| Topic: |
Science > Physics |
| User: |
"Edward Green" |
| Date: |
05 Nov 2003 07:41:39 PM |
| Object: |
Why is the wave function complex |
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
For example: the hydrogenic atom is a success story for
non-relativistic quantum mechanics. Has anybody looked at such things
carefully to see if we couldn't have gotten the same numbers out the
end from a purely real theory? What do complex numbers capture as
vehicles of a theory which real numbers cannot, given that when we are
done turning the crank, we expect to get real results?
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| User: "Old Man" |
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| Title: Re: Why is the wave function complex |
06 Nov 2003 03:53:53 PM |
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"Edward Green" <nulldev00@aol.com> wrote in message
news:2a0cceff.0311051741.5ce7c052@posting.google.com...
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
For example: the hydrogenic atom is a success story for
non-relativistic quantum mechanics. Has anybody looked at such things
carefully to see if we couldn't have gotten the same numbers out the
end from a purely real theory? What do complex numbers capture as
vehicles of a theory which real numbers cannot, given that when we are
done turning the crank, we expect to get real results?
Well .... , good question. The wave function has two characteristics
at every point in space and time: amplitude and phase. Complex math
is a convenient way of dealing with this. In scattering calculations, every
orbital angular momentum state has an energy dependent amplitude and
phase shift. The cross section can be most meaningfully and completely
described in terms of an energy dependent amplitude and phase shift for
each angular momentum scattering state wherein a resonance in one of
the phase shifts reveals the energy and spin state of an excited state in
the composite system (beam particle + target particle).
Interestingly, the scattering potential can be complex, U(r) = V(r) + iW(r),
wherein V(r) represents elastic scattering while W(r) represents reactions.
The imaginary part of the potential removes particles from the incident
particle beam via target particle breakup, such as d +d -> d + p + n,
reaction, such as d + d -> t + p, or absorption such as p + n -> d. In the
case of a complex potential, the phase shifts are also complex.
[Old Man]
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| User: "Edward Green" |
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| Title: Re: Why is the wave function complex |
07 Nov 2003 05:58:55 PM |
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"Old Man" <nomail@nomail.net> wrote in message news:<3fa97173_3@newsfeed.slurp.net>...
"Edward Green" <nulldev00@aol.com> wrote in message
news:2a0cceff.0311051741.5ce7c052@posting.google.com...
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
For example: the hydrogenic atom is a success story for
non-relativistic quantum mechanics. Has anybody looked at such things
carefully to see if we couldn't have gotten the same numbers out the
end from a purely real theory? What do complex numbers capture as
vehicles of a theory which real numbers cannot, given that when we are
done turning the crank, we expect to get real results?
Well .... , good question.
Yeah ... and I'm getting such good answers it's requiring an
uncomfortable amount of thought.
The wave function has two characteristics
at every point in space and time: amplitude and phase. Complex math
is a convenient way of dealing with this. In scattering calculations, every
orbital angular momentum state has an energy dependent amplitude and
phase shift. The cross section can be most meaningfully and completely
described in terms of an energy dependent amplitude and phase shift for
each angular momentum scattering state wherein a resonance in one of
the phase shifts reveals the energy and spin state of an excited state in
the composite system (beam particle + target particle).
Interestingly, the scattering potential can be complex, U(r) = V(r) + iW(r),
wherein V(r) represents elastic scattering while W(r) represents reactions.
The imaginary part of the potential removes particles from the incident
particle beam via target particle breakup, such as d +d -> d + p + n,
reaction, such as d + d -> t + p, or absorption such as p + n -> d. In the
case of a complex potential, the phase shifts are also complex.
I can't really respond to that, so I will digress. A species of
explanation not mentioned here appears in Sakurai's "Modern Quantum
Mechanics" in chap. 1.
The idea seems to be that in starting with two basis states in spin,
|z+>,|z-> we wish to construct two other pairs of basis states --
|y+>,|y-> and |x+>,|x-> -- and in order to do this in a sufficiently
interesting way, we need complex numbers! (in opposition to the case
of polarized light where, starting with a basis x^,y^, there is only
_one_ other particularly interesting basis to be had -- the one
rotated 45 degrees).
There may be a little more to it than that, but would you buy a horse
from this argument?
A recurrent theme here seems to be that complex numbers are damn good
at capturing things to do with phase -- like Fourier transforms (I do
make a quibble about 'each point having amplitude and phase' ... OK
.... I see what you mean: _all_ complex numbers are characterized by
amplitude and phase, via Z = s exp(it). In which case your statment
is equivalent to "at each point the wave function is a complex
number". ;-)
Complex numbers seem to be even more unreasonably effective than math
in general: particularly in the way the unite two of the most useful
functional forms in one: the sinusoid and the exponential. Cunning!
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| User: "Gregory L. Hansen" |
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| Title: Re: Why is the wave function complex |
07 Nov 2003 08:27:21 AM |
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In article <3fa97173_3@newsfeed.slurp.net>, Old Man <nomail@nomail.net> wrote:
"Edward Green" <nulldev00@aol.com> wrote in message
news:2a0cceff.0311051741.5ce7c052@posting.google.com...
Interestingly, the scattering potential can be complex, U(r) = V(r) + iW(r),
wherein V(r) represents elastic scattering while W(r) represents reactions.
The imaginary part of the potential removes particles from the incident
particle beam via target particle breakup, such as d +d -> d + p + n,
reaction, such as d + d -> t + p, or absorption such as p + n -> d. In the
case of a complex potential, the phase shifts are also complex.
[Old Man]
Just to expound a little, a complex index of refraction is also used in
classical optics.
--
"Let us learn to dream, gentlemen, then perhaps we shall find the
truth... But let us beware of publishing our dreams before they have been
put to the proof by the waking understanding." -- Friedrich August Kekulé
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| User: "FrediFizzx" |
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| Title: Re: Why is the wave function complex |
05 Nov 2003 08:24:00 PM |
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"Edward Green" <nulldev00@aol.com> wrote in message
news:2a0cceff.0311051741.5ce7c052@posting.google.com...
| What standard story is told when the student asks why we need complex
| numbers for the wave function of qm? Other, that is, than "that's the
| way it is, and it works".
|
| For example: the hydrogenic atom is a success story for
| non-relativistic quantum mechanics. Has anybody looked at such things
| carefully to see if we couldn't have gotten the same numbers out the
| end from a purely real theory? What do complex numbers capture as
| vehicles of a theory which real numbers cannot, given that when we are
| done turning the crank, we expect to get real results?
I think it has to do with orthagonality.
FrediFizzx
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| User: "Starblade Darksquall" |
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| Title: Re: Why is the wave function complex |
07 Nov 2003 01:38:25 AM |
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(Edward Green) wrote in message news:<2a0cceff.0311051741.5ce7c052@posting.google.com>...
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
For example: the hydrogenic atom is a success story for
non-relativistic quantum mechanics. Has anybody looked at such things
carefully to see if we couldn't have gotten the same numbers out the
end from a purely real theory? What do complex numbers capture as
vehicles of a theory which real numbers cannot, given that when we are
done turning the crank, we expect to get real results?
