Time evolution of a wave function

[Abhishek Sethi]

Hi,
I just completed my second year of my physics undergraduate degree. And recently did a course on Quantum Mechanics. I have a few questions regarding the basic theory and postulates, probably, because due to lack of full clarity.
So,
Consider a wave function ψ(x,o), which is well behaved and normalized. Which is the wave-function of a particular system initially.
Now, my question is, there are 2 postulates which determine the time evolution of any wave function.
One is the popular Schrodinger equation, which is described using the operator H(Hamiltonian).
The other postulate is, that upon "measurement", the wave function will collapse to one of the eigenvalues of the respective physical variable.
I do not understand the concept of "measurement" precisely, as I could not find that in my book. Is measurement taken as an instantaneous process? If not, then, is the "wave function collapse" postulate a consequence of the Schrodinger equation?
I think of measurement as an interaction with the system, which develops a time varying Hamiltonian, which makes the system evolve in accordance to Schrodinger equation, resulting in a wave function collapse?
If I am thinking the wrong way, then can we not use Schrodinger equation during the process of measurement?
If the wave function collapse is a consequence of Schrodinger equation, then why do es the wave function collapse seems to be a non-deterministic equation i.e is to say, the wave function cannot be uniquely determined after some time t(which is after measurement), while Schrodinger equation uniquely determines the wave function after a time t?
I need to be clear on this, even though it might be a trivial question. :)
Thank you.

 

[andrewkirk]

Hello Abishek. I'll offer a few suggestions that may help in thinking about this.

The other postulate is, that upon "measurement", the wave function will collapse to one of the eigenvalues of the respective physical variable.

I presume you meant 'eigenvectors' rather than 'eigenvalues' here. The eigenvalue is the amount read by the observer of the measuring equipment, not a representation of the post-measurement state.

A well-stated version of this postulate will not use the word 'collapse', because 'collapse' is an issue of interpretation that is not part of the science of QM itself.

A better way to say this is that, to make calculations about what happens after the measurement, we can assume that the state immediately after the measurement is represented by an eigenvector corresponding to the eigenvalue that was observed.

I do not understand the concept of "measurement" precisely, as I could not find that in my book. Is measurement taken as an instantaneous process?

The exact definition of what constitutes a measurement, and whether it is instantaneous, is one of interpretation, not core QM. It is not necessary to make a clear definition in terms of necessary and sufficient conditions, because all the measurements one is likely to make in practical science are well away from any fuzzy boundary regions between measurement and non-measurement. Nor do we need to decide whether it is instantaneous because, if it is not, it happens so fast that it makes no observable difference to the subsequent predictions. The most popular interpretation, involving decoherence, is that it is not instantaneous, but is very very fast.

If not, then, is the "wave function collapse" postulate a consequence of the Schrodinger equation?

Not in relation to the system being considered. I believe that, under decoherence theory, it may be a consequence of Schrodinger's equation applied to the larger system involving both the observed system and the measuring equipment, but others who are more familiar with the detail of decoherence would be able to comment better on that.

If the wave function collapse is a consequence of Schrodinger equation, then why does the wave function collapse seem to be a non-deterministic equation i.e is to say, the wave function cannot be uniquely determined after some time t(which is after measurement), while Schrodinger equation uniquely determines the wave function after a time t?

There are many things that seem to be non-deterministic (stochastic) that are not. The apparent stochasticity is really just epistemological - ie a lack of knowledge of the exact state. A coin flip is a classic example of this. Whether there is 'genuine' (non-epistomological) stochasticity involved is a matter of interpretation, not core-QM, and even defining what 'non-epistomological stochasticity' would be is problematic.

 

[Truecrimson]

I believe that, under decoherence theory, it may be a consequence of Schrodinger's equation applied to the larger system involving both the observed system and the measuring equipment, but others who are more familiar with the detail of decoherence would be able to comment better on that.

Precisely.

There are many things that seem to be non-deterministic (stochastic) that are not. The apparent stochasticity is really just epistemological - ie a lack of knowledge of the exact state. A coin flip is a classic example of this. Whether there is 'genuine' (non-epistomological) stochasticity involved is a matter of interpretation, not core-QM, and even defining what 'non-epistomological stochasticity' would be is problematic.

If we take collapse to be the process of decoherence, the stochasticity comes from not keeping track of environmental degrees of freedom in the larger system.

 

[bhobba]

If you want to understand the development of QM from the most elegant and transparent postulates (there is only two) see the first 3 chapters of Ballentine:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

You will understand collapse is not part of QM, the real basis of Schrodinger equation (its symmetry) and much more besides.

It had a strong effect on me, and I suspect it will likely be the same for you.

After that read:
https://arxiv.org/pdf/quant-ph/0101012v4.pdf

You may think that QM is pulled out of the air so to speak. It isn't really, but its a fun exercise figuring out what the physical assumptions are.

Thanks
Bill

 

[ShayanJ]

After that read:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

This second link also points to Ballentine's!

 

[bhobba]

This second link also points to Ballentine's!

Fixed

Thanks
Bill

 

[ShayanJ]

Fixed

Thanks
Bill

That paper again?! I wish I hadn't mentioned the wrong link!

 

[Abhishek Sethi]

I presume you meant 'eigenvectors' rather than 'eigenvalues' here. The eigenvalue is the amount read by the observer of the measuring equipment, not a representation of the post-measurement state.

Hi Andrew,
I actually meant eigenvectors and not eigenvalues. My apologies.

