Quantum Mechanics

Son Goku

Since this is a subject that raises a lot of questions and causes a lot of confusion, what better than a "Ask what you want" Sticky Thread?

The subject in general comes up quite often on boards, but because there is a lot of misformation (or vague information) around the place about the stuff I think that a thread like this would be a good place for (hopefully) decent answers.

humbert

Inner or scaler products I think I understand but outer products are causing me difficulty, eg. |x><y|
where
|x> = a1|i> + a2|j> + a3|k>
<y| = b1<i| + b2<j| + b3<k|

where i,j,k are an orthonormal basis, I get that it's like a column vector by a row vector and forms a matrix but exactly how to multiply them I don't know.
if you're given the component form is there an easier way to evaluate it?

Son Goku

Basically if you have:

|x> = a1|i> + a2|j> + a3|k>
<y| = b1<i| + b2<j| + b3<k|

Then |x><y| will take the form:
(Writing it as a matrix)

a1.b1 a1.b2 a1.b3
a2.b1 a2.b2 a2.b3
a3.b1 a3.b2 a3.b3

humbert

thanks! that's what I thought but wasn't sure if perhaps the non-diagonal entries were zero as a result of the orthogonal basis vectors. Is there an alternative way of expressing the outer product in terms of the components?

Son Goku

If this is what you want:

Outer-Product of two vectors a and b is:

Sum over i and j of: (a_i) (b_j)*

The star is there because you have to take the complex conjugate of b when turning it into a row vector.

By the way do you have a link to your course description?
I'm just wondering in case Shankar is "overkill" for the course.

humbert

There's a question on a previous exam paper I'm looking at that asks to show that the eigenvalues of a unitary matrix are of the form exp(ia) where a is just a real number of some sort and I can figure out how to do it?

Son Goku

Sorry only noticed this there now.....

Let U be a unitary matrix and let |u_i> and |u_j> be two different eigenvectors of U. i and j label which eigenvector we are talking about.
Also let's label the adjoint of U as A.

The definition of a Unitary Matrix is AU=I where I is the identity matrix.

U|u_i> = (u_i)|u_i>
<u_j|A = (u_j)*<u_j|

Where (u_i) and (u_j) are the respective eigenvalues of |u_i> and |u_j> and * denotes the complex conjugate.

Now take the expression:

<u_j|AU|u_i> = (u_i)(u_j)*<u_j|u_i> (from above)

However since AU = I we can rewrite this as:

<u_j|I|u_i> = (u_i)(u_j)*<u_j|u_i>
and since I does nothing (it's the identity):
<u_j|u_i> = (u_i)(u_j)*<u_j|u_i>

Rearranging:

{1-(u_i)(u_j)*}<u_j|u_i> = 0

When j=i, <u_j|u_i> = 1 and we get:

{1-(u_i)(u_i)*} = 0

(u_i)(u_i)* = 1

|(u_i)| = 1.

So all the eigenvalues of U are complex numbers of modulus 1 (or length 1) and therefore can be written as:
exp(ia)

humbert

Cool, thanks, I read that proof in the book and understood it. It's the leap from knowing it's a complex number of modulus one to proving it could be written as exp(ia). Probably maths I should have known. Actually when I think about it it's really just Euler's formula. Thanks.

Son Goku

humbert wrote:

Cool, thanks, I read that proof in the book and understood it. It's the leap from knowing it's a complex number of modulus one to proving it could be written as exp(ia). Probably maths I should have known. Actually when I think about it it's really just Euler's formula. Thanks.

Yeah, I even forgot that a complex number of modulus 1 can be written as exp(ia) needed to be proved, but of course it has to be proved. It is really just Euler's Formula. There is another way to prove it which is more rigorous, but takes ages.

Any other questions just feel free to ask.