We can use real numbers, but they'd have to be in a 2x2 real matrix.
For the number 1 we use Matrix A:
[1 0]
[0 1]
For the number i we use Matrix B:
[ 0 1]
[-1 0]
This works because matrix multiplication is distributive over
addition, and these particular matrices are commutative with respect
to themselves, eachother, and any, uh, whatever that word is for
'multiples, sum, sums of multiples or multiples of sums', the word
slips me for the moment, but there is a word for it. Also, matrix B
squared is -1 * matrix A.
So, yeah, it's quite possible to use a 2x2 real matrix in place of
complex numbers and get the exact same results.
(...Starblade Riven Darksquall...)
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| User: "zigoteau" |
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| Title: Re: Why is the wave function complex |
06 Nov 2003 02:44:44 AM |
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(Edward Green) wrote in message news:<2a0cceff.0311051741.5ce7c052@posting.google.com>...
Hi, Edward!
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
For example: the hydrogenic atom is a success story for
non-relativistic quantum mechanics. Has anybody looked at such things
carefully to see if we couldn't have gotten the same numbers out the
end from a purely real theory? What do complex numbers capture as
vehicles of a theory which real numbers cannot, given that when we are
done turning the crank, we expect to get real results?
Waves and complex numbers just go together. Perhaps I should say, sine
waves and complex numbers go together.
Electrical and electronic engineers deal with voltages that are always
real. However they both analyze the behavior of their circuits using
complex numbers. The complex number that they use to represent a
sinewave voltage has a real part which is in fact the voltage at time
t=0. The imaginary part is the voltage at 90° phase shift, i.e. after
a quarter period. Because of the mathematical properties of the sine
function:
sin(A+B) = sinA.cosB + cosA.sinB for all A and B
the voltage at any other time can be expressed uniquely in terms of
these two values. The voltage exists at infinitely many moments of
time, but you don't need as large a sheet of paper doing arithmetic
with only two values.
From relativity theory we know that an electron really has a nonzero
energy when it is at rest. When an electron and a positron recombine,
they mutually annihilate and produce two photons each of energy 0.51
MeV. Hence the 'rest' frequency of an electron has got to be 124 EHz
(that's exahertz; in full 124 000 000 000 000 000 000 Hz ±). In the
Schrödinger theory this 'true' oscillation is heterodyned down to
baseband, and you can't throw away the quadrature (imaginary) part. In
Klein-Gordon theory, which takes account of relativity theory, it's a
matter of taste whether you keep the imaginary part of the
wavefunction or not.
You might ask, but is it _really_ purely real or imaginary? Only those
amongst us who know the mind of God are qualified to give an answer.
There is none in terms of experiments you can do. An answer can
perhaps be given in terms of the relative simplicity or beauty of the
equations in the two cases. However beauty is in the eye of the
beholder.
Your next question is going to be, why sine waves and not square waves
or triangle waves or squiggly waves? The answer to that one has to do
with the fact that the equations of quantum mechanics (and of many
components in electronics and electrical engineering) are linear, and
have the form of eigenvalue equations. When two operators A and B
commute, it is possible to find a basis for the vector space, each
element of which is simultaneously an eigenvector of A and of B. The
equations of physics are invariant with respect to translation, i.e.
all places and times look the same. Hence the operator of the equation
commutes with d/dx, d/dy, d/dz and d/dt. The sensible eigenfunctions
of these operators are sinusoidal waves.
So sine rules the waves. Hence complex numbers.
Cheers,
Zigoteau.
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| User: "Eckard blumschein" |
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| Title: Re: Why is the wave function complex |
06 Nov 2003 06:46:41 AM |
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Hi,
you might be correct. However, I would like to add some gut feeling of
mine. After I looked into some papers by Feynman, I worried about his
rather speculative way of thinking. Admittedly, I dealt with the matter
in de.sci.physic some month ago. Also, I tried to address a related
question on R+ in sci.physics.research. John Baetz reacted angrily. Why?
Trained as an electrical engineer I grew up with complex calculus. Over
about 40 years, I realized that there are actually several flaws in
theory of signal processing in particular with respect to spectrogram
and understanding of auditory function. Meanwhile, I got convinced that
continuation of a causal function by means of zero padding is not
always adequate to reality. Having avoided arbitrariness, I managed to
calculate a real-valued "natural" spectrogram. My method does not
require any time window and consequently the result is not subject to
the notorious trade-off between spectral and temporal resolution being
often attributed to the uncertainty principle. Those who are interested
may ask me for a copy of a recently submitted pertaining paper of mine.
Perhaps I should say, sine waves and complex numbers go together.
That is correct. My real-valued frequency time analysis is based on
Fourier cosine transform and fully equivalent to complex magnitude-phase
analysis except for just one minor additional information contained
within the latter: The arbitrary moment of the begin of analysis.
This reference does definitely not exist in human hearing. It can
nonetheless be important in radar/sonar technology.
Electrical and electronic engineers deal with voltages that are always
real. However they both analyze the behavior of their circuits using
complex numbers.
They were instructed to always perform a return into the real domain
after they finished complex calculation. Transform into complex domain
means omitting one of the two rotating phasors constituting the cosine
function:
2 cos(x) = exp(jx) + exp(-jx)
is arbitrarily replaced by exp(jx)
(Mechanics and physics prefer to omit the other half instead.)
Return into reality is performed by omitting the imaginary part:
exp(jx) = cos(x) + jsin(x) is replaced by cos(x).
The complex number that they use to represent a
sinewave voltage has a real part which is in fact the voltage at time
t=0. The imaginary part is the voltage at 90° phase shift, i.e. after
a quarter period. Because of the mathematical properties of the sine
function:
sin(A+B) = sinA.cosB + cosA.sinB for all A and B
the voltage at any other time can be expressed uniquely in terms of
these two values.
Be cautious with any "other time". The world is not anticipatory.
Fourier himself investigated the temperature field within a closed loop.
Temporal asymmetry of the world is called causality, and physicist may
calculate how an apple falls but not why as long as they do so on an
abstract level where they feel entitled by Emily Noethers theorem to
shift time at will.
In the
Schrödinger theory this 'true' oscillation is heterodyned down to
baseband, and you can't throw away the quadrature (imaginary) part. In
Klein-Gordon theory, which takes account of relativity theory, it's a
matter of taste whether you keep the imaginary part of the
wavefunction or not.
"Throw away" is perhaps not a convincing suggestion, a strong word
rather than a detailed argument.
Fourier transform of a real signal always obeys Hermitian symmetry.
Does Hermitian symmetry also always hold in quantum physics?
I quote Carlson "Communication Systems" as an example for what I
consider a definitely wrong statement hidden behind telltale language:
"These considerations may appear somewhat academic in view of the fact
that ... is not a real function...". I consider Carlson wrong because he
ignores some more subtle aspects of causality. In the case of concern he
shifted frequency across zero and noticed that Hermitian symmetry was
lost.
You might ask, but is it _really_ purely real or imaginary? Only those
amongst us who know the mind of God are qualified to give an answer.
