Quantum amplitudes versus probability

[spaghetti3451]

Lately, I've been wondering that the sum rule from probability theory applies in all fields of study, expect for in quantum mechanics, where we need to apply the sum rule to probability amplitudes instead of to probabilities.

Is there any physical phenomenon that can explain this principle of quantum mechanics?

 

[mfb]

Where do you see the difference between "probability amplitudes" and "probabilities" in (probabilistic interpretations of) quantum mechanics?

To have something like a probability, probability has to be conserved in the time-evolution of the system - you always want 100% as sum of all probabilities. The squared amplitude is the easiest non-trivial function with that property.

 

[cosmik debris]

Probabilities are based on either the L1 or L2 norm as you have discovered. Probabilities based on other norms say L3 and up etc don't seem to be "interesting", at least not in physics. Somehow it seems to be related to the a^n + b^n = c^n only having integer solutions for 1 and 2. L2 norms allow negative probability amplitudes and for these amplitudes to cancel in some cases giving the phenomenon of interference. The squares of course are positive since a negative probability makes little sense.

I know this doesn't explain anything but a search will turn up quite a few papers on this.

 

[crissyb1988]

Probabilities are based on either the L1 or L2 norm as you have discovered. Probabilities based on other norms say L3 and up etc don't seem to be "interesting", at least not in physics. Somehow it seems to be related to the a^n + b^n = c^n only having integer solutions for 1 and 2. L2 norms allow negative probability amplitudes and for these amplitudes to cancel in some cases giving the phenomenon of interference. The squares of course are positive since a negative probability makes little sense.

I know this doesn't explain anything but a search will turn up quite a few papers on this.

The amplitudes can be negative or even complex, when you square them they become a probability out of 1 if the system is normalized.

 

[crissyb1988]

Lately, I've been wondering that the sum rule from probability theory applies in all fields of study, expect for in quantum mechanics, where we need to apply the sum rule to probability amplitudes instead of to probabilities.

Is there any physical phenomenon that can explain this principle of quantum mechanics?

Like mbf said we square the amplitudes and then add them as the sum of probabilities for each component of the system. Not quite sure I get what your asking?

 

[Fredrik]

we need to apply the sum rule to probability amplitudes instead of to probabilities.

Is there any physical phenomenon that can explain this principle of quantum mechanics?

It's part of the definition of quantum mechanics (or at least the path-integral version of QM). The only thing that can explain why QM is a good theory is a better theory.

 

[ENDLESSYOU]

The probability is one dimensional real number, while the probability ampilitude is two dimensional complex number. It is this two dimension makes it possible for the phenomenon like interference which needs the concept of phase. If we add two probability together, there won't be such interference. But when we add probability amplitude and then square them, we will get the cross terms which result the interference.

 

[bhobba]

Is there any physical phenomenon that can explain this principle of quantum mechanics?

Yes - entanglement. Check out the following very interesting paper:
http://arxiv.org/pdf/0911.0695v1.pdf

Thanks
Bill

 

[spaghetti3451]

All this discussion is unintelligible to me since I am a lowly undergraduate student who is struggling to grasp the basic principles of quantum mechanics.

My question was 'What physical phenomenon requires us to work with probability amplitudes in quantum mechanics'?

I think the answer is 'Quantum interference'. Let me illustrate with a simple example. In a two-slit interference experiment with electrons, the interference pattern is the sum of the intensities of the individual electrons and a superposition term. The superposition term is a mathematical manifestation of the quantum interference effect.

Comments are welcome.

P.S. I am a lowly undergraduate student who is struggling to grasp the basic principles of quantum mechanics, so please respond using concepts I might understand.

 

[vanhees71]

You are completely right! The superposition principle is one of the fundamental basic postulates in quantum theory. There is no "explanation" of it. As any theory in physics quantum theory is used, because it works in describing observations in nature (so far there is no observation which contradicts quantum theory).

 

[spaghetti3451]

You are completely right! The superposition principle is one of the fundamental basic postulates in quantum theory. There is no "explanation" of it. As any theory in physics quantum theory is used, because it works in describing observations in nature (so far there is no observation which contradicts quantum theory).

Thank you for your reply.

What I have understood from your reply boils down to this:

'The superposition principle applies to probability amplitudes and not to probabilities. No one has been able to provide a satisfying explanation for this postulate of quantum mechanics.

The principle is manifested in all physical phenomena as an effect which is known as quantum interference. For example, in the two-slit interference experiment with electrons, the interference pattern is the sum of the intensities of the individual electrons and a third cross term. The third term arises because the superposition principle applies to probability amplitudes and not to probabilities.'

Am I right?

 

[The_Duck]

Scott Aaronson has an interesting take on this: http://www.scottaaronson.com/democritus/lec9.html

That page is not about what experimental phenomena require probability amplitudes, but rather why probability amplitudes are natural mathematical objects.

 

[spaghetti3451]

Hmm! It's very amusing that although there's a far superior approach to introducing quantum mechanics to students, most universities still tend to adopt the older and traditional approach of teaching quantum mechanics.

Of course, we never learned Newtonian mechanics starting from the great experiments of Copernicus, Kepler and Galileo and then generalising their experimental observations to Newton's laws of motion. Instead, the theory of Newtonian mechanics is introduced to high school students with a set of postulates (which are explained using common observations in our daily life) and then the theory is applied to Galileo's free-fall experiment, Kepler's planetary motion, etc.

I wonder why quantum mechanics is still taught in the traditional approach. I think it's because the theory is less mature than the others and so physicists have a certain affliction to the history of the discovery of quantum mechanics. They feel that learning quantum mechanics in this way introduces students to the complex nature of theoretical and experimental discoveries of the highest calibre.

 

[bhobba]

Check out:
http://arxiv.org/pdf/0911.0695v1.pdf

Basically, accepting only some very reasonable axioms detailed in the above, there are just two choices - standard probability theory and Quantum Mechanics. What determines which should be used to model a given situation is two things (there may be others as well) either of which will determine which should be used. First if you require the outcomes of observations to be continuously connected to other outcomes as modelling physical systems would suggest then QM is required. Secondly only QM allows that weird and special thing called entanglement.

Thanks
Bill

 

[spaghetti3451]

Check out:
http://arxiv.org/pdf/0911.0695v1.pdf

Basically, accepting only some very reasonable axioms detailed in the above, there are just two choices - standard probability theory and Quantum Mechanics. What determines which should be used to model a given situation is two things (there may be others as well) either of which will determine which should be used. First if you require the outcomes of observations to be continuously connected to other outcomes as modelling physical systems would suggest then QM is required. Secondly only QM allows that weird and special thing called entanglement.

Thanks
Bill

The material in the paper is too advanced for me, given that I'm only an undergraduate student, but I can at least understand that all physical systems can be modeled by either the standard probability theory or quantum mechanics. What I can understand is that, for systems which can be modeled using standard probability theory, the predictions can be made more precise using the theories of classical physics. In the quantum case, such a precise prediction is not allowed.