Extremely elementary questions about QM

[okaythanksbud]

Im trying to learn QM on my own and I just want to clear some things up. I feel dumb writing some out but Id rather clear my confusion than believe im interpreting what I read correctly.

From what I've read, every measurement of a system gives us values that are the eigenvalues of a certain hermitian operator, which "corresponds" to the measurement. This seems like it was pulled out of thin air but its clear enough so ill take it. Now onto the questions:

1: Is it true that upon measurement the wave function will collapse into the eigenvector/eigenfunction corresponding to the eigenvalue we measured. Does this mean, for example, that if I measured energy E_n in a particle in a box, the wave function from that point forward would definitely be the solution to Hf=E_nf? (H is the hamiltonian operator with a potential of 0, and f(0)=f(L)=0--I discounted the time factor since it only affects the solution up to a phase)

2: If I were to shake the box after this measurement, im assuming it would go back to square 1 of being in an indeterminate state (assuming in 1) the wave function collapses into a determined state corresponding to E_n). Does the initial measurement have any bearing on the system now?

3: just to be sure, does a state refer to every possible description of a particle in some instant or just one description? If I interpret this correctly, the wave function is supposed to hold all the information about the state of our system. Does this mean there is only one wave function (aka only one state--which is a superposition of other states) for our system, or is there one for every measurement we can make (i.e. would the wave function of an electron in a box allow us to also predict, say, the spin of it in addition to its position, though not necessarily both simultaneously? or is there another wave function for that)?

4: Are states ever related--can making a measurement on one ever tell us anything about the other? Im assuming the answer must be yes--if so how can we tell? Is this maybe related to the commutator?

5: is it possible to collapse the wave function into a new state that is not definite, but a new "truncated" one? For example, is it possible to make the deduction, say for a particle in a box, that the energy is NOT E_n, giving us a new wave function corresponding to the superposition of every state besides the one corresponding to E_n? I have no idea about this one, im tempted to say no since Im unsure it would be possible to find Fourier coefficients to match a certain initial condition.

6: if A is some operator corresponding to a measurement, does Af actually represent anything (where f is a wave function for the system)? It makes sense that it would be used to calculate expectation values and in the case of the hamiltonian solve for the wavefunction due to its relation with the time derivative, but does it represent anything physical in our system? Its not like Af is a description of the state of our system or anything, correct?

Im sorry if these questions were very stupid or made no sense, its very hard to tell what inferences I can/should be making at the moment since learning about this is making me feel like im learning arithmetic for the first time in my life. I would greatly appreciate any help.

 

[PeterDonis]

@okaythanksbud please read this article (which is one of the sticky articles at the top of this forum):

https://www.physicsforums.com/threads/the-7-basic-rules-of-quantum-mechanics.971724/

 

[okaythanksbud]

@okaythanksbud please read this article (which is one of the sticky articles at the top of this forum):

https://www.physicsforums.com/threads/the-7-basic-rules-of-quantum-mechanics.971724/

Ah thanks for that, that’s definitely useful for me. However some of my questions aren’t addressed there (some aspects of 2-6 are mentioned but it doesn’t really allude to any answers, as far as I can see)

 

[PeterDonis]

some of my questions aren’t addressed there

Generally speaking, questions that aren't answered by one of the 7 basic rules might be questions of QM interpretation--i.e., different QM interpretations will give different answers. Discussion of interpretations belongs in the interpretations subforum.

That said, I'm not sure questions 2-6 aren't answered. Though in some cases it's your questions that need to be rethought. See below.

2: If I were to shake the box after this measurement, im assuming it would go back to square 1 of being in an indeterminate state (assuming in 1) the wave function collapses into a determined state corresponding to E_n). Does the initial measurement have any bearing on the system now?

Yes; your measurement prepared a new initial state, per Rule 7.

3: just to be sure, does a state refer to every possible description of a particle in some instant or just one description?

What does "description" mean?

If I interpret this correctly, the wave function is supposed to hold all the information about the state of our system.

See Rule 2. Pay particular attention to the distinction between the Hilbert space vector in general, and the position representation (the latter context is where the wave function comes in).