Leidenfrost

ah euler's formula, such a startingly beautiful symmetry.

what is it again euler's identity comes from his formula right.
the e^i(pie) +1=0 thing


oh and son Goku, what do you do if you don't mind my asking?

humbert

Hi, I'm quite stuck on the addition of angular momentum. I have an example of two particles with,
j1 = 1, j2 = 1/2 giving j = 1/2 or 3/2 and m from -3/2 to 3/2,

ψ(3/2,3/2) = φ(1,1)φ(1/2,1/2)

ψ(3/2,-3/2) = φ(1,-1)φ(1/2,-1/2)

then using ladder operators to get

sqrt(3)ψ(3/2,1/2) = sqrt(2)φ(1,0)φ(1/2,1/2) + φ(1,1)φ(1/2,-1/2)
I can see that the operator acts much like the product rule for differentiation but I don't know why, down to
ψ(3/2,-1/2)

Where I'm completely baffled is when it says to express ψ(1/2,1/2) as a linear sum

ψ(1/2,1/2) = aφ(1,1)φ(1/2,-1/2) + bφ(1,0)φ(1/2,1/2) where a and b are constants

it uses the raising operator to work out what a and b are

J+ψ(1/2,1/2) = 0 = aφ(1,1)φ(1/2,1/2) + b*sqrt(2)*φ(1,1)φ(1/2,1/2)

giving a = sqrt(2/3) and b = -1/3 but I don't follow how.

Any help would be greatly appreciated!

Son Goku

Try this: It was quicker than trying to type it here.

humbert

Sorry, got caught up in exams and forgot about this. that makes sense, thanks for all the help.

bogwalrus

So I'm just wondering. Does quantum mechanics prove that matter is the creation of the mind and everything is just space of potential energy? By prove i mean that it can't explain the processes through experiment anymore thus a dead end... (sorry if i am using the word space incorrectly).

Iv'e been reading up on quantum theory and some of the findings are amazing. Though quantum mechanics is being analysed on a completely experimental/physical basis it seems that physical experiment has come to a dead end and something else has to take over to explain certain phenomena such as that double slit experiment and the idea that nothing is really there until we observe.

This is by far the most interesting thing i have ever come across and i feel it will finally tie the soul with the mind and thus discover the way to true happiness for everyone. (sorry if i sound too hippy-ish there but its true, we live to be happy)

Professor_Fink

bogwalrus wrote:

So I'm just wondering. Does quantum mechanics prove that matter is the creation of the mind and everything is just space of potential energy?

No! It does nothing of the sort. A number of crackpots try to use quantum mechanics for a mechanism for telepathy, homeopathic medicine, faith healing, ghosts, and just about every other bit of nonsense out there. But as I've said, these people are crackpots, not scientists.

bogwalrus wrote:

Iv'e been reading up on quantum theory and some of the findings are amazing. Though quantum mechanics is being analysed on a completely experimental/physical basis it seems that physical experiment has come to a dead end and something else has to take over to explain certain phenomena such as that double slit experiment and the idea that nothing is really there until we observe.

Maybe you've misunderstood the book, or maybe the book is wrong, but either way 'the idea that nothing is really there until we observe' is not part of quantum mechanics. Basically the wave function, which describes the state of a particle, can place it in more than one location at a time, in what's known as a superposition.

When we make a measurement we will detect it to be one position. The reason for this is that position isn't a good description of the state of the particle. Just like if someone asks you whether purple is red or blue. It's neither, but if you have to answer one or the other, your choice will probably be random since purple is a mixture of red and blue.

The only thing that is a bit different in quantum mechanics is that our measurement rseults need to be consistent. So back to the colour example. Once you say red, the colour you have becomes red until you add some more blue to it. If you measure the position of a particle, then the particle will be in that position until it is spread out by some potential.

Virtually no serious scientists (there is one notable exception) believe that this has anything to do with humans or their brain. We do have a completely consistent picture of how this works with out ever having to take an anthropocentric view of things.

bogwalrus wrote:

This is by far the most interesting thing i have ever come across and i feel it will finally tie the soul with the mind and thus discover the way to true happiness for everyone. (sorry if i sound too hippy-ish there but its true, we live to be happy)

I'm afraid you may have been reading books by crackpots. I know it can be hard to tell, if your not already familiar with quantum mechanics. You might want to look at 'In search of Schroedinger's cat' by John Gribbin, as it is a pretty good pop-science account of quantum mechanics.

bogwalrus

I must admit i am quite impressionable and have probably been reading some crack-pot books=) I understand that i would need to learn a lot more about quantum mechanics before i can make any real deductions. I liked your example with the colours, made me think of superposition in a different way. But i still am curious about one or two things;

The particles that get measured in quantum mechanics are what exactly? Protons and neutrons or something smaller?