The exact definition of what constitutes a measurement, and whether it is instantaneous, is one of interpretation, not core QM. It is not necessary to make a clear definition in terms of necessary and sufficient conditions, because all the measurements one is likely to make in practical science are well away from any fuzzy boundary regions between measurement and non-measurement. Nor do we need to decide whether it is instantaneous because, if it is not, it happens so fast that it makes no observable difference to the subsequent predictions. The most popular interpretation, involving decoherence, is that it is not instantaneous, but is very very fast.

Hmmm. I understand your point. Not fully though. Can you please elaborate "the most popular interpretation"?
What I intended to ask was, what does measurement mean physically?
In other words, a measurement must ONLY affect the Hamiltonian operated on the wave function? I am not sure if I am correct here.
And if Hamiltonian is the only thing that changed, can I use Schrodinger equation to calculate the resultant wave function? Why or why not?

There are many things that seem to be non-deterministic (stochastic) that are not. The apparent stochasticity is really just epistemological - ie a lack of knowledge of the exact state. A coin flip is a classic example of this. Whether there is 'genuine' (non-epistomological) stochasticity involved is a matter of interpretation, not core-QM, and even defining what 'non-epistomological stochasticity' would be is problematic.

Hmm, are you essentially pointing out at the famous hidden variable theory? In a coin flip case, there was lack of information, that is to say, we are not aware of variables that influence the outcome of the result. I am not really sure what Bell's paper actually implied, but doesn't it eliminate the possibility of existing unknown hidden variable in QM?

Thank you Andrew.
Your response helped.
:)

 

[Abhishek Sethi]

Precisely.



If we take collapse to be the process of decoherence, the stochasticity comes from not keeping track of environmental degrees of freedom in the larger system.

Hi Truecrimson,
I did not fully understand what you mean by decoherence. I googled a bit, but could not understand clearly. Can you elaborate on this point or provide me a source to understand your point?
Thank you

 

[eudo]

Hi Andrew,
What I intended to ask was, what does measurement mean physically?

Simple question. Simple answer: We don't really know. This is a philosophical hanging thread in the theory. Measurements are needed to make sense of the theory, but it's hard to define what a measurement is. Look up "measurement problem in quantum mechanics" in Google. So, no, you're not stupid for wondering about this.

 

[Truecrimson]

Hi Truecrimson,
I did not fully understand what you mean by decoherence. I googled a bit, but could not understand clearly. Can you elaborate on this point or provide me a source to understand your point?
Thank you

Have you learned about density operators and mixed states yet? If you have, then this note might help.
https://arxiv.org/abs/quant-ph/0612118

Maximilian Schlosshauer's paper
http://arxiv.org/abs/quant-ph/0312059
and his book
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20
seem to be the standard references when it comes to the relation between decoherence and measurements in quantum theory.

I can say more when I have time, but I have to leave now.

Oh, and welcome to Physics Forums, Abhishek!

 

[Abhishek Sethi]

Have you learned about density operators and mixed states yet? If you have, then this note might help.
https://arxiv.org/abs/quant-ph/0612118

Maximilian Schlosshauer's paper
http://arxiv.org/abs/quant-ph/0312059
and his book
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20
seem to be the standard references when it comes to the relation between decoherence and measurements in quantum theory.

I can say more when I have time, but I have to leave now.

Oh, and welcome to Physics Forums, Abhishek!

Thank you Truecrimson. I will check these sources out.

 

[andrewkirk]

Can you please elaborate "the most popular interpretation"?

There was a survey a little while ago of physicists as to which interpretation of QM they favoured.
Sean Carroll wrote about it here.

What I intended to ask was, what does measurement mean physically?

I think it is generally agreed that a sufficient condition for a measurement is that a human reads an experimental result from some apparatus. Very few people believe that the participation of a human or other conscious organism is a necessary condition however. Those that do tend to gravitate towards the quantum mystic position that consciousness plays a role in what is loosely called 'wave function collapse'. The decoherence-based view doesn't need consciousness. It just says that what we call 'measurement' happens when the system being studied interacts with and hence becomes entangled with a macro-scale system, and a human operating a piece of scientific apparatus is certainly large enough to be considered macro scale.

I am not really sure what Bell's paper actually implied, but doesn't it eliminate the possibility of existing unknown hidden variable in QM?

Bell's paper says that, if the inequalities it states are violated - and there is compelling experimental evidence that they are - then there can be no local hidden variable theory. That does not rule out non-local theories such as Bohm's.

 

[Truecrimson]

In other words, a measurement must ONLY affect the Hamiltonian operated on the wave function? I am not sure if I am correct here. And if Hamiltonian is the only thing that changed, can I use Schrodinger equation to calculate the resultant wave function? Why or why not?

The Schrödinger equation applies only to systems that are, to a good approximation, uncorrelated with other systems. A system that has been measured has to at least correlate with the measuring device. Even if the system together with the measuring device is an isolated system, in general there won't be a differential equation for the wave function of the system. This is because the state of the system at any point in time may not depend only on the state of the system at previous time but also on the state of the measuring device.

But a macroscopic measuring device is not an isolated system and it correlates with lots of other stuffs whose states we don't and practically can't know. So even though the Schrödinger equation that governs everything that is now correlated with the measured system is deterministic, the lack of knowledge of all the relevant degrees of freedom translates to the unpredictability of the state of the measured system. This is called decoherence (which is distinguished from (reversible) dephasing or other terms in Hornberger's note that I linked to).