There is none in terms of experiments you can do. An answer can
perhaps be given in terms of the relative simplicity or beauty of the
equations in the two cases. However beauty is in the eye of the
beholder.
Instead you might look behind Heaviside's way of thinking. One can add
even and odd continuation in order to pretend causality. However,
imagination is not reality. Complex calculus is a nice tool in some
cases but not a panacea, and it should not be misused as a gospel.
Eckard Blumschein
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| User: "zigoteau" |
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| Title: Re: Why is the wave function complex |
09 Nov 2003 05:23:55 AM |
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Eckard blumschein <blumschein@et.uni-magdeburg.de> wrote in message news:<3FAA42B1.2080809@et.uni-magdeburg.de>...
Hi, Eckard!
<snip>
Be cautious with any "other time". The world is not anticipatory.
I think you're confusing maths and physics. A sinusoidal function is a
mathematical object which is defined for all values of the argument.
If you are a human being, with only limited knowledge about the state
of the real world, you have to sort out in your own mind how you get
useful answers. Luckily, most of physics is continuous, i.e. a small
error in our knowledge only leads to a small error in the resulting
calculations.
"Throw away" is perhaps not a convincing suggestion, a strong word
rather than a detailed argument.
?? I don't understand where you are coming from. I have allowed both
possibilities. I refrain from argument.
Fourier transform of a real signal always obeys Hermitian symmetry.
Does Hermitian symmetry also always hold in quantum physics?
I quote Carlson "Communication Systems" as an example for what I
consider a definitely wrong statement hidden behind telltale language:
"These considerations may appear somewhat academic in view of the fact
that ... is not a real function...". I consider Carlson wrong because he
ignores some more subtle aspects of causality. In the case of concern he
shifted frequency across zero and noticed that Hermitian symmetry was
lost.
But causality has to do with physics, and not with mathematics. Yes,
physical systems are causal, and this restricts the possible form of
response/propagator functions, etc. However I said nothing about the
form of the response function.
<snip>
Instead you might look behind Heaviside's way of thinking. One can add
even and odd continuation in order to pretend causality. However,
imagination is not reality. Complex calculus is a nice tool in some
cases but not a panacea, and it should not be misused as a gospel.
Sure. Garbage in, garbage out. If your maths does not correspond to
the structure of the universe, then the answers you get from it will
be wrong. However it is possible to do it right.
Cheers,
Zigoteau.
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| User: "Eckard blumschein" |
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| Title: Re: Why is the wave function complex |
10 Nov 2003 02:49:33 AM |
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Hi Zigoteau,
zigoteau wrote:
Be cautious with any "other time". The world is not anticipatory.
I think you're confusing maths and physics. A sinusoidal function is a
mathematical object which is defined for all values of the argument.
It would be possible to define time as purely mathematical. You just
didn't it.
If you are a human being, with only limited knowledge about the state
of the real world, you have to sort out in your own mind how you get
useful answers. Luckily, most of physics is continuous, ...
I wondered why no physicist responded to my post "(0,infty) or
[0,infty). A parallel discussion in de.sci.mathematik led to Cantor's
paradise.
"Throw away" is perhaps not a convincing suggestion, a strong word
rather than a detailed argument.
?? I don't understand where you are coming from. I have allowed both
possibilities. I refrain from argument.
Language is telltale.
Fourier transform of a real signal always obeys Hermitian symmetry.
Does Hermitian symmetry also always hold in quantum physics?
I quote Carlson "Communication Systems" as an example for what I
consider a definitely wrong statement hidden behind telltale language:
"These considerations may appear somewhat academic in view of the fact
that ... is not a real function...". I consider Carlson wrong because he
ignores some more subtle aspects of causality. In the case of concern he
shifted frequency across zero and noticed that Hermitian symmetry was
lost.
But causality has to do with physics, and not with mathematics. Yes,
physical systems are causal, and this restricts the possible form of
response/propagator functions, etc. However I said nothing about the
form of the response function.
Aren't we in sci.physics?
Instead you might look behind Heaviside's way of thinking. One can add
even and odd continuation in order to pretend causality. However,
imagination is not reality. Complex calculus is a nice tool in some
cases but not a panacea, and it should not be misused as a gospel.
Sure. Garbage in, garbage out. If your maths does not correspond to
the structure of the universe, then the answers you get from it will
be wrong. However it is possible to do it right.
I also hope so. Nonetheless, I do not just refer to apparently lacking
descriptivity of complex wave function. I rather doubt that group theory
is adequate to the infinitely small.
Cheers,
Eckard
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| User: "" |
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| Title: Re: Why is the wave function complex |
14 Nov 2003 11:41:14 AM |
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On Mon, 10 Nov 2003 09:49:33 +0100, Eckard blumschein
<blumschein@et.uni-magdeburg.de> wrote:
Hi Zigoteau,
zigoteau wrote:
Be cautious with any "other time". The world is not anticipatory.
I think you're confusing maths and physics. A sinusoidal function is a
mathematical object which is defined for all values of the argument.
It would be possible to define time as purely mathematical. You just
didn't it.
If you are a human being, with only limited knowledge about the state
of the real world, you have to sort out in your own mind how you get
useful answers. Luckily, most of physics is continuous, ...
I wondered why no physicist responded to my post "(0,infty) or
[0,infty). A parallel discussion in de.sci.mathematik led to Cantor's
paradise.
"Throw away" is perhaps not a convincing suggestion, a strong word
rather than a detailed argument.
?? I don't understand where you are coming from. I have allowed both
possibilities. I refrain from argument.
Language is telltale.
Fourier transform of a real signal always obeys Hermitian symmetry.
Does Hermitian symmetry also always hold in quantum physics?
I quote Carlson "Communication Systems" as an example for what I
consider a definitely wrong statement hidden behind telltale language:
"These considerations may appear somewhat academic in view of the fact
that ... is not a real function...". I consider Carlson wrong because he
ignores some more subtle aspects of causality. In the case of concern he
shifted frequency across zero and noticed that Hermitian symmetry was
lost.
But causality has to do with physics, and not with mathematics. Yes,
physical systems are causal, and this restricts the possible form of
response/propagator functions, etc. However I said nothing about the
form of the response function.
Aren't we in sci.physics?
Instead you might look behind Heaviside's way of thinking. One can add
even and odd continuation in order to pretend causality. However,
imagination is not reality. Complex calculus is a nice tool in some
cases but not a panacea, and it should not be misused as a gospel.
Sure. Garbage in, garbage out. If your maths does not correspond to
the structure of the universe, then the answers you get from it will
be wrong. However it is possible to do it right.
I also hope so. Nonetheless, I do not just refer to apparently lacking
descriptivity of complex wave function. I rather doubt that group theory
is adequate to the infinitely small.
Cheers,
Eckard
Any old cack will do, won't it?
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| User: "Bruce Scott TOK" |
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| Title: Re: Why is the wave function complex |
06 Nov 2003 08:27:21 AM |
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Ed Green asked:
|> What standard story is told when the student asks why we need complex
|> numbers for the wave function of qm? Other, that is, than "that's the
|> way it is, and it works".
Basically, the Fourier transform of a real function is a complex one.
The wave amplitude, which is the squared magnitude of the wave function,
is the real function which is the physical observable.
