)
a^dag(t) = a^dag(0) exp(iwt)
They're still complex conjugates of each other. a(0) and a^dag(0) are
sqrt(mw/2) x0 +- i sqrt(1/2mw) p0
And so a(0) and a^dag(0) contain the initial conditions. If, for
instance, p0=0, we have
x(t) = sqrt(1/2mw) (a(t) + a^dag(t))
= x0 cos wt
If x0=0 and we have a p0,
x(t) = p0/mw sin wt
Which looks a little odd, but
m*dx/dt = p(t) = p0 cos wt.
It works.
What a represents is a point (x,p) in a phase space with momentum along
the imaginary axis. If we write
H = (sqrt(w) a^dag) (sqrt(w) a)
and can give the axes units of sqrt(energy).
Maybe I'm easily amused, but I was just thrilled to see it all work out so
nicely in the classical case.
--
"For every problem there is a solution which is simple, clean and wrong."
-- Henry Louis Mencken
.
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| User: "" |
|
| Title: Re: The meaning of a^dag. |
07 Feb 2004 07:29:01 PM |
|
|
In article <c0412a$jae$2@hood.uits.indiana.edu>, (Gregory L. Hansen) writes:
I'm starting to pay a lot of attention to the harmonic oscillator. It
seems to be key to many things. So I thought, why not try the ladder
operators in classical mechanics?
a = sqrt(mw/2) x + i sqrt(1/2mw) p
a^dag = sqrt(mw/2) x - i sqrt(1/2mw) p
The Hamiltonian for the simple harmonic oscillator is then
H = p^2/2m + 1/2 m w^2 x^2 = w a^dag a
No 1/2 term because they commute in classical mechanics. The equations of
motion are, as usual, given by the Poisson brackets with the Hamiltonian.
da/dt = {a,H} = -i w a
da^dag/dt = {a^dag,H} = i w a^dag
With the solutions
a(t) = a(0) exp(-iwt)
a^dag(t) = a^dag(0) exp(iwt)
They're still complex conjugates of each other. a(0) and a^dag(0) are
sqrt(mw/2) x0 +- i sqrt(1/2mw) p0
And so a(0) and a^dag(0) contain the initial conditions. If, for
instance, p0=0, we have
x(t) = sqrt(1/2mw) (a(t) + a^dag(t))
= x0 cos wt
If x0=0 and we have a p0,
x(t) = p0/mw sin wt
Which looks a little odd, but
m*dx/dt = p(t) = p0 cos wt.
It works.
What a represents is a point (x,p) in a phase space with momentum along
the imaginary axis. If we write
H = (sqrt(w) a^dag) (sqrt(w) a)
and can give the axes units of sqrt(energy).
Maybe I'm easily amused, but I was just thrilled to see it all work out so
nicely in the classical case.
I recall my own thrill, many years ago, going through the same
"exercise". It is worthwhile. And, it illustrates an important
point: There are many things which we consider "quantum mechanical"
because we first encounter them when studying QM. This, however, is
often a matter of "historical accident". A specific formulation or
concept may exist in different theories, but be prominent in one while
obscure in another. Thus, I think that it greatly contributes to
one's understanding of QM if one goes back and checks which of the
things one learns apply equally well in classical physics. Only this
way one gets a good sense of what is uniquely QM and what isn't.
There was this (very recent) thread about spin of EM waves which was a
good illustration of this point.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
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|
| User: "Edward Green" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 04:08:21 PM |
|
|
wrote in message news:<xhgVb.64$_4.24615@news.uchicago.edu>...
<...>
I recall my own thrill, many years ago, going through the same
"exercise". It is worthwhile. And, it illustrates an important
point: There are many things which we consider "quantum mechanical"
because we first encounter them when studying QM.
I recognize the experience even abstracted from quantum mechanics:
can't recall exact instance, but computer programmer types may feel
that some things are intrinsically "computer science" because they
encountered them in the context of computer science courses, with of
course the consequence that you, the non-computer type, couldn't
possibly understand these things.
Some day ... sigh ... when some rare juxtiposition of mental and
physical and temporal liesure is mine, I make a mental note to sift
many recent threads you and Greg have participated in, because I
recognize there is some really excellent stuff going on here, even if
I have to let the pearls continue rolling just at this moment.
....
.
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| User: "" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 05:47:45 PM |
|
|
In article <2a0cceff.0402091408.60e4124b@posting.google.com>, (Edward Green) writes:
mmeron@cars3.uchicago.edu wrote in message news:<xhgVb.64$_4.24615@news.uchicago.edu>...
<...>
I recall my own thrill, many years ago, going through the same
"exercise". It is worthwhile. And, it illustrates an important
point: There are many things which we consider "quantum mechanical"
because we first encounter them when studying QM.
I recognize the experience even abstracted from quantum mechanics:
can't recall exact instance, but computer programmer types may feel
that some things are intrinsically "computer science" because they
encountered them in the context of computer science courses, with of
course the consequence that you, the non-computer type, couldn't
possibly understand these things.
Yes, have seen this too:-)
Some day ... sigh ... when some rare juxtiposition of mental and
physical and temporal liesure is mine, I make a mental note to sift
many recent threads you and Greg have participated in, because I
recognize there is some really excellent stuff going on here, even if
I have to let the pearls continue rolling just at this moment.
Well, I'm flattered. Thank you.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
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| User: "Gregory L. Hansen" |
|
| Title: Re: The meaning of a^dag. |
08 Feb 2004 07:42:05 AM |
|
|
In article <xhgVb.64$_4.24615@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c0412a$jae$2@hood.uits.indiana.edu>,
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
Maybe I'm easily amused, but I was just thrilled to see it all work out so
nicely in the classical case.
I recall my own thrill, many years ago, going through the same
"exercise". It is worthwhile. And, it illustrates an important
point: There are many things which we consider "quantum mechanical"
because we first encounter them when studying QM.
I'm convinced that not nearly as many things are uniquely quantum
mechanical as many people believe. In fact, I remember seeing a
formulation of classical mechanics that almost exactly parallels quantum;
including operators acting on vectors in something analogous to a Hilbert
space. I wish I could find it again.
Shankar, in the intro to his book on QM, did a very nice solution of the
coupled classical harmonic oscillator in the language of bras and kets.
This, however, is
often a matter of "historical accident". A specific formulation or
concept may exist in different theories, but be prominent in one while
obscure in another. Thus, I think that it greatly contributes to
one's understanding of QM if one goes back and checks which of the
things one learns apply equally well in classical physics. Only this
way one gets a good sense of what is uniquely QM and what isn't.
