Does quantum theory really undermine determinism?

[John Salerno]

I'm currently reading Stephen Hawking's book "A Briefer History of Time" and I was a little confused by the suggestion that the uncertainty principle undermines determinism, as stated in this sentence from the book:

 

The uncertainty principle signaled an end to Laplace's dream of a theory of science, a model of the universe that would be completely deterministic.

 

As explained in the book, the uncertainty principle states that the more accurately the position of a particle is measured, the less accurate the measurement will be of the particle's velocity. Therefore, the "initial conditions" of the system can never be accurately known in order to determine the past or future state of the universe.

 

This much is easy to understand, but it seems that the uncertainty principle really only undermines our ability to calculate these past or future states of the universe, not that it actually undermines the fact that the universe still is deterministic (not that it necessarily is, but the uncertainty principle as stated above doesn't seem to suggest otherwise), even if we can never calculate any given state of it because of this uncertainty.

 

However, I have seen elsewhere that perhaps the uncertainty principle suggests more than the simple explanation given in the book. That, in fact, it explicitly says that the position or velocity of a particle is not actually determined at all until it is measured. This seems to really hurt determinism, but is that really what the uncertainty principle says? That a particle's position or velocity essentially doesn't exist at all until we measure it, or is it simply that we can never know a particle's position or velocity for sure until the time of its measurement?

 

Basically, my question comes down to this: even given the uncertainty principle and the probabilistic nature of quantum mechanics, couldn't it still be the case that the universe is perfectly deterministic, even if we can't accurately make the measurements to determine these past or future states ourselves? Doesn't a particle still have a certain position and velocity at any given time, even if measuring one of these will then change the other?

 

Hawking hints at this idea:

 

We could still imagine that there is a set of laws that determine events completely for some supernatural being who, unlike us, could observe the present state of the universe without disturbing it. However, such models of our universe are not of much interest to us ordinary mortals.

 

Frankly, I think the mention of a "supernatural being" unnecessarily muddies the discussion and makes Hawking dismiss the idea too easily. It's not necessary to postulate anything supernatural in order to retain determinism.

 

The above quote could easily have been stated as something like this: "We could still imagine that there is a set of laws that determine events completely, despite our inability to measure the present state of the universe without disturbing it." Then the second sentence would be irrelevant, because it would be of interest to us to know that it is possible for the universe still to be perfectly deterministic, even if we can't (yet) discover all the laws.

Edited December 16, 2011 by John Salerno

[dimreepr]

I'm currently reading Stephen Hawking's book "A Briefer History of Time" and I was a little confused by the suggestion that the uncertainty principle undermines determinism, as stated in this sentence from the book:

 

 

 

As explained in the book, the uncertainty principle states that the more accurately the position of a particle is measured, the less accurate the measurement will be of the particle's velocity. Therefore, the "initial conditions" of the system can never be accurately known in order to determine the past or future state of the universe.

 

This much is easy to understand, but it seems that the uncertainty principle really only undermines our ability to calculate these past or future states of the universe, not that it actually undermines the fact that the universe still is deterministic (not that it necessarily is, but the uncertainty principle as stated above doesn't seem to suggest otherwise), even if we can never calculate any given state of it because of this uncertainty.

 

However, I have seen elsewhere that perhaps the uncertainty principle suggests more than the simple explanation given in the book. That, in fact, it explicitly says that the position or velocity of a particle is not actually determined at all until it is measured. This seems to really hurt determinism, but is that really what the uncertainty principle says? That a particle's position or velocity essentially doesn't exist at all until we measure it, or is it simply that we can never know a particle's position or velocity for sure until the time of its measurement?

 

Basically, my question comes down to this: even given the uncertainty principle and the probabilistic nature of quantum mechanics, couldn't it still be the case that the universe is perfectly deterministic, even if we can't accurately make the measurements to determine these past or future states ourselves? Doesn't a particle still have a certain position and velocity at any given time, even if measuring one of these will then change the other?

 

Hawking hints at this idea:

 

 

 

Frankly, I think the mention of a "supernatural being" unnecessarily muddies the discussion and makes Hawking dismiss the idea too easily. It's not necessary to postulate anything supernatural in order to retain determinism.

 

The above quote could easily have been stated as something like this: "We could still imagine that there is a set of laws that determine events completely, despite our inability to measure the present state of the universe without disturbing it." Then the second sentence would be irrelevant, because it would be of interest to us to know that it is possible for the universe still to be perfectly deterministic, even if we can't (yet) discover all the laws.