And the fact that these particles don't have any measurable position how does this effect the elements or matter it will form? "Is a gold bar not a gold bar when its smallest bits of matter are in a state of superposition"? And how come the observation by "us" concludes the position it will collapse into?

I know i'm out of my league here, i'll definitely read that book you mentioned and maybe get back to you in a few years;)

Professor_Fink

bogwalrus wrote:

The particles that get measured in quantum mechanics are what exactly? Protons and neutrons or something smaller?

The smaller the particle the more obvious quantum effects are. That said, everything is governed by quantum mechanics. But imagine an elephant in the room with you. It'd be virtually impossible not to know where it is (measuring it inadvertantly). A nucleon or electron can go much longer with out being effected by the environment. This process is generally known as decoherence.

bogwalrus wrote:

And the fact that these particles don't have any measurable position how does this effect the elements or matter it will form? "Is a gold bar not a gold bar when its smallest bits of matter are in a state of superposition"?

Generally the particles making up a gold bar are in a mass eigenstate, meaning that while they may not have a distinct position, the type of particle present is determined. So gold doesn't change into lead spontaneously.

It is possible for a superposition to exist of two different types of particle, provided they obey certain conservation rules, but due to decoherence, these tend to collapse very very quickly.

bogwalrus wrote:

And how come the observation by "us" concludes the position it will collapse into?

It's not us. Whenever a particle interacts strongly with something in the environment (maybe us or maybe a Hydrogen nucleus) it is prevented from undergoing destructive interference. This is because the two states of the first particle become entangled states of the two particles, and no longer describe only one particle. If we have access two only the first particle, then it's state is completely indistinguishable from if it had been measured.

bogwalrus wrote:

I know i'm out of my league here, i'll definitely read that book you mentioned and maybe get back to you in a few years;)

Feel free to ask any questions you have.

bogwalrus

Thanks=)

Dooom

Been pondering on this a while and haven't been able to find a (more or less) plain honest answer, or else i just wasn't looking hard enough.

If someone observes a particle, that person bumps the particle and disturbs it, or forces it to take just one of it's states, all from just observing it? And going on from there, how and why is it disturbed or forced to take a state just from looking at the thing?
I know the why bit might be overshooting a bit, but if anyone wanted to inform me on how i'd be a happier person. Cheers.

Professor_Fink

Dooom wrote: »

If someone observes a particle, that person bumps the particle and disturbs it, or forces it to take just one of it's states, all from just observing it? And going on from there, how and why is it disturbed or forced to take a state just from looking at the thing?

To understand this, you first need to be clear what a quantum superposition is. Any quantum state is described completely by it's state vector (or equivalently it's wave-function). By this, I mean that all of the dynamic properties (position, speed, momentum, energy, etc.) are encoded in this vector. The state vector is the weighted sum of a set of orthogonal 'basis' vectors, each of which is orthogonal (at right angles) to each other. The weight of each vector in the sum is called it's amplitude. Squaring the amplitude gives the probability of measuring the system to be in this state. It is impossible to deterministically distinguish to quantum states which are not orthogonal.

I haven't yet said what the basis vectors are, and that's because we are free to choose. They could be positions, in which case the state vector would be written as a weighted sum of positions (i.e. |S>= a |0cm>+b|1cm>+...). Notice that there is no momentum in the state vector. Equally we could choose momentum as our basis, in which case we would be writing the state vector as a sum of momenta, with no position terms.

The important thing to note here is that a particular value of momentum corresponds to a superposition of positions, and a position corresponds to a superposition of momenta. This is where the uncertainty principle comes from, as it is impossible to be in a state which is simultaneously in a particular position and has a fixed momentum.

When we measure a particle, the result will be one basis vector, rather than a superposition. Thus when we measure something, we lose all the terms in the original superposition which are not consistent with the measurement. This is how measurement affects the state. It's not immediately obvious that you've affected the state, since the measurement out come must have been part of the superposition anyway, but it has a big affect when you then try to measure another property of the superposition, since you have already collapsed it.