Does this mean there is only one wave function (aka only one state--which is a superposition of other states) for our system, or is there one for every measurement we can make (i.e. would the wave function of an electron in a box allow us to also predict, say, the spin of it in addition to its position, though not necessarily both simultaneously? or is there another wave function for that)?

If you want to represent both position (or more precisely "configuration", since you could use either the position representation, the momentum representation, or some other representation) and spin, the Hilbert space needs to include dimensions for both. So a full state in the full Hilbert space will be a linear combination of products of a configuration space state and a spin state.

It is true that the 7 Basic Rules article does not go into these kinds of details. But the rules still do apply to these kinds of states.

4: Are states ever related--can making a measurement on one ever tell us anything about the other?

States of what? What does "related" mean?

5: is it possible to collapse the wave function into a new state that is not definite, but a new "truncated" one? For example, is it possible to make the deduction, say for a particle in a box, that the energy is NOT E_n, giving us a new wave function corresponding to the superposition of every state besides the one corresponding to E_n?

In principle, yes, you could construct a measurement like this, since for every pure state there is a corresponding projection operator that represents the binary observable we would describe as "is the particle in this state, or not?" The "not" result projects the state into the portion of the Hilbert space that is orthogonal to what we were testing for (in this case the state E_n), but gives no other information about the state. In terms of Rule 5, this would be an operator whose spectrum has just two values, "yes" and "no".

In practice, actually constructing such a measurement is very difficult, and has not been done for most observables. Typically there is no reason to.

6: if A is some operator corresponding to a measurement, does Af actually represent anything (where f is a wave function for the system)?

Of course. See Rules 5 and 6. (Note, once again, that "state" is a more general term than "wave function"; as noted above, the latter assumes that you are using the position representation, but there is no need to assume this.)

does it represent anything physical in our system? Its not like Af is a description of the state of our system or anything, correct?

What do Rules 5 and 6 tell you?

 

[PeroK]

Ah thanks for that, that’s definitely useful for me. However some of my questions aren’t addressed there (some aspects of 2-6 are mentioned but it doesn’t really allude to any answers, as far as I can see)

Your questions seem a bit random to me. Try to focus on one thing at a time. For example, when you initially study a particle in a box, you are looking at the aspects of the spatial wavefunction. Studying the spin of a particle comes later. Then you'll learn how to combine the spatial and spin wavefunctions.

Question 4 also seems to me a fairly pointless, random question at this stage.

Instead, you need to focus 100% on what you are learning and not let your mind wander onto unrelated questions.

What textbook are you using?

 

[okaythanksbud]

Your questions seem a bit random to me. Try to focus on one thing at a time. For example, when you initially study a particle in a box, you are looking at the aspects of the spatial wavefunction. Studying the spin of a particle comes later. Then you'll learn how to combine the spatial and spin wavefunctions.

Question 4 also seems to me a fairly pointless, random question at this stage.

Instead, you need to focus 100% on what you are learning and not let your mind wander onto unrelated questions.

What textbook are you using?

The point of these questions is to clear up confusion about concepts I’ve read about. I think I worded them poorly. For example, in question 4 I was asking if making a measurement on one quantity (such as position) could tell us anything about another (such as momentum). In the case of these two quantities I believe the answer is no but in general I wouldn’t be surprised to learn two quantities that depend on each other (for example if the measurement position^2 had some sort of physical significance this measurement would give us information about the measurement of position).

I’m currently reading griffiths, I tried starting with Landau since I liked his classical mechanics text but the QM one seems like something that would be better to come back to after learning the fundamentals

 

[PeroK]

The point of these questions is to clear up confusion about concepts I’ve read about. I think I worded them poorly. For example, in question 4 I was asking if making a measurement on one quantity (such as position) could tell us anything about another (such as momentum). In the case of these two quantities I believe the answer is no but in general I wouldn’t be surprised to learn two quantities that depend on each other (for example if the measurement position^2 had some sort of physical significance this measurement would give us information about the measurement of position).

