Why is probability amplitude squared?

[Mandragonia]

I find this really fascinating. The Schroedinger Equation is a diffusion equation with an imaginary diffusion coefficient (or real diffusion in imaginary time?) Is that just a coincidence or is there some underlying process driving it? Why would it become more uncertain over time?

1. Is it just coincidence?
Yes, I think so. The main focus of Erwin Schroedinger and his colleagues was on building a model that describes the atomic structure and its properties, in particular emission and absorption spectra. The behaviour of free electrons was (presumably) less important.

2. Is there some underlying process driving it?
The spreading of a wave packet is regarded a consequence of Heisenberg's uncertainty principle.

3. Why would it become more uncertain over time?
A free electron can be in sharply localized state at t=0, with small uncertainty in position dx and some uncertainty in velocity dv. If you wait for some time, the wave packet evolves. The uncertainty in position increases due to the additional uncertainty in the velocity. Finally it becomes of the order t*dv, which can be much larger than the initial uncertainty dx.

 

[Jilang]

1. Is it just coincidence?
Yes, I think so. The main focus of Erwin Schroedinger and his colleagues was on building a model that describes the atomic structure and its properties, in particular emission and absorption spectra. The behaviour of free electrons was (presumably) less important.

2. Is there some underlying process driving it?
The spreading of a wave packet is regarded a consequence of Heisenberg's uncertainty principle.

3. Why would it become more uncertain over time?
A free electron can be in sharply localized state at t=0, with small uncertainty in position dx and some uncertainty in velocity dv. If you wait for some time, the wave packet evolves. The uncertainty in position increases due to the additional uncertainty in the velocity. Finally it becomes of the order t*dv, which can be much larger than the initial uncertainty dx.

Thanks for this. Does the uncertainty in the velocity stay the same or does that increase too?

 

[Mandragonia]

Thanks for this. Does the uncertainty in the velocity stay the same or does that increase too?

It stays the same for a free electron. Of course in a different setting it may increase too. I forgot to mention that the technical term for the gradual spreading of the wave packet is dispersion.

 

[teodorakis]

Hi i want to replenish this thread.
The wave function (not squared) describes the probability of a particle occupying a particular location in space - but this needs to be multiplied by the probability of a particle being detected at that same location in space.

By way of analogy, if Alice and Bob can be at one of four places, then the probability of Alice (or Bob) being at anyone place is 1:4 (the wave function) and the probability of Alice meeting Bob is also 1:4, BUT the probability of Alice meeting Bob at a specificied location is 1:16, ie. 4 squared.

That's how I interpret the wave function squared.
Carl Looper
8 December 2009
This is Carl's way of explaining and since I'm a bit bad with advanced math. This seems the most intuitive explanation to me. Is it that simple or do ı have to learn all the hamiltonian spaces states etc. to fully undersstand why this amplitude is squared.
By the way, squared amplitude is the intensity of a wave and probability is directly proportional to the intensity, and also compatible with the experimenatal results explanation does not give a relief to me...

 

[teodorakis]

And one other thing. Should i accept this as a postulate of the qm and do not attempt to look for further deeper explanation. In this case i will accept the probability of a particle is proportional to the intensity of the wave which is defined as the amplitude squared. Or should i interpret it as some joint probability of "meeting" the detector and detected.

 

[blue_leaf77]

BUT the probability of Alice meeting Bob at a specificied location is 1:16, ie. 4 squared.

What if there is only Alice in the system? How will you get the other 1/4 to give 1/16 after the multiplication?
The probability being modulus squared is a consequence of the basis representation of state posutlate. Expanding a state $|\psi \rangle$ into some complete orthonormal basis states $\left\{ |u_n\rangle \right\}$, and taking the inner product, you will get $\sum_n \langle u_n|u_n \rangle = 1$. This is where the modulus squaring of a state is interpreted as a probability because probabilities add up to unity. For the probability of a quantum object being found in a particular position, the representing basis states are the position basis state $|\mathbf{r}\rangle$

Should i accept this as a postulate of the qm

Yes, it is in fact one of the postulates of QM.

 

[lightarrow]

Also get a hold of a book by the great Lev Landau, IMHO an absolute classic, that will quite likely change your view of physics, called Mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20
...
The reason for the brevity is that, as pointed out by previous reviewers, Landau derives mechanics from symmetry. Historically, it was long after the main bulk of mechanics was developed that Emmy Noether proved that symmetries underly every important quantity in physics. So instead of starting from concrete mechanical case-studies and generalising to the formal machinery of the Hamilton equations, Landau starts out from the most generic symmetry and dervies the mechanics. The 2nd laws of mechanics, for example, is derived as a consequence of the uniqueness of trajectories in the Lagragian. For some, this may seem too "mathematical" but in reality, it is a sign of sophisitication in physics if one can identify the underlying symmetries in a mechanical system. Thus this book represents the height of theoretical sophistication in that symmetries are used to derive so many physical results.