Hope this helps.

Son Goku

Dooom wrote: »

Been pondering on this a while and haven't been able to find a (more or less) plain honest answer, or else i just wasn't looking hard enough.

If someone observes a particle, that person bumps the particle and disturbs it, or forces it to take just one of it's states, all from just observing it? And going on from there, how and why is it disturbed or forced to take a state just from looking at the thing?
I know the why bit might be overshooting a bit, but if anyone wanted to inform me on how i'd be a happier person. Cheers.

It is as of yet unknown what causes a particle to select one state in particular or in other words to lose the rest of the superposition. There are many different opinions on this, but quite a lot of people think it is an effect called decoherence, but it is unknown if that can explain it completely as of this point in time.

Professor_Fink

Son Goku wrote: »

It is as of yet unknown what causes a particle to select one state in particular or in other words to lose the rest of the superposition. There are many different opinions on this, but quite a lot of people think it is an effect called decoherence, but it is unknown if that can explain it completely as of this point in time.

I can't say I completely agree with this. When you measure the state of a particle, your measurement aparatus becomes entangled with the particle. Unless this process is completely reversed (which is virtually impossible for most devices) the elements of the original superposition can no longer interfere. This is why you get one state and loose the rest. This does decohere the particle since the phases become spread over more than one particle. Hence if you neglect the second system you are left with a mixed state rather than a pure state in a coherent superposition.

Decoherence is a rather general term which describes any process in which the coherence of a superposition is eroded. Decoherence is caused by an interaction with other quantum particles in the environment, be this wanted, as in the case of measurement, or unwanted as in the case of noise in quantum systems.

The real argument is over why we percieve one particular state as the outcome, and on whether the other components of the superposition really exist. Why aren't we aware of superpositions directly? Why is there a prefered basis? And so on.

Son Goku

Professor_Fink wrote: »

I can't say I completely agree with this.

I'm not advancing a particular opinion, all I'm saying is that there doesn't seem to be a general consensus across physics, although there is a general consensus within given areas of physics.

Unless I'm wrong I don't think there is a definitive answer as to why in a energy measurement for example you get one particular value and why this value corresponds to classical quantities and you have no contact with the the rest of the superposition that appears in the formalism.

For instance for a contrasting view of things in Algebraic QFT people take the view that you get a definitive eigenvalue because eigenvalues are properties of events and a measurement is an event. Bohr thought QM "knew" about classical mechanics in some vague sense. Doesn't make much sense to me, but it just shows the radically different opinions people have on this. Even people who think it is all one big unitary operation are divided into Many-Worlders and the other camp (can't think of their name).

For instance the "Heisenberg cut" criticism of this:

Unless this process is completely reversed (which is virtually impossible for most devices) the elements of the original superposition can no longer interfere. This is why you get one state and loose the rest.

would be, how does the apparatus know which state to entangle with and why is it a classical state? Why doesn't it entangle with a broad selection of states?

Maybe I'm being naive, I've just gotten so many conflicting ideas from people and books. Not only is there no agreement of the answer to the problem, there is no agreement on what is the problem. There is a lot of people who still think entanglement with the apparutus doesn't uniquely answer why the rest of the superposition is lost.

(N.B. Not that I disagree with, it's just one area in which I'm wary of sounding certain.)

Professor_Fink

Son Goku wrote: »

I'm not advancing a particular opinion, all I'm saying is that there doesn't seem to be a general consensus across physics, although there is a general consensus within given areas of physics.

Certainly the question of what constitutes a measurement and the ontological status of other parts of the superposition is something of a philosophical question, and one which is very prominant in philosophy of physics. But that's not really what I'm getting at here. We understand exactly why measurement prevents us from getting inteference with other elements of the superposition.

Certainly the very first atom in your TEM tip is quantum mechanical, or the photon you use to observe a particle, etc. Once this interacts with the particle, you produce an entangled state of the two particles. Tracing out the particle used in the measurement device yields a mixed state, and so there is loss of coherence. Thus, unless you can actually undo this interaction, you cannot get interference effects from the original particle.