I’m currently reading griffiths, I tried starting with Landau since I liked his classical mechanics text but the QM one seems like something that would be better to come back to after learning the fundamentals

Griffiths doesn't mention spin in the early chapters. So, there is no reason to know anything about spin at this stage.

I'd focus on the mathematics of wave mechanics,which I think you'll find a considerable challenge.

 

[vanhees71]

Well, I could have guessed the OP uses Griffith's book. Being confused is in this case for sure not the reader's fault. Better change your textbook sooner than later. My favorite for introductory QM still is Sakurai and Tuan, Modern Quantum Mechanics (Revised edition). The newer edition with Napolitano hasn't done too much damage to the book except of including a chapter on socalled "relativistic quantum mechanics", which is another topic which should not be taught anymore. The only consistent language about relativistic QT is relativistic QFT (for which I recommend Sidney Coleman's lectures onf QFT to start with).

 

[okaythanksbud]

Griffiths doesn't mention spin in the early chapters. So, there is no reason to know anything about spin at this stage.

I'd focus on the mathematics of wave mechanics,which I think you'll find a considerable challenge.

Spin was just an arbitrary example, just the first thing that come to mind when thinking about a measurement besides position/momentum

 

[PeroK]

Well, I could have guessed the OP uses Griffith's book. Being confused is in this case for sure not the reader's fault. Better change your textbook sooner than later. My favorite for introductory QM still is Sakurai and Tuan, Modern Quantum Mechanics (Revised edition). The newer edition with Napolitano hasn't done too much damage to the book except of including a chapter on socalled "relativistic quantum mechanics", which is another topic which should not be taught anymore. The only consistent language about relativistic QT is relativistic QFT (for which I recommend Sidney Coleman's lectures onf QFT to start with).

I knew that post was coming! There's nothing wrong with Griffiths as an introductory text. Students struggle with QM in any case. Sakurai is not an easy ride for a beginner

 

[vanhees71]

QM is not an easy ride, that's true, but you don't need to make it more difficult by using a confusing text, and given the frequency of students being confused by this text is "empirical evicence" enough for me to conclude that Griffiths is confusing.

 

[gentzen]

I knew that post was coming! There's nothing wrong with Griffiths as an introductory text. Students struggle with QM in any case. Sakurai is not an easy ride for a beginner

Maybe his alternative suggestions are not suited for everyone, but his base observation that Griffiths often has students confused probably holds water. In fact, I believe this is related to a trap that I personally still fall into all too often:

One trap I fell into especially when studying physics where those book-series like Landau-Lifschitz, Feynman, Thorsten Fließbach, (and many lower quality ones I forgot already or at least don't want to mention) ... The trap was that the series which helped me most during previous lectures was not necessarily the best for a lecture on a different topic. (And of course also that books not part of any series might have been a significantly superious choice to begin with, but I was not mature enough yet to understand that.)

In my case, it was the QM book by Thorsten Fließbach that kicked me out of QM for a very long time. It's not important whether it is good or bad on an absolute scale, simply the reasons why the student (i.e. me at that time) choose to use that book were inappropriate.
QM can be really impenetrable, so the time to find books that help you dig into it is really well spent. And if you select your QM book based on wrong criteria, you really risk getting seriously confused.

 

[PeroK]

Maybe his alternative suggestions are not suited for everyone, but his base observation that Griffiths often has students confused probably holds water. In fact, I believe this is related to a trap that I personally still fall into all too often:

In my case, it was the QM book by Thorsten Fließbach that kicked me out of QM for a very long time. It's not important whether it is good or bad on an absolute scale, simply the reasons why the student (i.e. me at that time) choose to use that book were inappropriate.
QM can be really impenetrable, so the time to find books that help you dig into it is really well spent. And if you select your QM book based on wrong criteria, you really risk getting seriously confused.

Eiher the OP gets his head down and studies Griffiths or buys a new textbook and studies that. Unless and until he finds Griffiths unworkable, it seems like a waste of time and money to start afresh.