 

I think quatum mechanics entirely muddies the scientific waters. The problem being it has no hypothesis. Its our atempt at retro-engineering the universe, consiquently it is filled with this type of uncertaincy

[questionposter]

Everything is built from the atomic realm, so how could the classical realm be deterministic if the atomic realm isn't?

[John Salerno]

Everything is built from the atomic realm, so how could the classical realm be deterministic if the atomic realm isn't?

 

You don't seem to understand my point. I'm saying, just because a particle's position and velocity can't be measured accurately simultaneously, does that really mean that there still isn't an objective, determined position and velocity by which we could still theoretically determine past or future events. I.e., how does the uncertainty principle really undermine determinism?

[imatfaal]

You don't seem to understand my point. I'm saying, just because a particle's position and velocity can't be measured accurately simultaneously, does that really mean that there still isn't an objective, determined position and velocity by which we could still theoretically determine past or future events. I.e., how does the uncertainty principle really undermine determinism?

 

HUP does not merely state that we cannot simultaneously measure position and momentum, it holds that particles cannot have absolutely defined positions and momentum. A particle has a volume of space it is likely to occupy - not an exact position. Determinism relies on being theoretically able to ascertain the position and momentum of all particles absolutely - as we cannot be sure of even one particle (even theoretically - let alone in measurement) then it fails.

[questionposter]

You don't seem to understand my point. I'm saying, just because a particle's position and velocity can't be measured accurately simultaneously, does that really mean that there still isn't an objective, determined position and velocity by which we could still theoretically determine past or future events. I.e., how does the uncertainty principle really undermine determinism?

 

Information isn't necessarily preserved through time, it get's destroyed and re-created, i.e. in the atomic world there is no past information you can use to accurately predict the future with. Whether or not something happens to be where it is or whether or not something will be perceived the specific way that it is, is chance. Things might seem deterministic because at large distances, probability is very minuscule, and there's also that at the large distances we see, we aren't just dealing with one atom, its gooplexes.

Edited December 19, 2011 by questionposter

[michel123456]

HUP does not merely state that we cannot simultaneously measure position and momentum, it holds that particles cannot have absolutely defined positions and momentum. A particle has a volume of space it is likely to occupy - not an exact position. Determinism relies on being theoretically able to ascertain the position and momentum of all particles absolutely - as we cannot be sure of even one particle (even theoretically - let alone in measurement) then it fails.

It fails to what?

 

When HUP states that there is no way to theoretically ascertain the position and momentum of a particle, what does that mean?:

A. that there is no position and momentum? that no exist?

B. that there are too many positions and momenta? that we cannot decide which is the good pair.

[imatfaal]

It fails to what?

 

When HUP states that there is no way to theoretically ascertain the position and momentum of a particle, what does that mean?:

A. that there is no position and momentum? that no exist?

B. that there are too many positions and momenta? that we cannot decide which is the good pair.

 

Determinism fails. That is to say that its basic premise is found to be incorrect - "if we could map the position and momentum of all the particles in the universe accurately enough..." we can't; end of story.

 

 

It means that there is no absolutely defined position or momentum within a universe modelled by quantum mechanics. The amazing power, accuracy, and testability of qm does not preclude a deeper understanding/model "underneath" qm, within which uncertainty is explained away; but at present our best understanding, models, and theories rely on qm and its strange restrictions and implications .

[michel123456]

It means that there is no absolutely defined position or momentum within a universe modelled by quantum mechanics.

 

I thought it was: when you know momentum you don't know position and when you know position you don't know momentum.

 

Or:

_when you know momentum many positions can do.

_when you know position many momenta can do.

[John Salerno]

HUP does not merely state that we cannot simultaneously measure position and momentum, it holds that particles cannot have absolutely defined positions and momentum.

 

Ok, this was my main question. If it's the former, then determinism can still hold. If it's the latter, like you say it is, then I see how it undermines determinism. But does HUP really say that particles don't have defined positions and momentums? I thought it was just the principle that they can't be measured together.

 

I know that *later* QM has come up with the idea (after the two-slit experiment) that particles act like waves and thus really only have a "range" of possible positions, but is that really a part of the HUP itself? And even if it's true that a particle has a range of possibilities for position and momentum, isn't this still just a problem with our measurements, and not necessarily of the absolute position or momentum of the particle itself?