In QM however there is another reason: the way the commutator brackets
are set up: ab - ba = i hbar for conjugate operators (e.g., p and x).
There is a lot on this in the text by Messiah, which still has my vote
on the best text for understanding such underlying subtleties.
--
cu,
Bruce
drift wave turbulence: http://www.rzg.mpg.de/~bds/
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| User: "Eckard blumschein" |
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| Title: Re: Why is the wave function complex |
06 Nov 2003 12:26:01 PM |
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Bruce Scott TOK wrote:
Basically, the Fourier transform of a real function is a complex one.
The Fourier spectrum of a real function always exhibits Hermitian
symmetry. Iff the real function under test is symmetrical with respect
to t=0, then the spectrum is either purely real or purely imaginary, not
a complex one.
Eckard Blumschein
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| User: "Jeff Relf" |
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| Title: . Imaginary Numbers . |
08 Nov 2003 04:20:41 AM |
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Hi Edward Green , You ask :
" What do complex numbers capture
as vehicles of a theory which real numbers cannot ? "
I think the imaginary numbers of Minkowskian spacetime
only apply to tiny particles . Am I wrong ?
I think the imaginary numbers are needed because
an extra dimension is needed
to describe probabilistic positions .
Stephen Hawking used imaginary numbers to toy with
various possible universes that
might have arose from the big bang .
His imaginary numbers basically added a fifth dimension .
( This is his bumpy : " Universe in a nutshell " )
In three sperate quotes , Hawking said :
" Imaginary time is
indistinguishable from directions in space .
[ Note : that means it's spatial , static , immutable ]
...
In relativity , there is no real distinction between
the space and time coordinates , just as there is
no difference between two space coordinates .
...
One could say :
' The boundary condition of the universe is that
it has no boundary . '
The universe would be completely self-contained
and not affected by anything outside itself .
It would neither be created nor destroyed .
It would just _ Be _ .
What place , then , for a creator ? "
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| User: "Edward Green" |
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| Title: Re: . Imaginary Numbers . |
09 Nov 2003 12:18:35 AM |
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Jeff Relf <__.Jeff-Relf@NCPlus.NET> wrote in an unusually long post
for him,
Hi Edward Green , You ask :
" What do complex numbers capture
as vehicles of a theory which real numbers cannot ? "
I think the imaginary numbers of Minkowskian spacetime
only apply to tiny particles . Am I wrong ?
Huh?
I think the imaginary numbers are needed because
an extra dimension is needed
to describe probabilistic positions .
Huh?
Stephen Hawking used imaginary numbers to toy with
various possible universes that
might have arose from the big bang .
Huh?
His imaginary numbers basically added a fifth dimension .
( This is his bumpy : " Universe in a nutshell " )
Etc.
In three sperate quotes , Hawking said :
" Imaginary time is
indistinguishable from directions in space .
[ Note : that means it's spatial , static , immutable ]
...
In relativity , there is no real distinction between
the space and time coordinates , just as there is
no difference between two space coordinates .
...
One could say :
' The boundary condition of the universe is that
it has no boundary . '
The universe would be completely self-contained
and not affected by anything outside itself .
It would neither be created nor destroyed .
It would just _ Be _ .
What place , then , for a creator ? "
Ahhhh.... right. I've been drinking cheap Chilean wine as I write
this, Mr. Relf. But I don't think I have drunk enough cheap Chilean
wine to reach where you are tonight, Jeff Relf!
So called imaginary numbers are simply multiples of i, which is
defined to be the sqrt(-1), as you well know. The name is
unfortunate. The damage was partially euphonically undone when they
were considered in combo with the reals and called complex, which
sounds less ficticious.
Why the logical structure reached in this way is so natural in so many
contexts is something I do not understand. But there is nothing
imaginary about the structure which was waiting there to be
discovered.
Maybe it would be better to think of the structure of complex numbers
as the fundamental thing, which appears in many contexts, and the
extension of the reals to include sqrt(-1) as only one context. That
is, maybe someone could have coined the "binarions" in connection with
electrical theory, and only later would somebody note that the second
element of the binarions acts as if it were the sqrt(-1), listed under
"applications of the binarions".
In contrast to the binarions, the quaternions seem to be more of a
niche thing, known more by reputation than in terms of the specific
algebra of their components.
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| User: "Jeff Relf" |
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| Title: An Orthogonal Dimension ? |
09 Nov 2003 12:32:02 AM |
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Hi Edward Green , You say :
" Huh ? Huh ? Huh ? "
Correct me if I'm wrong , but I thought imaginary numbers
implied the existence of an orthogonal dimension .
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| User: "Edward Green" |
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| Title: Re: An Orthogonal Dimension ? |
09 Nov 2003 07:39:04 PM |
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Jeff Relf <__.Jeff-Relf@NCPlus.NET> wrote in message news:<awype8zssgzg$.dlg@__.Jeff.Relf>...
Hi Edward Green , You say :
" Huh ? Huh ? Huh ? "
Correct me if I'm wrong , but I thought imaginary numbers
implied the existence of an orthogonal dimension .
You could express the thing that way, but the "implication" is purely
mathematical -- i.e., you are simply restating the fact that complex
numbers have two dissimilar components. And even there one could
nit-pick about "orthogonal": i and 1 are represented orthogonally on
the complex plane, but then x-hat and y-hat are represented
orthogonally on the real plane R^2. So what does "impl<y> the
existence of an orthogonal dimension" really mean or add to the
matter?
The mere existence of this mathematical structure implies nothing
about physical space, if that's what you mean.
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| User: "Jeff Relf" |
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| Title: Re: An Orthogonal Dimension ? |
09 Nov 2003 09:34:19 PM |
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Hi Edward Green , You say :
" The mere existence of this mathematical structure
implies nothing about physical space "
I wasn't talking about " physical space "
so much as spacetime as we perceive it .
e.g. In the case of _ Tiny _ particles that means :
Probabilistically and relativistically .
Hence the need for complex numbers .
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| User: "Eckard blumschein" |
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| Title: Re: Why is the wave function complex |
07 Nov 2003 02:57:32 AM |
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If electrical engineers are done turning the crank, then they performed
an inverse transform back into real domain where they started. In other
words, they know what slippy ground they were using whith complex
domain. Modal analysis, sometimes suffers from unphysical behavior of
complex modes. Isn't interpretation of the latter premature before the
due return into reality? I am not sure whether or not quantum physicists
are equally rigorous.
One more doubt. Negative frequency is very popular in signal processing
albeit it does not have a physical meaning. Look at spectra. They
exhibit striking symmetry. I would like to call it apparent symmetry due
to use of complex calculus. Such apparent symmetry does never brake.
Knowing this, I am highly alert in case of perfectly symmetrical looking
results of measurement. My last victim was a paper in PRL. For
physical reasons, I did not believe the putative symmetry of pressure
over time. Actually, streak camera measurement showed a quite different
asymmetrical picture.