There was this (very recent) thread about spin of EM waves which was a
good illustration of this point.
And there's a continuing opinion in that thread that if a technique was
first used in a quantum context, it can never again be considered
classical. Eh.
--
"'No user-serviceable parts inside.' I'll be the judge of that!"
.
|
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|
| User: "" |
|
| Title: Re: The meaning of a^dag. |
08 Feb 2004 09:25:19 PM |
|
|
In article <c05ebd$6j6$2@hood.uits.indiana.edu>, (Gregory L. Hansen) writes:
In article <xhgVb.64$_4.24615@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c0412a$jae$2@hood.uits.indiana.edu>,
(Gregory L. Hansen) writes:
Maybe I'm easily amused, but I was just thrilled to see it all work out so
nicely in the classical case.
I recall my own thrill, many years ago, going through the same
"exercise". It is worthwhile. And, it illustrates an important
point: There are many things which we consider "quantum mechanical"
because we first encounter them when studying QM.
I'm convinced that not nearly as many things are uniquely quantum
mechanical as many people believe.
Agreed. Heck, I've seen students believing (for a while) that
Legendre and Laguerre polynomials are "quantum mechanical":-)
In your exposition, above, you came upon one of those things that are
quantum mechanical, i.e. non-zero commutators. And note that any time
you come across a commutator in QM, it comes multiplied by h, so at
the classical limit it goes to zero.
In fact, I remember seeing a
formulation of classical mechanics that almost exactly parallels quantum;
including operators acting on vectors in something analogous to a Hilbert
space. I wish I could find it again.
Shankar, in the intro to his book on QM, did a very nice solution of the
coupled classical harmonic oscillator in the language of bras and kets.
Sure, can be done.
This, however, is
often a matter of "historical accident". A specific formulation or
concept may exist in different theories, but be prominent in one while
obscure in another. Thus, I think that it greatly contributes to
one's understanding of QM if one goes back and checks which of the
things one learns apply equally well in classical physics. Only this
way one gets a good sense of what is uniquely QM and what isn't.
There was this (very recent) thread about spin of EM waves which was a
good illustration of this point.
And there's a continuing opinion in that thread that if a technique was
first used in a quantum context, it can never again be considered
classical. Eh.
Yep. But then, would I've got a dollar for every stupid opinion
expressed on this NG, since I started following it, I would've been
ready to retire in luxury, by now:-)
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
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| User: "Edward Green" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 05:30:44 PM |
|
|
wrote in message news:<z4DVb.11$_4.6775@news.uchicago.edu>...
Yep. But then, would I've got a dollar for every stupid opinion
expressed on this NG, since I started following it, I would've been
ready to retire in luxury, by now:-)
They'd never be able to prove you got the money illicitly, but the IRS
would eventually get you for tax evasion.
.
|
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| User: "" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 06:08:29 PM |
|
|
In article <2a0cceff.0402091530.226891d5@posting.google.com>, (Edward Green) writes:
mmeron@cars3.uchicago.edu wrote in message news:<z4DVb.11$_4.6775@news.uchicago.edu>...
Yep. But then, would I've got a dollar for every stupid opinion
expressed on this NG, since I started following it, I would've been
ready to retire in luxury, by now:-)
They'd never be able to prove you got the money illicitly, but the IRS
would eventually get you for tax evasion.
yep:-(((
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
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| User: "Gregory L. Hansen" |
|
| Title: Re: The meaning of a^dag. |
10 Feb 2004 09:02:55 AM |
|
|
In article <1iVVb.32$_4.16743@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <2a0cceff.0402091530.226891d5@posting.google.com>,
nulldev00@aol.com (Edward Green) writes:
mmeron@cars3.uchicago.edu wrote in message
news:<z4DVb.11$_4.6775@news.uchicago.edu>...
Yep. But then, would I've got a dollar for every stupid opinion
expressed on this NG, since I started following it, I would've been
ready to retire in luxury, by now:-)
They'd never be able to prove you got the money illicitly, but the IRS
would eventually get you for tax evasion.
yep:-(((
If I got a dollar for every stupid opinion expressed on this NG, I'd pay
the taxes on it and I'd grin. Heck, I might even call myself
self-employed, as it would surely be my majority source of income.
--
"When the fool walks through the street, in his lack of understanding he
calls everything foolish." -- Ecclesiastes 10:3, New American Bible
.
|
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| User: "" |
|
| Title: Re: The meaning of a^dag. |
10 Feb 2004 02:06:15 PM |
|
|
In article <c0arqv$1rf$3@hood.uits.indiana.edu>, (Gregory L. Hansen) writes:
In article <1iVVb.32$_4.16743@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <2a0cceff.0402091530.226891d5@posting.google.com>,
nulldev00@aol.com (Edward Green) writes:
mmeron@cars3.uchicago.edu wrote in message
news:<z4DVb.11$_4.6775@news.uchicago.edu>...
Yep. But then, would I've got a dollar for every stupid opinion
expressed on this NG, since I started following it, I would've been
ready to retire in luxury, by now:-)
They'd never be able to prove you got the money illicitly, but the IRS
would eventually get you for tax evasion.
yep:-(((
If I got a dollar for every stupid opinion expressed on this NG, I'd pay
the taxes on it and I'd grin. Heck, I might even call myself
self-employed, as it would surely be my majority source of income.
And a good income it would be.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
|
|
|
|
|
| User: "Gregory L. Hansen" |
|
| Title: Re: The meaning of a^dag. |
08 Feb 2004 09:41:12 PM |
|
|
In article <z4DVb.11$_4.6775@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c05ebd$6j6$2@hood.uits.indiana.edu>,
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
In article <xhgVb.64$_4.24615@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c0412a$jae$2@hood.uits.indiana.edu>,
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
Maybe I'm easily amused, but I was just thrilled to see it all work out so
nicely in the classical case.
I recall my own thrill, many years ago, going through the same
"exercise". It is worthwhile. And, it illustrates an important
point: There are many things which we consider "quantum mechanical"
because we first encounter them when studying QM.
I'm convinced that not nearly as many things are uniquely quantum
mechanical as many people believe.
Agreed. Heck, I've seen students believing (for a while) that
Legendre and Laguerre polynomials are "quantum mechanical":-)
In your exposition, above, you came upon one of those things that are
quantum mechanical, i.e. non-zero commutators. And note that any time
you come across a commutator in QM, it comes multiplied by h, so at
the classical limit it goes to zero.