[imatfaal]

Ok, this was my main question. If it's the former, then determinism can still hold. If it's the latter, like you say it is, then I see how it undermines determinism. But does HUP really say that particles don't have defined positions and momentums? I thought it was just the principle that they can't be measured together.

 

I know that *later* QM has come up with the idea (after the two-slit experiment) that particles act like waves and thus really only have a "range" of possible positions, but is that really a part of the HUP itself? And even if it's true that a particle has a range of possibilities for position and momentum, isn't this still just a problem with our measurements, and not necessarily of the absolute position or momentum of the particle itself?

 

Not sure of the history/timeline. But Young's double slit experiment was very early - way before QM, in the 60s Jonsson performed it with electrons rather than light (or water waves etc) and caused a bit of a stir.

 

Uncertainty within our theories at the moment is not merely a measurement problem. Within the copenhagen interpretation of qm it comes from the fact that the matrices used to represent position and momentum do not commute - this implies the uncertainty, and it was later confirmed mathematically that uncertainty is necessary and the amount of this minimum amount of uncertainty was quantified . This is to say that it is deeply rooted within QM theory and is not merely an artefact of subs-standard measurement.

[John Salerno]

Not sure of the history/timeline. But Young's double slit experiment was very early - way before QM, in the 60s Jonsson performed it with electrons rather than light (or water waves etc) and caused a bit of a stir.

 

Uncertainty within our theories at the moment is not merely a measurement problem. Within the copenhagen interpretation of qm it comes from the fact that the matrices used to represent position and momentum do not commute - this implies the uncertainty, and it was later confirmed mathematically that uncertainty is necessary and the amount of this minimum amount of uncertainty was quantified . This is to say that it is deeply rooted within QM theory and is not merely an artefact of subs-standard measurement.

 

Hmm, very interesting. I know I need to read much more about QM than Hawking's brief summary, so that's a good start to get a sense of what it's saying. I need to find some other books for the general reader.

[DrRocket]

You don't seem to understand my point. I'm saying, just because a particle's position and velocity can't be measured accurately simultaneously, does that really mean that there still isn't an objective, determined position and velocity by which we could still theoretically determine past or future events. I.e., how does the uncertainty principle really undermine determinism?

 

You have succumbed to inaccurate descriptions of the uncertainty principle, common in popularizations.

 

What the uncertainty principle actually says is that for two complementary observables (position and momentum are complimentary) that if one takes particles prepared in identical quantum states and then does repeated measurements of position x followed by momentum p (or momentum p followed by position x) that

 

[math] \sigma_x \sigma_p \ge \frac{\hbar}{2} [/math]

 

where [math]\sigma_x[/math] and [math]sigma_p[/math] are the standard deviations associated with the random variables [math]x[/math] and [math]p[/math] respectively.

 

Inherent in this statement are that position and momentum are not deterministic. They are ranndom variables associated with a stochastic system.

 

It is NOT a statement that there will be any definite error in the measurement of either position or momentum. It is in fact a statement that there is no such thing as a definite measurable value of either position or momentum associated with a quantum particle.

[uncool]

This actually relates to a question I've had for some time:

 

Is it possible to determine the wavefunction for a particle without disturbing it? It wouldn't actually determine the position or momentum of the particle fully, and therefore wouldn't violate the uncertainty principle that way.

=Uncool-

[michel123456]

I succumbed too.

 

You have succumbed to inaccurate descriptions of the uncertainty principle, common in popularizations.

 

What the uncertainty principle actually says is that for two complementary observables (position and momentum are complimentary) that if one takes particles prepared in identical quantum states and then does repeated measurements of position x followed by momentum p (or momentum p followed by position x) that

 

[math] \sigma_x \sigma_p \ge \frac{\hbar}{2} [/math]

 

where [math]\sigma_x[/math] and [math]sigma_p[/math] are the standard deviations associated with the random variables [math]x[/math] and [math]p[/math] respectively.

(...)

 

Bolded mine.

So this is a statement about "measurements".

IIRC "standard deviations" is a term associated with measurements too.

 

Inherent in this statement are that position and momentum are not deterministic. They are ranndom variables associated with a stochastic system.

It is not trivial. To me it is not inherent at all.