A third aspect. Complex calculus is quite convenient and to some extent
close to natural relationship between periodic functions and their
derivatives. However, as to fill the void half, it requires continuation
substituting a real causal function in R+ by a sum of a function with
even symmetry and a function with odd symmetry in R. This superposition
is impractical because it is valid for just a single moment. In case of
Laplace transform this does not matter since this unilateral transform
has been based on a deterministic notion of causality. Is determinism
still adequate or also obsolete in quantum physics?
Eckard Blumschein
Edward Green wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
For example: the hydrogenic atom is a success story for
non-relativistic quantum mechanics. Has anybody looked at such things
carefully to see if we couldn't have gotten the same numbers out the
end from a purely real theory? What do complex numbers capture as
vehicles of a theory which real numbers cannot, given that when we are
done turning the crank, we expect to get real results?
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| User: "Jon Bell" |
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| Title: Re: Why is the wave function complex |
05 Nov 2003 10:56:02 PM |
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In article <2a0cceff.0311051741.5ce7c052@posting.google.com>,
Edward Green <nulldev00@aol.com> wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm?
We don't *have* to use a complex psi function. We can use two real
functions, call then alpha(x) and beta(x), corresponding to the real and
imaginary parts of psi(x). Then the Schroedinger equation (a single
differential equation for psi(x)) becomes two coupled differential
equations for alpha(x) and beta(x).
But of course alpha(x) and beta(x) wouldn't be independent of each other.
They'd have to be related in ways that correspond to the relationships
between the real and imaginary parts of an analytic complex function.
So we'd still have to ask, "why must alpha(x) and beta(x) be related in
this way?"
At least one of the QM books in my office discusses this at some length.
Tomorrow is my "slow day" on campus, so maybe I'll check it out and report
back. (I think it might be Morrison's "Understanding Quantum Physics"...
are you reading this, BAH?)
--
Jon Bell <jtbellap8@presby.edu> Presbyterian College
Dept. of Physics and Computer Science Clinton, South Carolina USA
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| User: "Steve Harris" |
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| Title: Re: Why is the wave function complex |
14 Nov 2003 06:09:39 PM |
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(Jon Bell) wrote in message news:<bock92$ms9$1@jtbell.presby.edu>...
In article <2a0cceff.0311051741.5ce7c052@posting.google.com>,
Edward Green <nulldev00@aol.com> wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm?
We don't *have* to use a complex psi function. We can use two real
functions, call then alpha(x) and beta(x), corresponding to the real and
imaginary parts of psi(x). Then the Schroedinger equation (a single
differential equation for psi(x)) becomes two coupled differential
equations for alpha(x) and beta(x).
But of course alpha(x) and beta(x) wouldn't be independent of each other.
They'd have to be related in ways that correspond to the relationships
between the real and imaginary parts of an analytic complex function.
So we'd still have to ask, "why must alpha(x) and beta(x) be related in
this way?"
At least one of the QM books in my office discusses this at some length.
Tomorrow is my "slow day" on campus, so maybe I'll check it out and report
back. (I think it might be Morrison's "Understanding Quantum Physics"...
are you reading this, BAH?)
COMMENT:
Feynman in his book QED claims he's left nothing out of the theory by
describing all results as path-added phase results, with the phases of
moving particles depicted by a single phasor-like extra variable
(Feynman uses the idea of a hand turning on a moving watch). If that's
true, the imaginary numbers are not needed for QED. You just need some
math which represents the changing phase of a particle over a path.
Imaginary numbers are most convenient for this when we're already
using 3 variables for our spacial dimentions and need a fourth
"dimension" for the phase of the matter wave. But we're all familiar
with using an extra spacial dimension to graph that kind of thing when
looking at 1-D and 2-D EM waves, right? It took me years to realize
that the wave they show you when trying to depict an EM wave is not
the wave itself, but a GRAPH of the wave, with E and B field strength
cribbed into one dimension where they really don't go (the vector
direction is real but the vector length isn't). But the tricky part is
that EM waves at least have the E and B fields pointing in some real
direction (one of the 3 standard dimentions), whereas a matter wave
has the phase vector of it rotating in some 4th dimension which has a
sort of "complex plane like" direction that you need to keep track of,
but doesn't have a direction in 3 D. So matter waves are like the
dials of Feynman's little particle-watches, where the hand points in a
direction which is irrevent to the 3 spacial dimentions for purposes
of calculation. All you need to do is make sure that all watches are
oriented the same when you add the phase vectors up. But you don't
absolutely need imaginary numbers for this-- all you really need at
minimum is a fourth dimension in which to keep your phase information.
The idea of using imaginary numbers and the complex direction to let
you have an extra dimension which is perpendicular to everything else,
is also of course used in relativity, where the extra dimension is
related to -ct. The amazing thing about the Dirac equations is that
they manage to encode the phase information of matter waves using the
imaginary i, but also manages to use the relativistic transformations
which make use of the same complex plane, at the same time. It's sort
of magical to me that the phase information in matter waves can be
mixed up with relativistic rotations. I suppose it has to do with the
fact that both of these things are proportional to changes in time.
Any of you math wonks want to try to explain this to me pictorially?
SBH
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| User: "Mark" |
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| Title: Re: Why is the wave function complex |
15 Nov 2003 12:53:03 PM |
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Edward Green <nulldev00@aol.com> wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm?
The phase is supposed to be "unobservable". What's actually meant by
that is that there is a gauge degree of freedom:
psi(x,y,z,t) -> exp(i phi(x,y,z,t) psi(x,y,z,t)
which is supposed to leave the equation of motion for psi invariant.
But, if you'll notice, the Schroedinger equation doesn't have that
property. Write it down as:
i h-bar @psi/@t = -h^2/2m Del^2 psi
Putting in exp(i phi), you'll get extra terms of the general forms:
(something) X psi
(something) X Del psi
In order for Schroedinger to be invariant, there have to be extra
terms of these forms
V psi
A.Del(psi)
to compensate. More generally, you need the derivatives to be replaced
by:
i h-bar @/@t --> i h-bar @/@t - V
-i h-bar Del --> -i h-bar Del + A
so that the (modified) Schroedinger equation reads:
i h-bar @psi/@t = (-h-bar^2/2m Del^2 + V + K/2m) psi
with
K = -i h-bar/2m Del.A - i h-bar/m A.Del + A^2/2m.
It can be generalized further starting with an equation of the form:
i h-bar @psi/@t = -h^2/2m S^{ab} @^2psi/@x^a@x^b
with
S^{ab} = delta^{ab} + sigma^{ab}
sigma^{ab} = -sigma^{ba}
summation convention used.
When the conversion is made, adding in V and A, you get an extra term
of the form proportional to:
sigma^{ab} (@A_b/@x^a - @A_a/@x^b)
= sigma.curl(A)
with the convention sigma^1 = sigma^{23}, sigma^2 = sigma^{31},
sigma^3 = sigma^{12}.
A transformation on psi -> U psi, U = exp(i phi) then requires one on V and
A such that
(i h-bar @/@t + V') U psi = U (i h-bar @/@t + V) psi
or
V -> V' = U V U^{-1} - i h-bar @U/@t U^{-1}
= V + h-bar @phi/@t
and
(-i h-bar Del + A') U psi = U (-i h-bar Del + A) psi
or
A -> A' = U A U^{-1} + i h-bar Del U U^{-1}
= A - h-bar Del(phi).