It's kind of a weird thing that commutators have the value of i*hbar times
the Poisson bracket, but when we see xp in a classical context and want
px, we don't apply the Poisson bracket to it.
In fact, I remember seeing a
formulation of classical mechanics that almost exactly parallels quantum;
including operators acting on vectors in something analogous to a Hilbert
space. I wish I could find it again.
Shankar, in the intro to his book on QM, did a very nice solution of the
coupled classical harmonic oscillator in the language of bras and kets.
Sure, can be done.
I'm appreciating Shankar more and more. I didn't really appreciate him
when I was taking classes, but when the student has trouble with his
homework I think it's easy to blame the textbook. But Shankar sis easy to
read, has a lot of white space for the amount of material he covers, and
he really seems to do QM right, starting from the fundamentals. Angular
momentum begins with a discussion of rotational symmetry, for instance.
Multiple degrees of freedom includes discussion of algebra on direct
product spaces. Now it's not a tricky concept to say X2|x1,x2>=x2|x1,x2>,
but seeing it in the language of direct product spaces gives a better
feeling of what's going on and how to generalize it.
Cohen-Tannoudji, on the other hand, I've liked less and less, and rarely
even look at now. It's amazingly dense for the amount of material
covered, it's a lot of work to get anything out of it, and they don't
really seem to repay the reader with special insight.
I'm going through Goldstein's chapter on small oscillations, and he's
putting that in a language of operators and vectors that has some amazing
parallels with quantum mechanics. And a kinetic energy matrix that plays
the role of the metric tensor in a Riemann space. This was all pretty
much developed in the 19th century. Sometimes I'm amazed at the
sophistication of their methods from way back then.
I don't think any student really appreciates, while he's taking classes,
why texts like Goldstein and Jackson are classics. But years after
classes are done, there are treasures to find. Things not covered, or at
least not comprehended, in class. For me, a distressing amount seems to
be the latter.
--
Irony: "Small businesses want relief from the flood of spam clogging their
in-boxes, but they fear a proposed national 'Do Not Spam' registry will
make it impossible to use e-mail as a marketing tool."
http://www.bizjournals.com/houston/stories/2003/11/10/newscolumn6.html
.
|
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| User: "" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 01:22:10 AM |
|
|
In article <c06vgo$ku3$3@hood.uits.indiana.edu>, (Gregory L. Hansen) writes:
In article <z4DVb.11$_4.6775@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c05ebd$6j6$2@hood.uits.indiana.edu>,
(Gregory L. Hansen) writes:
In article <xhgVb.64$_4.24615@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c0412a$jae$2@hood.uits.indiana.edu>,
(Gregory L. Hansen) writes:
Maybe I'm easily amused, but I was just thrilled to see it all work out so
nicely in the classical case.
I recall my own thrill, many years ago, going through the same
"exercise". It is worthwhile. And, it illustrates an important
point: There are many things which we consider "quantum mechanical"
because we first encounter them when studying QM.
I'm convinced that not nearly as many things are uniquely quantum
mechanical as many people believe.
Agreed. Heck, I've seen students believing (for a while) that
Legendre and Laguerre polynomials are "quantum mechanical":-)
In your exposition, above, you came upon one of those things that are
quantum mechanical, i.e. non-zero commutators. And note that any time
you come across a commutator in QM, it comes multiplied by h, so at
the classical limit it goes to zero.
It's kind of a weird thing that commutators have the value of i*hbar times
the Poisson bracket, but when we see xp in a classical context and want
px, we don't apply the Poisson bracket to it.
No. But, if we start with a classical wave formulation, the
relationship become apparent.
In fact, I remember seeing a
formulation of classical mechanics that almost exactly parallels quantum;
including operators acting on vectors in something analogous to a Hilbert
space. I wish I could find it again.
Shankar, in the intro to his book on QM, did a very nice solution of the
coupled classical harmonic oscillator in the language of bras and kets.
Sure, can be done.
I'm appreciating Shankar more and more. I didn't really appreciate him
when I was taking classes, but when the student has trouble with his
homework I think it's easy to blame the textbook. But Shankar sis easy to
read, has a lot of white space for the amount of material he covers, and
he really seems to do QM right, starting from the fundamentals. Angular
momentum begins with a discussion of rotational symmetry, for instance.
Multiple degrees of freedom includes discussion of algebra on direct
product spaces. Now it's not a tricky concept to say X2|x1,x2>=x2|x1,x2>,
but seeing it in the language of direct product spaces gives a better
feeling of what's going on and how to generalize it.
Cohen-Tannoudji, on the other hand, I've liked less and less, and rarely
even look at now. It's amazingly dense for the amount of material
covered, it's a lot of work to get anything out of it, and they don't
really seem to repay the reader with special insight.
I'm going through Goldstein's chapter on small oscillations, and he's
putting that in a language of operators and vectors that has some amazing
parallels with quantum mechanics. And a kinetic energy matrix that plays
the role of the metric tensor in a Riemann space. This was all pretty
much developed in the 19th century. Sometimes I'm amazed at the
sophistication of their methods from way back then.
Oh, they were very sophisticated. You may recall, I wrote few times
about the "gap". What I mean by this is the gap in the understanding
of the development of physics, by the laymen (especially by the cranks,
of course, but by decent laymen as well). The picture they get (from
High school physics, coffee table popularizations and the like) is as
follows:
1) First, there was Newton.
2) Then, for upwards of two centuries nothing really happened.
3) And then, came the 20th century with relativity and QM.
So, when viewed this way, it appears that relativity and QM are some
arbitrary creations pulled out of thin air. And the fault is with
point (2) above. It is plain false. During these two centuries, a
hell of a lot happened. True, there were no sudden revolutions, no
"paracrap chifts" which are the only thing considered worthy of
attention by those who don't know better. But, there was enormous,
steady development. The physics of the end of 19th century was vastly
different from this at the time of Newton's death. It included
analytical mechanics, statistical mechanics, EM theory. It had at its
disposal an enormous arsenal of mathematical tools (partial
differential equations, Fourier transforms, calculus of variations
etc.) which, though the direct descendants of Newton's work, just
weren't there, at hand, at Newton's time. Heck, most of the math we
use in physics nowadays is 19th century math.
And this, not the physics at the time of Newton's death, was the
landscape within which relativity and QM were born.