 

[vanhees71]

Maybe his alternative suggestions are not suited for everyone, but his base observation that Griffiths often has students confused probably holds water. In fact, I believe this is related to a trap that I personally still fall into all too often:

In my case, it was the QM book by Thorsten Fließbach that kicked me out of QM for a very long time. It's not important whether it is good or bad on an absolute scale, simply the reasons why the student (i.e. me at that time) choose to use that book were inappropriate.
QM can be really impenetrable, so the time to find books that help you dig into it is really well spent. And if you select your QM book based on wrong criteria, you really risk getting seriously confused.

That's intersting. I usually like Fließbach's book pretty much, and I don't see, where his QM book is particularly flawed. Of course chpt. 1 is a combination of all didactical sins you can commit when introducing to the subject, but otherwise I found it pretty clearly written using the "wave-mechanics approach". Of course, not everybody likes the same books or find them good for learning a new subject.

Landau-Lifshitz vol. 3 is a pretty good intro to QM too, if you like the "wave-mechanics approach". For me the eye opener was the textbook by E. Fick, which uses the abstract Dirac formalism. Unfortunately this brillant book seems not to be translated to English.

The Feynman lectures are of course also great, but more as an additional textbook to a more "conventional" one.

 

[PeterDonis]

in question 4 I was asking if making a measurement on one quantity (such as position) could tell us anything about another (such as momentum)

When you asked question 4 in your OP you didn't say "quantity", you said "state". Now you're using the word "quantity", but since you give position and momentum as examples, it's evident that what you actually mean is "observable".

What this illustrates is that you will be much better at asking questions if you (a) learn the correct terminology, and (b) think carefully about what you are asking about before you ask it.

The answer to your properly worded question 4, whether making a measurement of one observable can tell you anything about another, is that it depends on the observables and the specific experiment.

In your example of position and momentum, these observables have a Heisenberg uncertainty relation between them, which means, roughly speaking, that the more accurately you measure one, the more uncertainty there is in the other after the measurement.

If we consider position and spin, OTOH, in general there is no relationship between these two observables. However, we can set up particular experiments that correlate them--for example, a Stern-Gerlach magnet correlates a particle's momentum and spin, so that if we later measure the momentum, that gives us information about the spin.

 

[romsofia]

4: Are states ever related--can making a measurement on one ever tell us anything about the other? Im assuming the answer must be yes--if so how can we tell? Is this maybe related to the commutator?

https://farside.ph.utexas.edu/teaching/qmech/Quantum/node41.html The last part of this section address this (but i'm sure you'll find the whole section useful).

 

[vanhees71]

When you asked question 4 in your OP you didn't say "quantity", you said "state". Now you're using the word "quantity", but since you give position and momentum as examples, it's evident that what you actually mean is "observable".

What this illustrates is that you will be much better at asking questions if you (a) learn the correct terminology, and (b) think carefully about what you are asking about before you ask it.

The answer to your properly worded question 4, whether making a measurement of one observable can tell you anything about another, is that it depends on the observables and the specific experiment.

In your example of position and momentum, these observables have a Heisenberg uncertainty relation between them, which means, roughly speaking, that the more accurately you measure one, the more uncertainty there is in the other after the measurement.

This is also a misconception, unfortunately spread rather widely in the literature. The quantum state does not tell you something about the measurement you do on a system but something on how it is prepared before the measurement. The usual Heisenberg uncertainty relation, you learn in the first few lectures in the QM 1 course describes properties of "possible preparations of a system" with respect to any pair of observables, $A$ and $B$. These observables are reprsented by self-adjoint operators $\hat{A}$ and $\hat{B}$ on the Hilbert space describing the system within quantum mechanics. Then you define the expectation value and standard deviation of the observable, given the state of the system. For a pure state you can use a normalized Hilbert-space vector $|\psi \rangle$. Then the expectation value of the observable $A$ is
$$\langle A \rangle = \langle \psi|\hat{A} \psi \rangle,$$
and the standard deviation $\Delta A$ is defined by
$$\Delta A^2 = \langle A^2 \rangle-\langle A \rangle^2.$$
Then the Heisenberg uncertainty relation for the observables $A$ and $B$ reads
$$\Delta A \Delta B \geq \frac{1}{2} |\langle \mathrm{i} [\hat{A},\hat{B}] \rangle|^2.$$
For position and momentum in the same direction you thus find
$$\Delta x \Delta p_x \geq \frac{\hbar}{2}.$$
Now you can measure either $A$ with as much precision as you like on an ensemble of systems all equally prepared in the said state and determine $\langle A \rangle$ as well as $\Delta A$. Then you can also measure $B$ with high precision and determine $\langle B \rangle$ and $\Delta B$. Then the uncertainty relation predicts something about $\Delta A$ and $\Delta B$ due to the preparation of the state. To resolve the quantum-mechanical uncertainty you must ensure that your measurement device to measure $A$ and $B$ have a much higer resolution than what's predicted by QM. Otherwise you rather determine the resolution of your apparati than to test the QM uncertainty relation.