So, the phase factor can be understood as representing an extra degree
of freedom that directly relates to a gauge degree of freedom of both
a potential V AND a vector potential A that exhibit the gauge degrees
of freedom of the electromagnetic field.
The phases can be more complex. It can be in U(2), which then makes
psi not just a complex number in C, but a pair of complex numbers in
C^2. In this case, the sigma's could be elements of U(2), that
mix the two complex components of psi around.
So, there's really nothing special about the complex numbers being
what psi is of; psi can be of something more general, with the general
phase giving you a more complex set of gauge degrees of freedom.
.
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| User: "Mark" |
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| Title: Re: Why is the wave function complex |
15 Nov 2003 12:19:14 PM |
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Edward Green <nulldev00@aol.com> wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm?
jtbellq2f@presby.edu (Jon Bell) wrote:
We don't *have* to use a complex psi function. We can use two real
functions, call then alpha(x) and beta(x), corresponding to the real and
imaginary parts of psi(x). Then the Schroedinger equation (a single
differential equation for psi(x)) becomes two coupled differential
equations for alpha(x) and beta(x).
The question is answered more revealingly by starting out with just
rho(x,y,z,t) = psi(x,y,z,t)* psi(x,y,z,t) and writing out a self-contained
dynamic system from it.
From the Schroedinger equation (generalized for the sake of simplicity):
i h-bar @_t psi = -(h-bar)^2/2 W^ab >a >b psi + V psi
[W invertible and symmetric; W^{-1} = m = mass matrix]
@_t = partial time derivative
>a = partial derivative with respect to x^a applied to the right
<a partials applied to the left
sumation convention used
you get evolution equations separately for psi and psi*
@_t psi = i h-bar/2 W^ab >a >b psi - iV/h-bar psi
@_t psi* = -i h-bar/2 W^ab psi* <a <b + iV/h-bar psi
so that
@_t rho = i h-bar/2 W^ab psi* (>a >b - <a <b) psi
= i h-bar/2 W^ab >a (psi* (>b - <b) psi), since W^ab = W^ba
= ->a (i h-bar/2 W^ab psi* (<b - >b) psi)
which is the continuity equation
@_t rho + >a J^a = 0
with the current
J^a = i h-bar/2 W^ab psi* (<b - >b) psi
So, now taking the time derivative of J^a, you get:
@_t J^a
= i h-bar/2 W^ab @_t (psi* (<b - >b) psi)
= (h-bar/2)^2 W^ab W^cd psi* (<c<d (<b - >b) - (<b - >b) >c>d) psi
+ 1/2 W^ab psi* ((<b - >b) V - V (<b - >b)) psi
The last term is just -W^ab >bV psi* psi = rho f^a, where
f^a = -W^ab >bV = Force/unit mass
The first term is actually a divergence, since:
b (W^ac W^bd psi* (<c<d + >c>d) psi)
= W^ac W^bd psi* (<b<c<d + <c<d>b + <b>c>d + >b>c>d) psi
and
2 >b (W^ab W^cd psi* <c>d psi)
= >b (W^ab W^cd psi* (<c>d + <d>c) psi)
= >c (W^ac W^bd psi* (<b>d + <d>b) psi)
= W^ac W^bd psi* (<b<c>d + <b>c>d + <c<d>b + <d>b>c) psi
= W^ac W^bd psi* (<d<c>b + <b>c>d + <c<d>b + <d>d>c) psi
= 2 W^ac W^bd psi* (<c<d>b + <b>c>d) psi
so
b (W^ac W^bd psi* (<c<d + >c>d) psi - 2 W^ab W^cd psi* <c>d psi)
= W^ac W^bd psi* (<b<c<d - <c<d>b - <b>c>d + >b>c>d) psi
= W^ac W^bd psi* (<c<d (<b - >b) - (<b - >b) >c>d) psi
Thus
@_t J^a + >b T^ab = rho f^a
where
T^ab = (h-bar/2)^2 psi* (2 W^ab W^cd <c>d - W^ac W^bd (<c<d + >c>d)) psi
is the stress tensor. In fact, it can be written as:
T^ab = rho v^a v^b + P^ab
where
v^a = J^a/rho
and P^ab is a stress of a purely quantum origin which can be expressed in
terms of a potential Q as:
@_a P^ab = rho W^ab @_a Q.
And that potential is just:
Q = -h-bar^2/2 W^ab (Del R)/R; R^2 = rho
the Quantum Potential, a' la Bohm.
(There might be some sign or factor errors somewhere in here).
.
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| User: "Keith F. Lynch" |
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| Title: Re: Why is the wave function complex |
20 Nov 2003 10:11:20 PM |
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Steve Harris <sbharris@ix.netcom.com> wrote:
Feynman in his book QED claims he's left nothing out of the theory
by describing all results as path-added phase results, with the
phases of moving particles depicted by a single phasor-like extra
variable (Feynman uses the idea of a hand turning on a moving watch).
Nitpick: He says he did leave something out: polarization.
If that's true, the imaginary numbers are not needed for QED.
Of course they aren't. Anything you can do with imaginary numbers,
you can do without. It's just more unwieldy.
You don't even need a full range of real numbers. You can fake it
with just the integers.
You don't even need integers, other than 0 and 1. Or any operations
other than AND, OR, and NOT.
You don't even need the AND, OR, and NOT operations. They can be
emulated with just the NAND operation.
Similarly, you don't need power tools to build a house. Muscle power
suffices. But I'd rather use the power tools, even if they require
a little more training.
And I'd rather use complex numbers for QED. Despite the name, they
make the equations simpler.
--
Keith F. Lynch - - http://keithlynch.net/
I always welcome replies to my e-mail, postings, and web pages, but
unsolicited bulk e-mail (spam) is not acceptable. Please do not send me
HTML, "rich text," or attachments, as all such email is discarded unread.
.
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| User: "Steve Harris" |
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| Title: Re: Why is the wave function complex |
21 Nov 2003 08:32:49 PM |
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"Keith F. Lynch" <kfl@KeithLynch.net> wrote in message news:<bpk398$d8c$1@panix3.panix.com>...
Steve Harris <sbharris@ix.netcom.com> wrote:
Feynman in his book QED claims he's left nothing out of the theory
by describing all results as path-added phase results, with the
phases of moving particles depicted by a single phasor-like extra
variable (Feynman uses the idea of a hand turning on a moving watch).
Nitpick: He says he did leave something out: polarization.
If that's true, the imaginary numbers are not needed for QED.
Of course they aren't. Anything you can do with imaginary numbers,
you can do without. It's just more unwieldy.
You don't even need a full range of real numbers. You can fake it
with just the integers.
You don't even need integers, other than 0 and 1. Or any operations
other than AND, OR, and NOT.
You don't even need the AND, OR, and NOT operations. They can be
emulated with just the NAND operation.
COMMENT:
Good point. I knew all that, but somehow hadn't put it all together.
Duh. A computer does everything with 1's and 0's and chains of very
simple logical operations thereon (NAND is a set of 4 rules).