I don't think any student really appreciates, while he's taking classes,
why texts like Goldstein and Jackson are classics. But years after
classes are done, there are treasures to find.
Well, that's pretty much the definition of a "classic". Something you
can keep coming back to and always find it a worthwhile experience.
Things not covered, or at
least not comprehended, in class. For me, a distressing amount seems to
be the latter.
Nothing to be distressed about, this is quite natural. When
encountering a totally alien terrain, at first pass you don't even
know what matters and what is worthy of attention. If you're bright
and diligent then, by the end of the first pass, you may be in
position to start studying seriously.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
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| User: "Gregory L. Hansen" |
|
| Title: Re: The meaning of a^dag. |
13 Feb 2004 03:02:31 PM |
|
|
wrote in message news:<
Oh, they were very sophisticated. You may recall, I wrote few times
about the "gap". What I mean by this is the gap in the understanding
of the development of physics, by the laymen (especially by the cranks,
of course, but by decent laymen as well). The picture they get (from
High school physics, coffee table popularizations and the like) is as
follows:
Update for sci.physics discussions...
0) Aristotle philosophized about nature, but fucked it up.
0.5) Aristotle remains the pinnacle of science for a thousand years.
0.6) They laughed Galileo, but Galileo was right. Only Galileo was right.
1) First, there was Newton.
2) Then, for upwards of two centuries nothing really happened.
3) And then, came the 20th century with relativity and QM.
4) And nothing else happened from the 1920's to the present day.
.
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| User: "" |
|
| Title: Re: The meaning of a^dag. |
13 Feb 2004 07:28:14 PM |
|
|
In article <8ce5c97e.0402131302.450ffe64@posting.google.com>, (Gregory L. Hansen) writes:
mmeron@cars3.uchicago.edu wrote in message news:<
Oh, they were very sophisticated. You may recall, I wrote few times
about the "gap". What I mean by this is the gap in the understanding
of the development of physics, by the laymen (especially by the cranks,
of course, but by decent laymen as well). The picture they get (from
High school physics, coffee table popularizations and the like) is as
follows:
Update for sci.physics discussions...
0) Aristotle philosophized about nature, but fucked it up.
0.5) Aristotle remains the pinnacle of science for a thousand years.
0.6) They laughed Galileo, but Galileo was right. Only Galileo was right.
1) First, there was Newton.
2) Then, for upwards of two centuries nothing really happened.
3) And then, came the 20th century with relativity and QM.
4) And nothing else happened from the 1920's to the present day.
Yep, that's pretty much the complete version:-) Especially point (4).
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
|
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| User: "Gregory L. Hansen" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 09:22:10 AM |
|
|
In article <CyGVb.16$_4.8290@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c06vgo$ku3$3@hood.uits.indiana.edu>,
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
In article <z4DVb.11$_4.6775@news.uchicago.edu>,
I'm going through Goldstein's chapter on small oscillations, and he's
putting that in a language of operators and vectors that has some amazing
parallels with quantum mechanics. And a kinetic energy matrix that plays
the role of the metric tensor in a Riemann space. This was all pretty
much developed in the 19th century. Sometimes I'm amazed at the
sophistication of their methods from way back then.
Oh, they were very sophisticated. You may recall, I wrote few times
about the "gap". What I mean by this is the gap in the understanding
of the development of physics, by the laymen (especially by the cranks,
of course, but by decent laymen as well). The picture they get (from
High school physics, coffee table popularizations and the like) is as
follows:
1) First, there was Newton.
2) Then, for upwards of two centuries nothing really happened.
3) And then, came the 20th century with relativity and QM.
So, when viewed this way, it appears that relativity and QM are some
arbitrary creations pulled out of thin air. And the fault is with
Yeah, that seems about right. But sad to say, I think physics classes
modify that view only by sticking Maxwell's equations somewhere in the
middle of point 2. (Maxwell's equations are, of course, completely
ignored by historians who like to attach specific names to specific
inventions like the radio or the electrical generator.) Some of the
mathematical apparatus may be dimly remembered from the linear algebra
section of the calculus sequence, but a lot of it is introduced for the
first time in quantum mechanics, and only in quantum mechanics. And
differential geometry is not covered at all, except by a few students
that learn it in a course in general relativity and then are completely
astonished that a matrix of kinetic energies can be considered the metric
tensor in a many-body Riemann phase space.
I suppose some of that is inevitable, as physics advances and new material
must be added to the curriculum. Students are exposed to operator methods
in quantum mechanics because they don't do rigid body mechanics. Previous
students may have been astounded at applications of operator methods that
don't involve rotations.
Things not covered, or at
least not comprehended, in class. For me, a distressing amount seems to
be the latter.
Nothing to be distressed about, this is quite natural. When
encountering a totally alien terrain, at first pass you don't even
know what matters and what is worthy of attention. If you're bright
and diligent then, by the end of the first pass, you may be in
position to start studying seriously.
It was always a problem for me because I never liked to go on if I didn't
think I understood the fundamentals, and we never spent much time in class
on that. Like this thing about quantizing the field as if it were
harmonic oscillators. I Just Didn't Get It. It's starting to make sense
now.
I don't know if that makes me careful, or just slow.
--
Irony: "Small businesses want relief from the flood of spam clogging their
in-boxes, but they fear a proposed national 'Do Not Spam' registry will
make it impossible to use e-mail as a marketing tool."
http://www.bizjournals.com/houston/stories/2003/11/10/newscolumn6.html
.
|
|
|
| User: "" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 03:35:31 PM |
|
|
In article <c088j2$37k$1@hood.uits.indiana.edu>, (Gregory L. Hansen) writes:
In article <CyGVb.16$_4.8290@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c06vgo$ku3$3@hood.uits.indiana.edu>,
(Gregory L. Hansen) writes:
In article <z4DVb.11$_4.6775@news.uchicago.edu>,
I'm going through Goldstein's chapter on small oscillations, and he's
putting that in a language of operators and vectors that has some amazing
parallels with quantum mechanics. And a kinetic energy matrix that plays
the role of the metric tensor in a Riemann space. This was all pretty
much developed in the 19th century. Sometimes I'm amazed at the
sophistication of their methods from way back then.
Oh, they were very sophisticated. You may recall, I wrote few times
about the "gap". What I mean by this is the gap in the understanding
of the development of physics, by the laymen (especially by the cranks,
of course, but by decent laymen as well). The picture they get (from
High school physics, coffee table popularizations and the like) is as
follows:
1) First, there was Newton.