Ok, Landau - Mechanics is a great book, very insightful.
But some things are probably too semplicistic, like the derivation of energy, momentum, angular momentum conservation from the lagrangian symmetry properties: Noether theorem is a bit less trivial than how he proves it, do you agree?

--
lightarrow

 

[teodorakis]

What if there is only Alice in the system? How will you get the other 1/4 to give 1/16 after the multiplication?
The probability being modulus squared is a consequence of the basis representation of state posutlate. Expanding a state $|\psi \rangle$ into some complete orthonormal basis states $\left\{ |u_n\rangle \right\}$, and taking the inner product, you will get $\sum_n \langle u_n|u_n \rangle = 1$. This is where the modulus squaring of a state is interpreted as a probability because probabilities add up to unity. For the probability of a quantum object being found in a particular position, the representing basis states are the position basis state $|\mathbf{r}\rangle$

Yes, it is in fact one of the postulates of QM.

Thanks but this looks like we found the number 1 from this inner product so it must be probability i am sorry for my ignorance but all i want is to understand this square thing in a more deeper level.

 

[Nugatory]

Hi i want to replenish this thread.
The wave function (not squared) describes the probability of a particle occupying a particular location in space - but this needs to be multiplied by the probability of a particle being detected at that same location in space.

By way of analogy, if Alice and Bob can be at one of four places, then the probability of Alice (or Bob) being at anyone place is 1:4 (the wave function) and the probability of Alice meeting Bob is also 1:4, BUT the probability of Alice meeting Bob at a specificied location is 1:16, ie. 4 squared.

That's how I interpret the wave function squared.
Carl Looper
8 December 2009
This is Carl's way of explaining and since I'm a bit bad with advanced math. This seems the most intuitive explanation to me. Is it that simple or do ı have to learn all the hamiltonian spaces states etc. to fully undersstand why this amplitude is squared.
By the way, squared amplitude is the intensity of a wave and probability is directly proportional to the intensity, and also compatible with the experimenatal results explanation does not give a relief to me...

It only seems intuitive to you because you're still trying to reinterpret QM to agree with your classical intuition.

QM says that the squared modulus of the wave function at a given position gives you the probability of finding the particle at that position if you perform a position measurement. It does not follow that you can break that probability down into "probability of particle being there not yet detected" and "probability of detecting it when it is there" - indeed, any attempt to identify the former quantity will lead to contradictions in the math and with experiment.

(There's another problem here in that the wave function is generally a complex number so cannot be the probability of anything. I'm assuming that you mean to interpret the unsquared magnitude of the wave function, as opposed to its value, as a probability. But if you do that, superpositions stop displaying interference... and we know that's not how the universe works).

 

[Vanadium 50]

all i want is to understand this square thing in a more deeper level.

Then it is better to ask questions than to try and push your own theories. Not only is the latter unproductive, it's against PF rules.

 

[bhobba]

i am sorry for my ignorance but all i want is to understand this square thing in a more deeper level.

I have refreshed my memory of what I said in earlier posts of this thread. I gave quite a few reasons - but the deepest reason of all is that to model physical systems if it goes from one state to another in say one second it must go through some other state in half a second. This is the requirement of continuity which is a standard condition that is assumed of physical systems. As the following shows that requirement, plus some reasonableness assumptions, leads to QM:
http://arxiv.org/pdf/quant-ph/0101012.pdf

That's the why of the formalism - what it means of course is a whole new ball qame and is argued about all the time.

Thanks
Bill

 

[bhobba]

Ok, Landau - Mechanics is a great book, very insightful. But some things are probably too semplicistic, like the derivation of energy, momentum, angular momentum conservation from the lagrangian symmetry properties: Noether theorem is a bit less trivial than how he proves it, do you agree?

Landau is like that. He is terse and you get the impression it's a bit simplistic. It isn't - its very deep - its just his style.

Most definitely Noether's theorem provides the underlying reason for all this stuff that Landau avoids. But to start with its good to see it doesn't really require such high powered machinery - you can look into that later eg:
https://www.amazon.com/dp/0801896940/?tag=pfamazon01-20

Even the above is not the deepest view which lies in Lie algebras - but slowly grasshopper, slowly

Thanks
Bill

 

[teodorakis]

Then it is better to ask questions than to try and push your own theories. Not only is the latter unproductive, it's against PF rules.

I am not trying to push my own theories, no need to be so aggressive. I am just asking questions. I already stated that my knowledge is restircted and i want to learn more.

 

[gill1109]

I was told this story: there are two natural ways in geometry to generate d non-negative numbers which add up to 1. One way is by taking a stick of length 1, and breaking it into d pieces. The other way is by taking a sphere of radius 1 in d-dimensional space and looking at the squares of the coordinates of a point on the surface. Pythagoras theorem. Quantum physics uses Pythagoras. Ordinary probability uses stick-breaking.

 

[cosmik debris]

Scott Aaronson discusses this and gives some references for further reading.

http://www.scottaaronson.com/democritus/lec9.html