Including MATHEMATICA manipulation of equation symbols. And including
any calculation in QED or QCD. To calculate in QCD, the Feynman
diagrams get reduced to functions which get reduced to simpler
functions, and so on down to 1's, 0's and information on what sets of
logical operations must be performed on them. Anything a brain can put
down in math or text as "science" can be emulated by a computer, or
Universal Turing Machine.
I once read a physics history that claimed that the real genius of
Shroedinger was to understand that the psi function "really was
complex." But of course that's silly. It's complex only if you must
represent it with an exponential function, so you can run it through a
DE. If your computer code gives you actual results from such a thing,
it all goes down to 1's and 0's. Current either flows or not-- no
imaginary stuff there.
.
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| User: "Ken Muldrew" |
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| Title: Re: Why is the wave function complex |
21 Nov 2003 01:38:27 PM |
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"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
Similarly, you don't need power tools to build a house. Muscle power
suffices. But I'd rather use the power tools, even if they require
a little more training.
I think you got that one backwards.
Ken Muldrew
kmuldrezw@ucalgazry.ca
(remove all letters after y in the alphabet)
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| User: "" |
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| Title: Re: Why is the wave function complex |
06 Nov 2003 05:49:16 AM |
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In article <bock92$ms9$1@jtbell.presby.edu>,
(Jon Bell) wrote:
In article <2a0cceff.0311051741.5ce7c052@posting.google.com>,
Edward Green <nulldev00@aol.com> wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm?
We don't *have* to use a complex psi function. We can use two real
functions, call then alpha(x) and beta(x), corresponding to the real and
imaginary parts of psi(x). Then the Schroedinger equation (a single
differential equation for psi(x)) becomes two coupled differential
equations for alpha(x) and beta(x).
But of course alpha(x) and beta(x) wouldn't be independent of each other.
They'd have to be related in ways that correspond to the relationships
between the real and imaginary parts of an analytic complex function.
So we'd still have to ask, "why must alpha(x) and beta(x) be related in
this way?"
At least one of the QM books in my office discusses this at some length.
Tomorrow is my "slow day" on campus, so maybe I'll check it out and report
back. (I think it might be Morrison's "Understanding Quantum Physics"...
are you reading this, BAH?)
Life interrupted. I've been intending to comment on a sentence in
the first chapter that leaped off the page and whacked me with
a virtual baseball bat. I figured out one aspect of the
declaration but am having a hell of a time trying to figure
out how it applies to springs. I still have the thread active
in my newslist. E-kicking me in the e-butt occasionally will
help me to remember to get back to that study. I do seem to
get side-tracked easily these days.
I have Bruce's post to me carefully saved away. Just to let
you know that I've not ignored the work you did to write the
posts.
/BAH
Subtract a hundred and four for e-mail.
.
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| User: "Gregory L. Hansen" |
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| Title: Re: Why is the wave function complex |
21 Nov 2003 09:18:39 PM |
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In article <2a0cceff.0311051741.5ce7c052@posting.google.com>,
Edward Green <nulldev00@aol.com> wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
For example: the hydrogenic atom is a success story for
non-relativistic quantum mechanics. Has anybody looked at such things
carefully to see if we couldn't have gotten the same numbers out the
end from a purely real theory? What do complex numbers capture as
vehicles of a theory which real numbers cannot, given that when we are
done turning the crank, we expect to get real results?
Harrison, in his book Applied Quantum Mechanics, says
"In fact, a single amplitude, such as sin(kx), is not enough because it
does not tell which way the wave is moving. Thus for a water wave we need
not only the height of the water surface, but the velocity of the water
motion; for a light wave we need both the electric field and the magnetic
field. We require two amplitudes to describe the electron and we choose
to make them the real and imaginary part of a complex amplitude psi(x,t),
as could be done with a light wave or a water wave."
I'm not sure what to think of that, I haven't wrapped my mind around it
just yet.
--
"Let us learn to dream, gentlemen, then perhaps we shall find the
truth... But let us beware of publishing our dreams before they have been
put to the proof by the waking understanding." -- Friedrich August Kekulé
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| User: "Edward Green" |
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| Title: Re: Why is the wave function complex |
22 Nov 2003 09:48:23 AM |
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(Gregory L. Hansen) wrote in message news:<bpmkif$skl$1@hood.uits.indiana.edu>...
In article <2a0cceff.0311051741.5ce7c052@posting.google.com>,
Edward Green <nulldev00@aol.com> wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
Harrison, in his book Applied Quantum Mechanics, says
"In fact, a single amplitude, such as sin(kx), is not enough because it
does not tell which way the wave is moving. Thus for a water wave we need
not only the height of the water surface, but the velocity of the water
motion; for a light wave we need both the electric field and the magnetic
field. We require two amplitudes to describe the electron and we choose
to make them the real and imaginary part of a complex amplitude psi(x,t),
as could be done with a light wave or a water wave."
I'm not sure what to think of that, I haven't wrapped my mind around it
just yet.
Aha. I remember thinking about the "which way is it moving question".
I just can't remember what I thought now. ;-)
For the one dimensional wave equation a general solution describes
arbitrary shapes running in either direction:
A(x,t) = F(x-ct) + E(x+ct)
At first glance I was tempted to take this as evidence against the
assertion that we need a second component -- after all we only have
one amplitude here, and apparently the system remembers what it is
doing!
However, given an instantaneous snapshot we would be unable to say
what was going to happen in the next instant ... there is no way to
decode the contributions of E and F; the solution must remember for
us. It seems the innocent looking simple wave equation for a single
amplitude must be incomplete as a description of any physical system:
a maximal present state description doesn't allow us to extrapolate
the future.
(Should we take this as a supporting example of the "two component"
hypothesis, the components being E and F? Before we get too excited
about this, we should remember in three dimensions we would have to
specify the propagating shape in all wave vectors on a sphere! So the
two-ness of E and F in one dimension is a coincidence).
I'm not sure about the "two-ness" thing. Morrison uses EM as an
example ... but even here merely knowing E and B at a point leaves us
in the dark what is going to happen at t + dt ... we need a
neighborhood ... for what's that worth. And given a neighborhood
dependent property, do we really need two components? Well the 1-d
wave equation also uses a neighborhood property to propagate
(derivatives), but is apparently a wave component short of a six-pack.
Hmm ... this needs more thought.
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| User: "Gregory L. Hansen" |
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| Title: Re: Why is the wave function complex |
22 Nov 2003 01:52:01 PM |
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In article <2a0cceff.0311220748.6edf3c00@posting.google.com>,
Edward Green <nulldev00@aol.com> wrote:
glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) wrote in message
news:<bpmkif$skl$1@hood.uits.indiana.edu>...
In article <2a0cceff.0311051741.5ce7c052@posting.google.com>,
Edward Green <nulldev00@aol.com> wrote:
What standard story is told when the student asks why we need complex
numbers for the wave function of qm? Other, that is, than "that's the
way it is, and it works".