This is true even in the Copenhagen interpretation, once we consider a real apparatus. Why we get one value rather than another is really the measurement problem, not why we get eigenstates of the measurement operator.

Son Goku wrote: »

Unless I'm wrong I don't think there is a definitive answer as to why in a energy measurement for example you get one particular value and why this value corresponds to classical quantities and you have no contact with the the rest of the superposition that appears in the formalism.

See above.

Son Goku wrote: »

For instance for a contrasting view of things in Algebraic QFT people take the view that you get a definitive eigenvalue because eigenvalues are properties of events and a measurement is an event. Bohr thought QM "knew" about classical mechanics in some vague sense. Doesn't make much sense to me, but it just shows the radically different opinions people have on this. Even people who think it is all one big unitary operation are divided into Many-Worlders and the other camp (can't think of their name).

Again, these are views largely on why we get certain results, and are somewhat linked to human perception, rather than why we don't get interference.

Son Goku wrote: »

For instance the "Heisenberg cut" criticism of this:
would be, how does the apparatus know which state to entangle with and why is it a classical state? Why doesn't it entangle with a broad selection of states?

I'm not sure how an open quantum systems approach has a heisenberg cut criticism, but I'll try to answer your questions. The apparatus becomes entangled with the quantum system you are measuring in a way specificied by the interaction. For an Ising interaction, this will be by Z basis states picking up different phases, where as for an exchange interaction it will be a little more complicated, perhaps implimenting a controlled-NOT type interaction.

The states are only classical in that they are eigenstates of the measurement operator. They can quite easily be superpositions of positions, or momenta, or spin or what ever other property you are interested in. For example measuring photon polarization at different angles is measuring some particular superposition of horizontally and vertically (or clockwise and anti-clockwise) polarizations.

Son Goku wrote: »

Maybe I'm being naive, I've just gotten so many conflicting ideas from people and books. Not only is there no agreement of the answer to the problem, there is no agreement on what is the problem. There is a lot of people who still think entanglement with the apparutus doesn't uniquely answer why the rest of the superposition is lost.

Well, i suppose it depends what you've been reading. There is certainly a mystique surrounding measurement in quantum mechanics, and many people who express an opinion are often uninformed about the details of open quantum systems, for example. Pop sci books are often particularly bad at this.

The fact is, that if we are allowed to consider even one particle in the measurement apparatus quantum mechanically, we can fully explain why you don't get interference. We can also mimic this using single particles as the measurement apparatus (and actually do for quantum error correction).

Son Goku

Professor_Fink wrote: »

Well, i suppose it depends what you've been reading.

Most of the stuff I've read on the measurement problem are scientific texts, but they date from 1940 - 1985. Maybe people know more now and my understanding is out of date. You have to remember you work in an area that really closely links with this stuff, where as my area doesn't so I might be saying naive things. However I'd never read Pop-science.

For instance:

This is true even in the Copenhagen interpretation, once we consider a real apparatus. Why we get one value rather than another is really the measurement problem, not why we get eigenstates of the measurement operator.

Bohr explicitly states both of these are a problem. The conflicting opinions I've heard are from physicists. For instance have you ever heard what Schwinger thought or Omnes? Their opinions are really confusing.

the ontological status of other parts of the superposition

For instance I thought Cooper experiments and other such things in condensed matter showed that the superposition exists.

Perhaps I should be more explicit:

This is true even in the Copenhagen interpretation, once we consider a real apparatus. Why we get one value rather than another is really the measurement problem, not why we get eigenstates of the measurement operator.

I understand this, but a priori, if you get what I mean why are the eigenvalues classical quantities? This might sound vague, so let me be more concrete. Bohr and more recently Haag have pointed out that we write superpositions in terms of superposed "classical" states. For instance take a state |P> = a|E1> + b|E2>, where |E1> and |E2> are two states of definite energy. However at the quantum level there is just |P>, which could be expanded in any way. How is it that an energy measuring device can find the E-basis?
Basically why is it that if you want to do QM, you need to "already" know CM in a a priori sense? For instance I can understand your argument only in so far as we know classical measurment apparatus respond to a basis appropriate to them.