2) Then, for upwards of two centuries nothing really happened.
3) And then, came the 20th century with relativity and QM.
So, when viewed this way, it appears that relativity and QM are some
arbitrary creations pulled out of thin air. And the fault is with
Yeah, that seems about right. But sad to say, I think physics classes
modify that view only by sticking Maxwell's equations somewhere in the
middle of point 2. (Maxwell's equations are, of course, completely
ignored by historians who like to attach specific names to specific
inventions like the radio or the electrical generator.)
Yes, too many historians confuse science and technology.
Some of the
mathematical apparatus may be dimly remembered from the linear algebra
section of the calculus sequence, but a lot of it is introduced for the
first time in quantum mechanics, and only in quantum mechanics. And
differential geometry is not covered at all, except by a few students
that learn it in a course in general relativity and then are completely
astonished that a matrix of kinetic energies can be considered the metric
tensor in a many-body Riemann phase space.
Aha. And that, the equations of trajectories in *classical
mechanics*, given a force proportional to m (i.e. classical gravity)
are formally the same as equations of geodesics in an appropriate
space. Yes.
I suppose some of that is inevitable, as physics advances and new material
must be added to the curriculum. Students are exposed to operator methods
in quantum mechanics because they don't do rigid body mechanics. Previous
students may have been astounded at applications of operator methods that
don't involve rotations.
Some may be inevitable but I think that there is too much "rushing
forward". Especially, I believe that a good solid course of
analytical mechanics should be an absolute "must" before even touching
QM.
Things not covered, or at
least not comprehended, in class. For me, a distressing amount seems to
be the latter.
Nothing to be distressed about, this is quite natural. When
encountering a totally alien terrain, at first pass you don't even
know what matters and what is worthy of attention. If you're bright
and diligent then, by the end of the first pass, you may be in
position to start studying seriously.
It was always a problem for me because I never liked to go on if I didn't
think I understood the fundamentals, and we never spent much time in class
on that. Like this thing about quantizing the field as if it were
harmonic oscillators. I Just Didn't Get It. It's starting to make sense
now.
The thing is, you might've believed that for others it did make sense
on first pass. No, it didn't:-)
I don't know if that makes me careful, or just slow.
There is a fine balance one has to strike here. If you rush ahead too
fast, leaving too much stuff that is not understood behind you, you
get nowhere. If you insist on never moving ahead till everything is
clear you, well, get nowhere. The proper balance point is individual,
everybody has to find it for himself. But it is a good idea, after
you've finished learning a given topic (not immediately after, have to
give it some time to stew in your brain) to get back to it and study
it at leisure. I remember being shocked, at times, to find out to
what extent I did not understand things I was sure I do understand.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
|
| User: "Ken Muldrew" |
|
| Title: Re: The meaning of a^dag. |
10 Feb 2004 01:36:14 PM |
|
|
wrote:
There is a fine balance one has to strike here. If you rush ahead too
fast, leaving too much stuff that is not understood behind you, you
get nowhere. If you insist on never moving ahead till everything is
clear you, well, get nowhere. The proper balance point is individual,
everybody has to find it for himself. But it is a good idea, after
you've finished learning a given topic (not immediately after, have to
give it some time to stew in your brain) to get back to it and study
it at leisure. I remember being shocked, at times, to find out to
what extent I did not understand things I was sure I do understand.
The great thing about teaching is discovering how little you
understood of the subject that you thought you had mastered. Unless,
of course, you have sharper knives at your throat and can't dedicate
any time to thinking about what you're teaching and how best to do it,
in which case you just regurgitate everything the way you learned it.
Ken Muldrew
kmuldrezw@ucalgazry.ca
(remove all letters after y in the alphabet)
.
|
|
|
| User: "" |
|
| Title: Re: The meaning of a^dag. |
10 Feb 2004 03:50:46 PM |
|
|
In article <402931e2.95454418@news.ucalgary.ca>, (Ken Muldrew) writes:
mmeron@cars3.uchicago.edu wrote:
There is a fine balance one has to strike here. If you rush ahead too
fast, leaving too much stuff that is not understood behind you, you
get nowhere. If you insist on never moving ahead till everything is
clear you, well, get nowhere. The proper balance point is individual,
everybody has to find it for himself. But it is a good idea, after
you've finished learning a given topic (not immediately after, have to
give it some time to stew in your brain) to get back to it and study
it at leisure. I remember being shocked, at times, to find out to
what extent I did not understand things I was sure I do understand.
The great thing about teaching is discovering how little you
understood of the subject that you thought you had mastered.
Indeed, the ultimate way of learning is teaching.
Unless,
of course, you have sharper knives at your throat and can't dedicate
any time to thinking about what you're teaching and how best to do it,
in which case you just regurgitate everything the way you learned it.
And neither you, nor your students, benefit in any way from such
process.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
|
| User: "Ken Muldrew" |
|
| Title: Re: The meaning of a^dag. |
10 Feb 2004 06:08:25 PM |
|
|
wrote:
And neither you, nor your students, benefit in any way from such
process.
Administrators seem quite content with it. :-(
Ken Muldrew
kmuldrezw@ucalgazry.ca
(remove all letters after y in the alphabet)
.
|
|
|
|
| User: "" |
|
| Title: Re: The meaning of a^dag. |
10 Feb 2004 07:32:50 PM |
|
|
In article <40297259.14238357@news.ucalgary.ca>, (Ken Muldrew) writes:
mmeron@cars3.uchicago.edu wrote:
And neither you, nor your students, benefit in any way from such
process.
Administrators seem quite content with it. :-(
Yes, I know:-( Administrators are primarily concerned with everything
being done according to regulations. As to whether anything useful is
being done, they couldn't care less.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
|
|
|
|
| User: "Gregory L. Hansen" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 04:13:18 PM |
|
|
In article <D2TVb.25$_4.15405@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c088j2$37k$1@hood.uits.indiana.edu>,
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
In article <CyGVb.16$_4.8290@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c06vgo$ku3$3@hood.uits.indiana.edu>,
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
In article <z4DVb.11$_4.6775@news.uchicago.edu>,
I'm going through Goldstein's chapter on small oscillations, and he's
putting that in a language of operators and vectors that has some amazing
parallels with quantum mechanics. And a kinetic energy matrix that plays
the role of the metric tensor in a Riemann space. This was all pretty
much developed in the 19th century. Sometimes I'm amazed at the
sophistication of their methods from way back then.