Harrison, in his book Applied Quantum Mechanics, says
"In fact, a single amplitude, such as sin(kx), is not enough because it
does not tell which way the wave is moving. Thus for a water wave we need
not only the height of the water surface, but the velocity of the water
motion; for a light wave we need both the electric field and the magnetic
field. We require two amplitudes to describe the electron and we choose
to make them the real and imaginary part of a complex amplitude psi(x,t),
as could be done with a light wave or a water wave."
I'm not sure what to think of that, I haven't wrapped my mind around it
just yet.
Aha. I remember thinking about the "which way is it moving question".
I just can't remember what I thought now. ;-)
For the one dimensional wave equation a general solution describes
arbitrary shapes running in either direction:
A(x,t) = F(x-ct) + E(x+ct)
At first glance I was tempted to take this as evidence against the
assertion that we need a second component -- after all we only have
one amplitude here, and apparently the system remembers what it is
doing!
However, given an instantaneous snapshot we would be unable to say
what was going to happen in the next instant ... there is no way to
decode the contributions of E and F; the solution must remember for
us. It seems the innocent looking simple wave equation for a single
amplitude must be incomplete as a description of any physical system:
a maximal present state description doesn't allow us to extrapolate
the future.
(Should we take this as a supporting example of the "two component"
hypothesis, the components being E and F? Before we get too excited
about this, we should remember in three dimensions we would have to
specify the propagating shape in all wave vectors on a sphere! So the
two-ness of E and F in one dimension is a coincidence).
I'm not sure about the "two-ness" thing. Morrison uses EM as an
example ... but even here merely knowing E and B at a point leaves us
in the dark what is going to happen at t + dt ... we need a
neighborhood ... for what's that worth. And given a neighborhood
dependent property, do we really need two components? Well the 1-d
wave equation also uses a neighborhood property to propagate
(derivatives), but is apparently a wave component short of a six-pack.
Hmm ... this needs more thought.
Think of a snapshot of the electric field of a plane wave,
E(z,0) = E0 cos(kz) e_x
But which way is it going? We could have have
E(z,t) = E0 cos(kz-wt) e_x
or
E(z,t) = E0 cos(kz+wt) e_x = E0 cos(-kz-wt) e_x
E0 is the electric field at z=t=0 and needn't be positive unless we adopt
some convention. The direction is determined by the relative sign of kz
versus wt. Let's just call it k, may be positive or negative, w
postive, and
E(z,t) E0 cos(kz-wt) e_x
Meanwhile,
dH/dt = -curl E
= k E0 sin(kz-wt) e_y
H = (k/w) E0 cos(kz-wt) e_y
And H(z,t)=H(0,0) = E0 k/w
with k either positive or negative, and H(0,0) either has the same or
opposite sign as E(0,0). So H(0,0) contains an extra peice of
information-- which direction is the wave going?
You could get the same information with dE/dt, in closer analogy to a wave
on a string, except you might want to look at a point slightly away from
zero then.
And then the possibility leaps to mind that the wavefunction might be
describe as something analogous to an electric and magnetic wave rather
than a comples wave, and the probability density would be something
analogous to the Poynting vector. But it's a very nice day and I don't
want to sit around inside trying to figure it out.
(Why yes, life has gotten much better since classes and homework ended.)
--
"Let us learn to dream, gentlemen, then perhaps we shall find the
truth... But let us beware of publishing our dreams before they have been
put to the proof by the waking understanding." -- Friedrich August Kekulé
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| User: "Edward Green" |
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| Title: Re: Why is the wave function complex |
22 Nov 2003 08:16:29 PM |
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(Gregory L. Hansen) wrote in message news:<bpoep1$ecq$1@hood.uits.indiana.edu>...
In article <2a0cceff.0311220748.6edf3c00@posting.google.com>,
Edward Green <nulldev00@aol.com> wrote:
....
I'm not sure about the "two-ness" thing. Morrison uses EM as an
example ... but even here merely knowing E and B at a point leaves us
in the dark what is going to happen at t + dt ... we need a
neighborhood ... for what's that worth. And given a neighborhood
dependent property, do we really need two components? Well the 1-d
wave equation also uses a neighborhood property to propagate
(derivatives), but is apparently a wave component short of a six-pack.
Hmm ... this needs more thought.
BTW: I had another idea here. The 1-d wave equation doesn't remember
which way it is going given a snapshot of the amplitude, because it
contains only second derivatives, which are invariant under spatial
reflection?
Ooooh.... :-)
Think of a snapshot of the electric field of a plane wave,
I had an idea that was what Morrison was thinking of, though in
general you would want to know how an arbitrary field was going to
evolve ... which could just perhaps be decomposed into plane waves.
E(z,0) = E0 cos(kz) e_x
But which way is it going? We could have have
E(z,t) = E0 cos(kz-wt) e_x
or
E(z,t) = E0 cos(kz+wt) e_x = E0 cos(-kz-wt) e_x
E0 is the electric field at z=t=0 and needn't be positive unless we adopt
some convention. The direction is determined by the relative sign of kz
versus wt.
Right. Just as in the case F(x-ct) vs. E(x+ct). Which by the way
shows that this approach isn't limited to sinusoids.
Let's just call it k, may be positive or negative, w
postive, and
E(z,t) E0 cos(kz-wt) e_x
Meanwhile,
dH/dt = -curl E
= k E0 sin(kz-wt) e_y
H = (k/w) E0 cos(kz-wt) e_y
And H(z,t)=H(0,0) = E0 k/w
with k either positive or negative, and H(0,0) either has the same or
opposite sign as E(0,0). So H(0,0) contains an extra peice of
information-- which direction is the wave going?
Ok ... so at least in the case of plane waves, the extra component
tells us precisely what Morrison claimed it told us.
You could get the same information with dE/dt, in closer analogy to a wave
on a string, except you might want to look at a point slightly away from
zero then.
Which maybe ties into my comment about second derivatives vs. first
derivatives.
[I've noticed, again and again and again, a fascinating phenomenon on
sci.physics: think about any general point and it immediately surfaces
again in other seemingly unrelated threads. Like physics is way
interconnected, man. (Hear this read in a Joe Friday Monotone). I was
thinking about the information necessary to propagate a solution in a
precessing top, and had problems seeing how we differentiated between
a top already precessing and one started from rest. _Something_ was
the same about these cases ... which showed that this something was
insufficient to propagate the solution.]
And then the possibility leaps to mind that the wavefunction might be
describe as something analogous to an electric and magnetic wave rather
than a comples wave, and the probability density would be something
analogous to the Poynting vector.
The same thought had occured to me ... which leads us back to my
original question: is there anything fundamentally complex about the
wave function as opposed to, say, EM, or could we as easily have
formulated a coupled set of real wave equations, and, OTOH, a complex
EM?
Obviously at least the first part is true: whatever can we abbreviated
in complex notation can also be written out in coupled real
components.
I wonder ... could we recast Maxwell's equations in terms of one
complex field? Exercise for a rainy day.
But it's a very nice day and I don't
want to sit around inside trying to figure it out.
Excellent choice ... I, even I, went to the park with the dog, though
I had some stray thoughts about wave equations, while she lusted after
squirrel blood.
Complex numbers must somehow encapsulate a very common form of two
component systems ... it remains to figure out just why this form is
so common.
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