If possible could you explain what goes on the following experiment:
Helium-3 before it undergoes gamma decay is in the state
|P> = a|Decayed> + b|Undecayed>. Now a decayed Helium-3 reflects a certain frequency of light that I'll call U-light, where as an undecayed helium-3 doesn't interact with U-light. In the Xeno experiment you shine U-light at the atom with a detector behind it. There is a screen far away which detects U-light. If the screen detects U-light then you know the atom is undecayed as it hasn't interacted with the light. These means the state is now projected on to |Undecayed>. If you keep emitting the light fast enough the state never has a chance to get back to the |P> = a|Decayed> + b|Undecayed> state as you are continuously projecting it on to the |Undecayed> state.

Now here is the crucial thing, you are preventing the atom from decaying by having light not interact with it. I don't understand this at all, perhaps you could explain what is causing the loss of the superposition here.

Professor_Fink

I'll try to come back to the other points later, as I'm tight on time, but let me just try to give a quick answer to this question:

Son Goku wrote: »

I understand this, but a priori, if you get what I mean why are the eigenvalues classical quantities? This might sound vague, so let me be more concrete. Bohr and more recently Haag have pointed out that we write superpositions in terms of superposed "classical" states. For instance take a state |P> = a|E1> + b|E2>, where |E1> and |E2> are two states of definite energy. However at the quantum level there is just |P>, which could be expanded in any way. How is it that an energy measuring device can find the E-basis?
Basically why is it that if you want to do QM, you need to "already" know CM in a a priori sense? For instance I can understand your argument only in so far as we know classical measurment apparatus respond to a basis appropriate to them.

You get E1 or E2 when you measure the energy of the system, because you use the Hamiltonian for that system as the measurement operator. The Hamiltonian has specific eigenvectors with corresponding eigenvalues. If the system has a degenerate energy manifold (for example as is the case with the ground state of a ferro-magnet), then you simply project onto that vectorspace, but don't collapse superpositions within it. So you get E1 or E2 because they are inherantly encoded in the Hamiltonian of the system. Choosing a perturbed Hamiltonian, the new eigenstates could be |E1>+|E2> and |E1> - |E2>.

Professor_Fink

Son Goku wrote: »

Most of the stuff I've read on the measurement problem are scientific texts, but they date from 1940 - 1985. Maybe people know more now and my understanding is out of date. You have to remember you work in an area that really closely links with this stuff, where as my area doesn't so I might be saying naive things. However I'd never read Pop-science.

Yeah, I guessed you weren't big on popular science. The fact is though that even a lot of physicists aren't really up on the relevant quantum mechanics.

For instance:

Son Goku wrote: »

For instance I thought Cooper experiments and other such things in condensed matter showed that the superposition exists.

Oh, superpositions do indeed exist. What I meant was that when you do something that you have no feasable way of reversing, you can really argue all you want about how real the other parts of the wave function are, or whether something may happen afterwards which collapses the wave function.

Son Goku wrote: »

If possible could you explain what goes on the following experiment:
Helium-3 before it undergoes gamma decay is in the state
|P> = a|Decayed> + b|Undecayed>. Now a decayed Helium-3 reflects a certain frequency of light that I'll call U-light, where as an undecayed helium-3 doesn't interact with U-light. In the Xeno experiment you shine U-light at the atom with a detector behind it. There is a screen far away which detects U-light. If the screen detects U-light then you know the atom is undecayed as it hasn't interacted with the light. These means the state is now projected on to |Undecayed>. If you keep emitting the light fast enough the state never has a chance to get back to the |P> = a|Decayed> + b|Undecayed> state as you are continuously projecting it on to the |Undecayed> state.

Now here is the crucial thing, you are preventing the atom from decaying by having light not interact with it. I don't understand this at all, perhaps you could explain what is causing the loss of the superposition here.

I'll do my best.

The particle starts off in the state |P> = a|Decayed> + b|Undecayed>. I'll assume that you are using a single photon source and that it is either perfectly reflected or transmitted depending on the state of the particle. This isn't strictly necessary, but might make things a bit clearer than using coherent states.