[O'Nero Samuel]

I'm currently reading Stephen Hawking's book "A Briefer History of Time" and I was a little confused by the suggestion that the uncertainty principle undermines determinism, as stated in this sentence from the book:

 

 

 

As explained in the book, the uncertainty principle states that the more accurately the position of a particle is measured, the less accurate the measurement will be of the particle's velocity. Therefore, the "initial conditions" of the system can never be accurately known in order to determine the past or future state of the universe.

 

This much is easy to understand, but it seems that the uncertainty principle really only undermines our ability to calculate these past or future states of the universe, not that it actually undermines the fact that the universe still is deterministic (not that it necessarily is, but the uncertainty principle as stated above doesn't seem to suggest otherwise), even if we can never calculate any given state of it because of this uncertainty.

 

However, I have seen elsewhere that perhaps the uncertainty principle suggests more than the simple explanation given in the book. That, in fact, it explicitly says that the position or velocity of a particle is not actually determined at all until it is measured. This seems to really hurt determinism, but is that really what the uncertainty principle says? That a particle's position or velocity essentially doesn't exist at all until we measure it, or is it simply that we can never know a particle's position or velocity for sure until the time of its measurement?

 

Basically, my question comes down to this: even given the uncertainty principle and the probabilistic nature of quantum mechanics, couldn't it still be the case that the universe is perfectly deterministic, even if we can't accurately make the measurements to determine these past or future states ourselves? Doesn't a particle still have a certain position and velocity at any given time, even if measuring one of these will then change the other?

 

Hawking hints at this idea:

 

 

 

Frankly, I think the mention of a "supernatural being" unnecessarily muddies the discussion and makes Hawking dismiss the idea too easily. It's not necessary to postulate anything supernatural in order to retain determinism.

 

The above quote could easily have been stated as something like this: "We could still imagine that there is a set of laws that determine events completely, despite our inability to measure the present state of the universe without disturbing it." Then the second sentence would be irrelevant, because it would be of interest to us to know that it is possible for the universe still to be perfectly deterministic, even if we can't (yet) discover all the laws.

 

John,

 

 

keep in mind that velocity and position are determined relative to the position of the observer. The uncertainty principle highlights this relativity to framer of reference and practically shows that there is no other frame of reference that is relatively in inertia with reference to other frames of references. In this sense, then, getting consistency and precision in the measurement of the position and velocity of a body is determined by its relative consistency with the integral of that due to other frame of reference. This is the underlying truth of the Uncertainty principle.

 

And for determinism, it is simply trumped by time relativity and orbits. Hawking did not dismiss this concept by terming it in the supernatural domain...he only accepted his limit. There is what i call back timing...this is what time use to trump all form of determinism...it creates an unending constant change. Keep in mind that this change is relative to the perspective of the observer....and position and velocity are measure with the human time...to me this is a really narrow time frame.

 

 

[John Salerno]

keep in mind that velocity and position are determined relative to the position of the observer. The uncertainty principle highlights this relativity to framer of reference and practically shows that there is no other frame of reference that is relatively in inertia with reference to other frames of references. In this sense, then, getting consistency and precision in the measurement of the position and velocity of a body is determined by its relative consistency with the integral of that due to other frame of reference. This is the underlying truth of the Uncertainty principle.

 

And for determinism, it is simply trumped by time relativity and orbits. Hawking did not dismiss this concept by terming it in the supernatural domain...he only accepted his limit. There is what i call back timing...this is what time use to trump all form of determinism...it creates an unending constant change. Keep in mind that this change is relative to the perspective of the observer....and position and velocity are measure with the human time...to me this is a really narrow time frame.

 

Hmm, but you seem to be framing the discussion of determinism (or lack thereof) within general relativity, which preceded the uncertainty principle. Are you suggesting that GR and the UP are saying the same thing about determinism? It seems to me that the UP is making a much stronger claim about uncertainty and non-determinism than the consequences of GR suggest.

[DrRocket]

 

Bolded mine.

So this is a statement about "measurements".

IIRC "standard deviations" is a term associated with measurements too.

 

 

 

Actually, he standard deviation is associated with the concept of a random variable and it has nothing to do with measuurements.

 

There is a notion of a "sample standard deviation", which is slightly different from the actual standard deviation (aka "population standard deviation"). You find the sample deviation discussed in the context of a measurement from a subset of the total population and it is the optimal non-biased estimate of the "population standard deviation" from the sample taken. But the standard deviation itself is associated with a random variable.

 

 

The reason that randomness is inherent in the statement of the Heisenberg uncertainty principle is that it is in fact a statement about random variables. The statement makes no sense in the context of a deterministic system.