Oh, they were very sophisticated. You may recall, I wrote few times
Just a thought, but maybe we shouldn't expect students of today to cover
the same methods that the graybeards of yestercentury used. There's a lot
that the 19th century students didn't cover, too.
about the "gap". What I mean by this is the gap in the understanding
of the development of physics, by the laymen (especially by the cranks,
of course, but by decent laymen as well). The picture they get (from
High school physics, coffee table popularizations and the like) is as
follows:
1) First, there was Newton.
2) Then, for upwards of two centuries nothing really happened.
3) And then, came the 20th century with relativity and QM.
So, when viewed this way, it appears that relativity and QM are some
arbitrary creations pulled out of thin air. And the fault is with
Yeah, that seems about right. But sad to say, I think physics classes
modify that view only by sticking Maxwell's equations somewhere in the
middle of point 2. (Maxwell's equations are, of course, completely
ignored by historians who like to attach specific names to specific
inventions like the radio or the electrical generator.)
Yes, too many historians confuse science and technology.
Some of the
mathematical apparatus may be dimly remembered from the linear algebra
section of the calculus sequence, but a lot of it is introduced for the
first time in quantum mechanics, and only in quantum mechanics. And
differential geometry is not covered at all, except by a few students
that learn it in a course in general relativity and then are completely
astonished that a matrix of kinetic energies can be considered the metric
tensor in a many-body Riemann phase space.
Aha. And that, the equations of trajectories in *classical
mechanics*, given a force proportional to m (i.e. classical gravity)
are formally the same as equations of geodesics in an appropriate
space. Yes.
I suppose some of that is inevitable, as physics advances and new material
must be added to the curriculum. Students are exposed to operator methods
in quantum mechanics because they don't do rigid body mechanics. Previous
students may have been astounded at applications of operator methods that
don't involve rotations.
Some may be inevitable but I think that there is too much "rushing
forward". Especially, I believe that a good solid course of
analytical mechanics should be an absolute "must" before even touching
QM.
I don't know if I'd say it's a must before a first exposure. Students
are understandably eager to get to QM, and it takes multiple exposures for
it to sink in. But the first course should be a lot more leisurely, the
mechanical and mathematical apparatus developed within it rather than
relying on pre-requisits and saying "Just write down the energy and call
it the Hamiltonian", and little effort should be made to find useful
results. For instance, I don't think perturbation theory has any business
in a first QM course. Or maybe even higher-order corrections to the
hydrogen atom. Quantum computing definitely belongs in a first QM course,
because it's a simple application of matrix methods.
Things not covered, or at
least not comprehended, in class. For me, a distressing amount seems to
be the latter.
Nothing to be distressed about, this is quite natural. When
encountering a totally alien terrain, at first pass you don't even
know what matters and what is worthy of attention. If you're bright
and diligent then, by the end of the first pass, you may be in
position to start studying seriously.
It was always a problem for me because I never liked to go on if I didn't
think I understood the fundamentals, and we never spent much time in class
on that. Like this thing about quantizing the field as if it were
harmonic oscillators. I Just Didn't Get It. It's starting to make sense
now.
The thing is, you might've believed that for others it did make sense
on first pass. No, it didn't:-)
I don't know if that makes me careful, or just slow.
There is a fine balance one has to strike here. If you rush ahead too
fast, leaving too much stuff that is not understood behind you, you
get nowhere. If you insist on never moving ahead till everything is
clear you, well, get nowhere. The proper balance point is individual,
everybody has to find it for himself. But it is a good idea, after
you've finished learning a given topic (not immediately after, have to
give it some time to stew in your brain) to get back to it and study
it at leisure. I remember being shocked, at times, to find out to
what extent I did not understand things I was sure I do understand.
It's pretty much just as you say. Many times I've gone through
fundamentals at the beginning, and wondered why I should care, they meant
nothing to me. I need to see applications, let the relevance start to
sink in. The problem is that in class, we never went *back* to the
fundamentals. And I never really had all my homework completed at any
given time, and so never really saw an opportunity go back by myself.
--
"The preferred method of entering a building is to use a tank main gun
round, direct fire artillery round, or TOW, Dragon, or Hellfire missile to
clear the first room." -- THE RANGER HANDBOOK U.S. Army, 1992
.
|
|
|
| User: "" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 06:03:15 PM |
|
|
In article <c090lu$dmt$2@hood.uits.indiana.edu>, (Gregory L. Hansen) writes:
In article <D2TVb.25$_4.15405@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c088j2$37k$1@hood.uits.indiana.edu>,
(Gregory L. Hansen) writes:
In article <CyGVb.16$_4.8290@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c06vgo$ku3$3@hood.uits.indiana.edu>,
(Gregory L. Hansen) writes:
In article <z4DVb.11$_4.6775@news.uchicago.edu>,
I'm going through Goldstein's chapter on small oscillations, and he's
putting that in a language of operators and vectors that has some amazing
parallels with quantum mechanics. And a kinetic energy matrix that plays
the role of the metric tensor in a Riemann space. This was all pretty
much developed in the 19th century. Sometimes I'm amazed at the
sophistication of their methods from way back then.
Oh, they were very sophisticated. You may recall, I wrote few times
Just a thought, but maybe we shouldn't expect students of today to cover
the same methods that the graybeards of yestercentury used. There's a lot
that the 19th century students didn't cover, too.
For sure. The tricky part is to decide what is the *core* that must
be covered.
...
I suppose some of that is inevitable, as physics advances and new material
must be added to the curriculum. Students are exposed to operator methods
in quantum mechanics because they don't do rigid body mechanics. Previous
students may have been astounded at applications of operator methods that
don't involve rotations.
Some may be inevitable but I think that there is too much "rushing
forward". Especially, I believe that a good solid course of
analytical mechanics should be an absolute "must" before even touching
QM.
I don't know if I'd say it's a must before a first exposure. Students
are understandably eager to get to QM, and it takes multiple exposures for
it to sink in. But the first course should be a lot more leisurely, the
mechanical and mathematical apparatus developed within it rather than
relying on pre-requisits and saying "Just write down the energy and call
it the Hamiltonian", and little effort should be made to find useful
results. For instance, I don't think perturbation theory has any business
in a first QM course. Or maybe even higher-order corrections to the
hydrogen atom. Quantum computing definitely belongs in a first QM course,
because it's a simple application of matrix methods.