So the first photon interacts with the atom and we get a state a|Decayed,+> + b|Undecayed,->. Here I'm using a + to indicate a forward moving photon and - to indicate a reflected photon. If we trace out the photons, the density matrix for the atom is |a|^2|Decayed><Decayed| + |b|^2|Undecayed><Undecayed|, so clearly the coherence has gone. Assuming no photon loss, if we fail to be measure a photon hitting the back screen, then we must have the state |Undecayed>.

After a small time interval, the state of the atom is then |Undecayed> + eps|Decayed> (up to some normalization). The photon scattering process is repeated, and again tracing out the photon we get |Undecayed><Undecayed| + |eps|^2|Decayed><Decayed| (again up to normalization).

Notice that the probability of getting a decayed atom is proportional to |eps|^2. The evolution of the system is exp(-i k*X*t), where k is some constant and X swaps between decayed and undecayed. (Actually this isn't really true, since the decay requires the emission of a particle which makes it irreversible, but this is a pretty good toy model for how the Xeno effect works). So you get cos(kt)|Undecayed> + sin(kt)|Decayed>. For small kt, then, we have eps proportional to t.

So in a unit time we have a probability of the atom decaying of \sum_{i=0}^{1/tau} (k tau)^2, if we take tau as the time between incident photons. This probability can be made arbitrarily small by decreasing tau. Thus the system can be prevented from decaying.

Hope this helps.

Son Goku

Professor_Fink wrote: »

Both Posts

You see I understand this as it is just basic QM. I can easily see and work out what you've written above. What I'm trying to ask or understand is why is it the eigenvectors physically? What is it about classical measurement that selects these states? What is the mechanism?

I'll try and ask my main query as swiftly as possible using a specific phrasing from the 1940s. I've always the same problem when I raise this question because what I'm asking sounds trivial unless I explain myself correctly. Hopefully you'll understand what I'm asking, but I'll phrase it in four different ways:
What is the mechanism of the modern correspondance principle, á la Lif****z?

Why is the language of QM always simply the language of classical mechanics with an L^2 measure on top?

Why is there no formulation of QM that doesn't refer implicitly to CM?

Or what occurs during the R process/projection, do you think it is one U process in disguise?

Sorry for this, it's just I have great difficulty articulating this problem.

Son Goku

Actually you're alright, I think I might need to go away and have a think about this, as everytime I try and say it it sounds like I'm asking basic QM questions. I'll try and get my thoughts together, although reading Schwinger's measurement algebra treatment of QM has helped.

On a complete tangent have you ever read Sydney Coleman's Aspects of Symmetry? I just finished it today. A fantastic book. Or Weinberg's Three Volume collection on Quantum Field Theory?

Professor_Fink

Don't worry, I know these aren't really trivial questions. I think perhaps you are just looking at quantum mechanics from different perspective, so perhaps my responses aren't terribly enlightening.

I'm wondering if perhaps I'm not understanding what you mean by a classical state. I presume you mean an eigen state of an operator which corresponds to an observable for which there is a classical equivalent. So for position, a classical state might is one which is a delta f uction in space. Is this correct?

If so, I think perhaps the confusion is steming from the fact that you can make any quantum state into a 'classical state' by choosing an appropriate operator for measurement. There is a whole formalism for describing quantum systems which has grown out of this idea, known as the stabilizer formalism, which leads to some pretty cool results about what systems we can simulate efficiently on classical computers.

I will hold off on answering your question, until I know what you mean, as I don't want to add to the confusion.

bogwalrus

Does the idea of infinity come into calculations in QM? whats smaller then the Quantum particle?? What i mean is that you guys are talking about is measuring a particle that is in superposition and making calculations on the whole of this particle......but what state is the smallest part of that particle in and so on and on.....

I think what i am asking is when you measure the atoms in front of you and keep going down until you reach the quantum way of measuring who is to say that there is some other form of measurement to deal with the quantum quantum particles.....or do such things exists?

great unwashed

oh no, someone's going to mention 'quarks'

I wonder about the phenomenology of QM sometimes - how we know this stuff or how it appears to us. I'm just about to start Stephen Weinbergs book about the discovery of subatomic particles again and the early experiments to measure the mass of the electron and shooting beta particles through gold foil to find out about the subatomic world are nostalgic now but still very important too.

I have a few basic questions.

Can any of you give me the best description of charge you've ever heard without mentioning quarks?