 

Hmm, but you seem to be framing the discussion of determinism (or lack thereof) within general relativity, which preceded the uncertainty principle. Are you suggesting that GR and the UP are saying the same thing about determinism? It seems to me that the UP is making a much stronger claim about uncertainty and non-determinism than the consequences of GR suggest.

 

 

General relativity is a completely deterministic theory. It has absolutely nothing to do with the uncertainty principle.

 

In fact the most fundamental conflict between general relativity and quantum theory is that the former is completely deterministic while tthe latter is stochastic.

[michel123456]

@Samuel

John,

 

 

keep in mind that velocity and position are determined relative to the position of the observer. The uncertainty principle highlights this relativity to framer of reference and practically shows that there is no other frame of reference that is relatively in inertia with reference to other frames of references. In this sense, then, getting consistency and precision in the measurement of the position and velocity of a body is determined by its relative consistency with the integral of that due to other frame of reference. This is the underlying truth of the Uncertainty principle.

That looks like a fair interpretation. Do you have some reliable source for this?

 

@Dr Rocket

Here below the introduction of the wiki article on the Uncertainty Principle with bolded, underline, color mine

 

In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known. In layman's terms, the more precisely one property is measured, the less precisely the other can be controlled, determined, or known.

 

In his Nobel Laureate speech, Max Born said:

 

...To measure space coordinates and instants of time, rigid measuring rods and clocks are required. On the other hand, to measure momenta and energies, devices are necessary with movable parts to absorb the impact of the test object and to indicate the size of its momentum. Paying regard to the fact that quantum mechanics is competent for dealing with the interaction of object and apparatus, it is seen that no arrangement is possible that will fulfill both requirements simultaneously...[1]

 

 

Published by Werner Heisenberg in 1927, the uncertainty principle was a key discovery in the early development of quantum theory. It implies that it is impossible to simultaneously measure the present position while also determining the future motion of a particle, or of any system small enough to require quantum mechanical treatment.[2] Intuitively, the principle can be understood by considering a typical measurement of a particle. It is impossible to determine both momentum and position by means of the same measurement, as indicated by Born above. Assume that its initial momentum has been accurately calculated by measuring its mass, the force applied to it, and the length of time it was subjected to that force. Then to measure its position after it is no longer being accelerated would require another measurement to be done by scattering light or other particles off of it. But each such interaction will alter its momentum by an unknown and indeterminable increment, degrading our knowledge of its momentum while augmenting our knowledge of its position. So Heisenberg argues that every measurement destroys part of our knowledge of the system that was obtained by previous measurements.[3] The uncertainty principle states a fundamental property of quantum systems, and is not a statement about the observational success of current technology.[2]

 

What I understand is that the UP is about measurement. The last sentence in blue states that it has nothing to do with technology, it has to do with the interaction needed to collect knowledge. But it is all about measurement.

Edited December 21, 2011 by michel123456

[DrRocket]

@Dr Rocket

Here below the introduction of the wiki article on the Uncertainty Principle with bolded, underline, color mine

 

 

 

What I understand is that the UP is about measurement. The last sentence in blue states that it has nothing to do with technology, it has to do with the interaction needed to collect knowledge. But it is all about measurement.

 

Get a better source than a Wiki explanation in "layman's terms".

 

What I told you is correct. The Heisenberg uncertainty principle is a statement regarding the random variables that are observables in quantum mechanics. It is not tied to measurement, but its effects are, of course, seen in the sample paths of random variables that are the result of measurement.

 

The important point is that quantum mechanics is inherently stochastic and the uncertainty principle is a statement about random variables. The uncertainty principle is not the source of the stochastic nature of quantum mechanics but is simply an important statement about what are inherently random variables.

Edited December 22, 2011 by DrRocket

[michel123456]

Get a better source than a Wiki explanation in "layman's terms".

 

What I told you is correct. The Heisenberg uncertainty principle is a statement regarding the random variables that are observables in quantum mechanics. It is not tied to measurement, but its effects are, of course, seen in the sample paths of random variables that are the result of measurement.

 

The important point is that quantum mechanics is inherently stochastic and the uncertainty principle is a statement about random variables. The uncertainty principle is not the source of the stochastic nature of quantum mechanics but is simply an important statement about what are inherently random variables.

 

Who's talking?

You introduce yourself as a better source than Wiki & Max Born (1954 Nobel laureate).

