Superposition state and wavefunction collapse

[dyn]

Hi. My question is a general one but I will use an infinite well as an example.

Without knowing details of the exact wavefunction I presume it can exist as a linear superposition of an infinite number of energy eigenstates ? Without knowing the exact wavefunction ; does that mean that when the energy is measured all energy eigenvalues are equally likely ? Even the infinite energy value ?

When the energy is measured the wavefunction collapses to the associated eigenfunction. But then what happens ? Does it stay in that state forever ?
Thanks

 

[Vanadium 50]

Without knowing the exact wavefunction ; does that mean that when the energy is measured all energy eigenvalues are equally likely ?

No.

 

[bhobba]

No.

You need to learn some actual QM. Since this is an I level thread you should be able to understand the following:
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20
https://www.amazon.com/dp/0465062903/?tag=pfamazon01-20

But the answer is it depends on what's called the Born Rule:
https://en.wikipedia.org/wiki/Born_rule

Thanks
Bill

 

[PeroK]

Hi. My question is a general one but I will use an infinite well as an example.

Without knowing details of the exact wavefunction I presume it can exist as a linear superposition of an infinite number of energy eigenstates ? Without knowing the exact wavefunction ; does that mean that when the energy is measured all energy eigenvalues are equally likely ? Even the infinite energy value ?

When the energy is measured the wavefunction collapses to the associated eigenfunction. But then what happens ? Does it stay in that state forever ?
Thanks

There are two mathematical problems with your question. If you have an infinite number of states, then they cannot all be equally likely - unless that likelihood is zero. This is because the total probability must add up to at most 1.

There is no "infinite" energy state. All energy states represent a finite energy level

Finally, unless you have some information about how a particular state was prepared, you can't make any assumptions about the probability of any energy level being measured.

 

[bhobba]

I thought one of the aims of this forum was to help and educate not to insult or patronize. If any of my questions are beneath any members then maybe they would be better off not replying.

I am sorry for any insult etc you may have felt - it was not intended - I assure you. Not only would doing such be against forum rules it simply is not a decent thing to do. So for that I apologize.

However you will find a lot of responses here do not directly answer questions. This is for a number of reasons:
1. The answer would be long and if standard textbook stuff you would be better simply reading the book.
2. What you nut out for yourself you remember and understand better.

Plus I also pointed you to the direct answer as to why its not equal probability - the Born Rule.

I am sorry its like this in QM but unfortunately questions like you asked require actually studying QM. I did give the answer - the Born Rule - but until you understand QM its not likely to make sense which is why I directed you to some texts on basic QM first. If you already have that knowledge then the Born Rule answers all your queries.

Now the next question is likely why the Born Rule - that is tied up with something called Gleason's Theorem which is an advanced topic, but I will give a link to anyway - see post 137:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Thanks
Bill

 

[atyy]

Hi. My question is a general one but I will use an infinite well as an example.

Without knowing details of the exact wavefunction I presume it can exist as a linear superposition of an infinite number of energy eigenstates ? Without knowing the exact wavefunction ; does that mean that when the energy is measured all energy eigenvalues are equally likely ? Even the infinite energy value ?

No. The probability of obtaining a given energy eigenvalue is given by the Born rule.
https://mcgreevy.physics.ucsd.edu/w15/130C-2015-chapter01.pdf (p1-19, Axiom 4)

When the energy is measured the wavefunction collapses to the associated eigenfunction. But then what happens ? Does it stay in that state forever ?

Yes. This is true in general for systems whose time evolution is governed by a time-independent Hamiltonian. https://mcgreevy.physics.ucsd.edu/w15/130C-2015-chapter01.pdf (p1-18, Axiom 3)

 

[dyn]

[Moderator's note: off topic comment deleted.]

There are two mathematical problems with your question. If you have an infinite number of states, then they cannot all be equally likely - unless that likelihood is zero. This is because the total probability must add up to at most 1.

There is no "infinite" energy state. All energy states represent a finite energy level

Finally, unless you have some information about how a particular state was prepared, you can't make any assumptions about the probability of any energy level being measured.

As far as I know the energy levels in an infinite well are given by E_n = n²π²ħ²/(2mL²) and as quoted in University Physics by Young & Freedman " there are an infinite number of levels ; even a particle of infinitely great kinetic energy is confined within the box"

I am sorry for any insult etc you may have felt - it was not intended - I assure you. Not only would doing such be against forum rules it simply is not a decent thing to do. So for that I apologize.

However you will find a lot of responses here do not directly answer questions. This is for a number of reasons:
1. The answer would be long and if standard textbook stuff you would be better simply reading the book.
2. What you nut out for yourself you remember and understand better.

Plus I also pointed you to the direct answer as to why its not equal probability - the Born Rule.

I am sorry its like this in QM but unfortunately questions like you asked require actually studying QM. I did give the answer - the Born Rule - but until you understand QM its not likely to make sense which is why I directed you to some texts on basic QM first. If you already have that knowledge then the Born Rule answers all your queries.

Now the next question is likely why the Born Rule - that is tied up with something called Gleason's Theorem which is an advanced topic, but I will give a link to anyway - see post 137:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Thanks
Bill

Thanks for your reply. I have studied the Born Rule and as far as I understand it - it gives the probability of obtaining the measured eigenvalue for each eigenfunction as | < φ_n | ψ > |² but to perform this calculation requires knowing the original wavefunction. My original question was concerned with if the original wavefunction was not known

 

[PeroK]

As far as I know the energy levels in an infinite well are given by En = n2π2ħ2/(2mL2) and as quoted in University Physics by Young & Freedman " there are an infinite number of levels ; even a particle of infinitely great kinetic energy is confined within the box"

All those energy levels have finite energy. There is no such thing as a particle of infinitely great kinetic energy. You could say "no matter how much energy a particle has, it stays within the box".

 

[dyn]

[Moderator's note: off topic comment deleted.]

All those energy levels have finite energy. There is no such thing as a particle of infinitely great kinetic energy. You could say "no matter how much energy a particle has, it stays within the box".

If the energy varies as n² and n can go to infinity , how is that not infinite energy ?

 

[PeroK]

If the energy varies as n² and n can go to infinity , how is that not infinite energy ?

For what value of $n$ is the energy infinite?

 

[PeterDonis]

Thread closed for moderation.

Edit: Some off topic posts and portions of posts have been deleted. Please keep the discussion focused on the physics. If you feel someone else's post violates the rules or is disrespectful, please report it; do not respond to it.

Thread reopened.

 

[PeterDonis]

there are an infinite number of levels

This is true, but it is not the same as saying that the energy eigenvalue of any level is infinite. All of the energy eigenvalues are finite.

n can go to infinity

This is sloppy phrasing which is leading you astray. Better phrasing would be: n can be any arbitrarily large finite positive integer. See the difference?

 

[PeterDonis]

even a particle of infinitely great kinetic energy is confined within the box

If this is a quote from the actual textbook, it makes me like that textbook less. This is sloppy phrasing as well, and it means the textbook itself has led you astray. Unfortunately that does happen.

 

[PeterDonis]

My original question was concerned with if the original wavefunction was not known

If it isn't known, then of course you can't compute probabilities. But you do know that any such wave function can be expressed as a properly normalized linear combination of energy eigenfunctions, which are known. And there are an infinite number of them, one for each finite positive integer. Can you construct such a properly normalized linear combination that has equal coefficients for each term? If you can't, then that shows that there is no possible wave function that has equal probabilities for every energy. And @PeroK has already given you a reason why you can't.