There is, of course, also the other aspect of measurements, i.e., the fact that, if you measure a "small system", e.g., something on a single elementary particle like an electron, it's impossible to do so without significantly perturbing this system, but to describe this by theory is much more involved than using the above discussed uncertainty relation, since here you have to analyze the interaction between the measurement device and the measured system.

If we consider position and spin, OTOH, in general there is no relationship between these two observables. However, we can set up particular experiments that correlate them--for example, a Stern-Gerlach magnet correlates a particle's momentum and spin, so that if we later measure the momentum, that gives us information about the spin.

Indeed in non-relativistic QM the spin is compatible with both position and momentum. Note that this is not the case in relativistic physics!

 

[PeterDonis]

The quantum state does not tell you something about the measurement you do on a system but something on how it is prepared before the measurement.

The measurement itself can be considered a new preparation, so the restrictions on the possible states that can be prepared, which is what the Heisenberg uncertainty principle actually describes, are also restrictions on the possible states that can result from a measurement.

 

[vanhees71]

It is preparation! A usual measurement is not a preparation. That's the case only for very idealized von Neumann measurements. The entire confusion about "measurement", "observables", and states comes from this confusing language! The standard Heisenberg uncertainty relation describes constraints on the possibility to prepare a quantum state with regard to the degree of common "determinability" of two arbitrary observables.

It has nothing to do with the ability to measure the one or the other or in some sense both observables precisely, nor is it about the disturbance of the system by the measurement with regard to the measured observables under consideration. Both questions must be analyzed for the given concrete measurement apparatus, acting on the system in measuring the one or the other observable or in some sense both of them, i.e., there's no generally valid conclusion you can draw concerning these other questions, while the standard HUP is a generally valid constraint on the "preparability" of quantum systems.

 

[PeterDonis]

It is preparation! A usual measurement is not a preparation.

I was referring to the 7 Basic Rules Insights article, which has already been referenced in this thread. Rule 7 of that article specifically describes what measurements can be considered as preparations. Those are the kinds of measurements I was referring to.

 

[vanhees71]

Here's an ancient thread, discussing the important difference between the standard HUP (more precisely called the Heisenberg-Robertson uncertainty relation) about preparations of quantum systems vs. the more complicated issue of the unavoidable disturbance of quantum systems by measurement discussing a pretty easy to understand experimental paper on this subject:

https://www.physicsforums.com/threa...elation-vs-noise-disturbance-measures.664972/

Here's the updated link to the article discussed:

https://doi.org/10.1038/nphys2194

 

[PeterDonis]

I don't have serious issues with the article.

Then please do not make posts like this in a public thread:

I do not agree with everything in this Insights article, because it perpetuates these confusing imprecisions in language! We are doing physics not theological exegetics of some holy scriptures!

 

[PeterDonis]

Can we get back to physics now?

If you want to keep the discussion focused on physics, don't derail it.

 

[vanhees71]

But my point IS about the physics, and it's the key issue about this question: The standard HUP is about the "preparability" of quantum systems not about "measurability". That's all I was saying. You are the one who is disgressing, because you insist to perpetuate precisely this misunderstanding about the HUP!

 

[jedishrfu]

It seems now is a good time to close this thread. The Op has gotten some good responses and now its time to move on to other questions.

Thank you all for contributing here.

Jedi