 

This is a larger part of Born's Nobel lecture p. 265, 266 (p10, 11 pdf) bolded mine

Thus two questions arise: what prevents us, in spite of the theoretical assertion,

to measure both quantities to any desired degree of accuracy by refined

experiments? Secondly, if it really transpires that this is not feasible, are we

still justified in applying to the electron the concept of particle and therefore

the ideas associated with it?

Referring to the first question, it is clear that if the theory is correct - and

we have ample grounds for believing this - the obstacle to simultaneous

measurement of position and motion (and of other such pairs of so-called

conjugate quantities) must lie in the laws of quantum mechanics themselves.

In fact, this is so. But it is not a simple matter to clarify the situation. Niels

Bohr himself has gone to great trouble and ingenuity25 to develop a theory

of measurements to clear the matter up and to meet the most refined and

ingenious attacks of Einstein, who repeatedly tried to think out methods of

measurement by means of which position and motion could be measured

simultaneously and accurately. The following emerges: to measure space

coordinates and instants of time, rigid measuring rods and clocks are required.

On the other hand, to measure momenta and energies, devices are

necessary with movable parts to absorb the impact of the test object and to

indicate the size of its momentum. Paying regard to the fact that quantum

mechanics is competent for dealing with the interaction of object and apparatus,

it is seen that no arrangement is possible that will fulfil both require-

ments simultaneously. There exist, therefore, mutually exclusive though

complementary experiments which only as a whole embrace everything

which can be experienced with regard to an object.

 

But you refuted this source.

What about this one?

http://galileo.phys.virginia.edu/classes/252/uncertainty_principle.html

Can we take it as common ground ?

[DrRocket]

Who's talking?

You introduce yourself as a better source than Wiki & Max Born (1954 Nobel laureate).

 

This is a larger part of Born's Nobel lecture p. 265, 266 (p10, 11 pdf) bolded mine

 

 

But you refuted this source.

What about this one?

http://galileo.phys...._principle.html

Can we take it as common ground ?

 

Things are a hell of a lot better understood today than in 1959.

 

Go get that better source.

 

While you are at it understand your own references better. I don't see any significant conflict Born's statements with what I said earlier. But your last reference give the common, incorrect, over-simplfied statement of the uncertainty principle that one finds in popularizations and too-simple undergraduate texts. What I told you in terms of standard deviations is correct.

 

The fact that the Heisenberg uncertainty principle is reflected in the results of experiments, which obviously require measurements, does not imply that the fundamental issue is measurement or that the uncertainty principle is the source of the stochastic nature of quantum mechanics.

Edited December 22, 2011 by DrRocket

[michel123456]

Things are a hell of a lot better understood today than in 1959.

 

Ah. Born was wrong?

 

 

Go get that better source.

Which one?

 

While you are at it understand your own references better. I don't see any significant conflict Born's statements with what I said earlier.

So you must have become blind between your last 2 posts. I thought Born was wrong, now you say he was right. Is that it?

 

But your last reference give the common, incorrect, over-simplfied statement of the uncertainty principle that one finds in popularizations and too-simple undergraduate texts.

I hope professor M.Fowler of the University of Virginia, author of my last link, does not follow this Forum. You said "Incorrect"? Are you an academic?

 

 

The fact that the Heisenberg uncertainty principle is reflected in the results of experiments, which obviously require measurements, does not imply that the fundamental issue is measurement or that the uncertainty principle is the source of the stochastic nature of quantum mechanics.

That's the question.

Edited December 22, 2011 by michel123456

[DrRocket]

Ah. Born was wrong?

 

 

 

 

 

Which one?

 

Any good book oon quantum mechanics -- the books by Griffiths or,Messiah are pretty standard.,

 

 

So you must have become blind between your last 2 posts. I thought Born was wrong, now you say he was right. Is that it?

 

I am neither blind nor stupid. What you think is your problem.

 

I hope professor M.Fowler of the University of Virginia, author of my last link, does not follow this Forum. You said "Incorrect"? Are you an academic?

 

I don't give a damn what Fowler thinks or what he follows. Why would I ? I provided you with the proper statement of the Heisenberg Uncertainty Principle.

 

I am no longer an academic. But that is not important. What is important is the proper statement of the principle and understanding of that statement.

[michel123456]

I am neither blind nor stupid. What you think is your problem.

You are not stupid Dr.

I whish you a merry christmas.