Well, yes, there is always lots of stuff that's a result of inertia.
You know, "I learned it this way, was good enough for me, so should be
good enough for everybody.":-)
The thing to consider is the following. Physics student nowadays
spends about as much time on his studies as his counterpart circa 1900
did. But, the amount of stuff to learn is vastly different. So, some
judicious choices are required regarding what should be studied and
how. So far, I haven't seen that much attention to this issue.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
|
| User: "Herman Trivilino" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 10:31:48 PM |
|
|
<mmeron@cars3.uchicago.edu> wrote ...
The thing to consider is the following. Physics student nowadays
spends about as much time on his studies as his counterpart circa 1900
did. But, the amount of stuff to learn is vastly different. So, some
judicious choices are required regarding what should be studied and
how. So far, I haven't seen that much attention to this issue.
Not at the junior and senior level where the student encounters an
introduction to quantum mechanics, no. But certainly at the freshman and
sophomore level (where the introductory sequence is taught) there has at
least been some attention to this issue. Of course, some point to the
Feynman Lectures of the early 60's as the beginning of this process of
paying attention to amount of stuff that ought to be taught. They also then
point to the time when the decline in the number of physics majors began in
American universities. It was at about the same time!
-----= Posted via Newsfeeds.Com, Uncensored Usenet News =-----
http://www.newsfeeds.com - The #1 Newsgroup Service in the World!
-----== Over 100,000 Newsgroups - 19 Different Servers! =-----
.
|
|
|
| User: "" |
|
| Title: Re: The meaning of a^dag. |
09 Feb 2004 11:41:18 PM |
|
|
In article <40285eb3$1_8@corp.newsgroups.com>, "Herman Trivilino" <physhead@kingwoodREMOVECAPScable.com> writes:
<mmeron@cars3.uchicago.edu> wrote ...
The thing to consider is the following. Physics student nowadays
spends about as much time on his studies as his counterpart circa 1900
did. But, the amount of stuff to learn is vastly different. So, some
judicious choices are required regarding what should be studied and
how. So far, I haven't seen that much attention to this issue.
Not at the junior and senior level where the student encounters an
introduction to quantum mechanics, no. But certainly at the freshman and
sophomore level (where the introductory sequence is taught) there has at
least been some attention to this issue. Of course, some point to the
Feynman Lectures of the early 60's as the beginning of this process of
paying attention to amount of stuff that ought to be taught. They also then
point to the time when the decline in the number of physics majors began in
American universities. It was at about the same time!
An interesting observation.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
|
|
|
| User: "Gregory L. Hansen" |
|
| Title: Re: The meaning of a^dag. |
10 Feb 2004 09:07:26 AM |
|
|
In article <7dVVb.31$_4.16364@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c090lu$dmt$2@hood.uits.indiana.edu>,
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
Just a thought, but maybe we shouldn't expect students of today to cover
the same methods that the graybeards of yestercentury used. There's a lot
that the 19th century students didn't cover, too.
For sure. The tricky part is to decide what is the *core* that must
be covered.
...
I suppose some of that is inevitable, as physics advances and new material
must be added to the curriculum. Students are exposed to operator methods
in quantum mechanics because they don't do rigid body mechanics. Previous
students may have been astounded at applications of operator methods that
don't involve rotations.
Some may be inevitable but I think that there is too much "rushing
forward". Especially, I believe that a good solid course of
analytical mechanics should be an absolute "must" before even touching
QM.
I don't know if I'd say it's a must before a first exposure. Students
are understandably eager to get to QM, and it takes multiple exposures for
it to sink in. But the first course should be a lot more leisurely, the
mechanical and mathematical apparatus developed within it rather than
relying on pre-requisits and saying "Just write down the energy and call
it the Hamiltonian", and little effort should be made to find useful
results. For instance, I don't think perturbation theory has any business
in a first QM course. Or maybe even higher-order corrections to the
hydrogen atom. Quantum computing definitely belongs in a first QM course,
because it's a simple application of matrix methods.
Well, yes, there is always lots of stuff that's a result of inertia.
You know, "I learned it this way, was good enough for me, so should be
good enough for everybody.":-)
The thing to consider is the following. Physics student nowadays
spends about as much time on his studies as his counterpart circa 1900
did. But, the amount of stuff to learn is vastly different. So, some
judicious choices are required regarding what should be studied and
how. So far, I haven't seen that much attention to this issue.
Not so many months ago, Physics Today gave some attention to that issue.
One school tried an approach of many short units, each devoted to a
specific topic like solving the wave equation. And then it would be
applied to electromagnetism, gravity, quantum mechanics, acoustics...
across a whole spectrum of physics. Apparantly the students appreciated
the approach and were better able to apply the lessons they'd learned to
novel problems.
--
"Suppose you were an idiot... And suppose you were a member of
Congress... But I repeat myself." - Mark Twain
.
|
|
|
| User: "" |
|
| Title: Re: The meaning of a^dag. |
10 Feb 2004 02:27:02 PM |
|
|
In article <c0as3e$1rf$4@hood.uits.indiana.edu>, (Gregory L. Hansen) writes:
In article <7dVVb.31$_4.16364@news.uchicago.edu>,
<mmeron@cars3.uchicago.edu> wrote:
In article <c090lu$dmt$2@hood.uits.indiana.edu>,
(Gregory L. Hansen) writes:
Just a thought, but maybe we shouldn't expect students of today to cover
the same methods that the graybeards of yestercentury used. There's a lot
that the 19th century students didn't cover, too.
For sure. The tricky part is to decide what is the *core* that must
be covered.
...
I suppose some of that is inevitable, as physics advances and new material
must be added to the curriculum. Students are exposed to operator methods
in quantum mechanics because they don't do rigid body mechanics. Previous
students may have been astounded at applications of operator methods that
don't involve rotations.
Some may be inevitable but I think that there is too much "rushing
forward". Especially, I believe that a good solid course of
analytical mechanics should be an absolute "must" before even touching
QM.
I don't know if I'd say it's a must before a first exposure. Students
are understandably eager to get to QM, and it takes multiple exposures for
it to sink in. But the first course should be a lot more leisurely, the
mechanical and mathematical apparatus developed within it rather than
relying on pre-requisits and saying "Just write down the energy and call
it the Hamiltonian", and little effort should be made to find useful
results. For instance, I don't think perturbation theory has any business
in a first QM course. Or maybe even higher-order corrections to the
hydrogen atom. Quantum computing definitely belongs in a first QM course,
because it's a simple application of matrix methods.
Well, yes, there is always lots of stuff that's a result of inertia.
You know, "I learned it this way, was good enough for me, so should be
good enough for everybody.":-)
The thing to consider is the following. Physics student nowadays
spends about as much time on his studies as his counterpart circa 1900
did. But, the amount of stuff to learn is vastly different. So, some
judicious choices are required regarding what should be studied and
how. So far, I haven't seen that much attention to this issue.
Not so many months ago, Physics Today gave some attention to that issue.
One school tried an approach of many short units, each devoted to a
specific topic like solving the wave equation. And then it would be
applied to electromagnetism, gravity, quantum mechanics, acoustics...
across a whole spectrum of physics. Apparantly the students appreciated
the approach and were better able to apply the lessons they'd learned to
novel problems.
Hmm, may work. Though, it may cause other problems. But it is
certainly worth following up. Possibly few different methods can be
tried in different settings and compared, over time.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
|
|
|
| User: "Timo Nieminen" |
|
| Title: Re: The meaning of a^dag. |
11 Feb 2004 04:36:50 AM |
|
|
On Tue, 10 Feb 2004 wrote:
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
Not so many months ago, Physics Today gave some attention to that issue.
One school tried an approach of many short units, each devoted to a
specific topic like solving the wave equation. And then it would be
applied to electromagnetism, gravity, quantum mechanics, acoustics...
across a whole spectrum of physics. Apparantly the students appreciated
the approach and were better able to apply the lessons they'd learned to
novel problems.
Hmm, may work. Though, it may cause other problems. But it is
certainly worth following up. Possibly few different methods can be
tried in different settings and compared, over time.
Basically why they made us do maths courses on DEs, linear algebra,
analysis, etc. Didn't work, though.
Uni of Sydney recently set up a very interesting computational science
course (as in a field one can major in). Learn computational and
mathematical tools, learn to apply to a wide variety of problems. No
previous computer programming experience required, no maths beyond high
school maths required.
Needless to say, computer science students taking the computational
science subjects for easy credit sometimes found it not so easy.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
.
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| User: "Gregory L. Hansen" |
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| Title: Re: The meaning of a^dag. |
10 Feb 2004 07:56:02 PM |
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In article <Pine.LNX.4.50.0402111026210.14768-100000@localhost>,
Timo Nieminen <timo@physics.uq.edu.au> wrote:
On Tue, 10 Feb 2004 wrote:
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
Not so many months ago, Physics Today gave some attention to that issue.
One school tried an approach of many short units, each devoted to a
specific topic like solving the wave equation. And then it would be
applied to electromagnetism, gravity, quantum mechanics, acoustics...
across a whole spectrum of physics. Apparantly the students appreciated
the approach and were better able to apply the lessons they'd learned to
novel problems.
Hmm, may work. Though, it may cause other problems. But it is
certainly worth following up. Possibly few different methods can be
tried in different settings and compared, over time.
Basically why they made us do maths courses on DEs, linear algebra,
analysis, etc. Didn't work, though.
I went through linear algebra in the usual calculus sequence. When I took
a quantum mechanics course, years later, I had to learn it all over from
the beginning, while trying to meet the deadline for my physics homework.
I'd gotten one of those little white and green outline books, mainly
because I'd completely forgotten that material even existed in my math
texts. At the time, it looked like something I'd never done before.
Uni of Sydney recently set up a very interesting computational science
course (as in a field one can major in). Learn computational and
mathematical tools, learn to apply to a wide variety of problems. No
previous computer programming experience required, no maths beyond high
school maths required.
Needless to say, computer science students taking the computational
science subjects for easy credit sometimes found it not so easy.
Ah, yes, the "sleeper" course.
--
"The average person, during a single day, deposits in his or her underwear
an amount of fecal bacteria equal to the weight of a quarter of a peanut."
-- Dr. Robert Buckman, Human Wildlife, p119.
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| User: "Timo Nieminen" |
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| Title: Re: The meaning of a^dag. |
11 Feb 2004 07:11:17 AM |
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On Wed, 11 Feb 2004, Gregory L. Hansen wrote:
Timo Nieminen <timo@physics.uq.edu.au> wrote:
I went through linear algebra in the usual calculus sequence. When I took
a quantum mechanics course, years later, I had to learn it all over from
the beginning, while trying to meet the deadline for my physics homework.
I'd gotten one of those little white and green outline books, mainly
because I'd completely forgotten that material even existed in my math
texts. At the time, it looked like something I'd never done before.
Well, not like one learns that much as an undergrad, beyond learning how
to cram for exams. Eventually you need to use something, and then you
learn it properly.
A nice set of well-designed maths for physics subjects might be good for a
small fraction of the students, but I expect that most would simply ignore
it during the semester, cram for the exam, and promptly purge from
memory.
One alternative would be to have a single full-time, 6 semester course.
Not practical, of course, but it would work, and work well. Lots of
assignments, fewer exams.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
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| User: "" |
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| Title: Re: The meaning of a^dag. |
10 Feb 2004 07:04:33 PM |
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In article <Pine.LNX.4.50.0402111026210.14768-100000@localhost>, Timo Nieminen <timo@physics.uq.edu.au> writes:
On Tue, 10 Feb 2004 wrote:
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
Not so many months ago, Physics Today gave some attention to that issue.
One school tried an approach of many short units, each devoted to a
specific topic like solving the wave equation. And then it would be
applied to electromagnetism, gravity, quantum mechanics, acoustics...
across a whole spectrum of physics. Apparantly the students appreciated
the approach and were better able to apply the lessons they'd learned to
novel problems.
Hmm, may work. Though, it may cause other problems. But it is
certainly worth following up. Possibly few different methods can be
tried in different settings and compared, over time.
Basically why they made us do maths courses on DEs, linear algebra,
analysis, etc. Didn't work, though.
Well, I wouldn't quite say that it didn't work but certainly big part
of this was wasted since, at the time you learn it, it is divorced
from applications. And by the time you get to applications, you
already forgot what you've learned.
Uni of Sydney recently set up a very interesting computational science
course (as in a field one can major in). Learn computational and
mathematical tools, learn to apply to a wide variety of problems. No
previous computer programming experience required, no maths beyond high
school maths required.
Needless to say, computer science students taking the computational
science subjects for easy credit sometimes found it not so easy.
I can believe this.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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