Is there an explanation for the exclusion priniciple - why these numbers of electrons and not others?

When did physicists start talking about particle spin and what use does it have?

Professor_Fink

bogwalrus wrote: »

Does the idea of infinity come into calculations in QM? whats smaller then the Quantum particle?? What i mean is that you guys are talking about is measuring a particle that is in superposition and making calculations on the whole of this particle......but what state is the smallest part of that particle in and so on and on.....

Well, basically in as far as we know, the standard model (quarks, electrons, muons, taons and their neutrinos, and exchange particles etc. are as far down as you can go. In reality, though, you really need quantum field theory to describe things accurately, and in that even the number of particles isn't fixed.

bogwalrus wrote: »

I think what i am asking is when you measure the atoms in front of you and keep going down until you reach the quantum way of measuring who is to say that there is some other form of measurement to deal with the quantum quantum particles.....or do such things exists?

I have no idea what a quantum quantum particle would be, since I don't see what adding the extra quantum- prefix could mean. Basically all particles obey quantum mechanics. Even big things, like footballs. The reason we don't have an intuition for quantum mechanics is that because at a macroscopic scale, the probability of having superpositions which do not interact with the environment is exponentially supressed, and so never happens. At smaller scales superpositions can survive longer without interacting with the rest of the environment, and so quantum effects are more noticable.

We don't have a theory which supercedes quantum theory, and we have no reason to believe that one exists. Even string theory is just a quantum field theory, albeit one in which the particles are extnded objects

Professor_Fink

great unwashed wrote: »

Can any of you give me the best description of charge you've ever heard without mentioning quarks?

Charge is not intrinsically related to quarks. Quarks just happen to have a charge. Some leptons also have a charge, for example the electron.

Charge is simply a way of quantifying how particles interact with photons, of quantifying the strength of the electric field surrounding a particle.

great unwashed wrote: »

Is there an explanation for the exclusion priniciple - why these numbers of electrons and not others?

Yes, and it's actually very simple (although somewhat mathematical).

We call particles which have an anti-symmetric wave function Fermions. Particles with symmetric wave functions are called Bosons. Bare in mind that both these particles are identicle.

So, for fermions, a two particle state is
1/sqrt(2) {|psi_1> |psi_2> - |psi_2> |psi_1>}
while for a boson it is
1/sqrt(2) {|psi_1> |psi_2> + |psi_2> |psi_1>}
where |psi_1> and |psi_2> are the two states, one of which is occupied by each particle, and the position indicates which particle is in that state.

So, if the two particles are in the same state, then |psi_1>=|psi_2> and so the Fermions are in the state 0. This is not a valid state, since that probability of measuring the particle to be in any state is 0 (probabilities always add up to 1), and so it is impossible for this to ever happen. Hence the Pauli exclusion principle.

great unwashed wrote: »

When did physicists start talking about particle spin and what use does it have?

Particle spin is just the quantum equivalent of angular momentum of the free particle. Each particle is like a spinning top, and the rate at which is turns is it's spin. This is what gives rise to the magnetic field around charged particles with non-zero spin.

It was first proposed in 1925, and so has been around almost as long as quantum mechanics.

bogwalrus

When you say in Quantum Mechanics "the observer is the observed" what does this mean? It is a thing David Bohm said on something i read and i don't know what he means by it but sounds quite interesting. Something about there is no separation from the observing apparatus and the observed????

Professor_Fink

bogwalrus wrote: »

When you say in Quantum Mechanics "the observer is the observed" what does this mean?

I'm afraid I don't know what Bohm was refering to. He has a rather unconventional approach to measurement in quantum mechanics, so I'm not sure if he was simply elluding to a specific interpretation, or was trying to capture something about quantum mechanics in general. It's a nice sound bite, but not very informative.

Kareir

Hey,

I'm sort of an interested amateur in physics, i'll be doing it next year, though. I've read various books, but i was wondering if someone could explain the basic points of Quantum Mechanics to me. I get relativity pretty well, but I've managed to let pieces of QM be forgotten, somehow.

Thanks a lot,
_Kar

Professor_Fink

Kareir wrote: »

I've read various books, but i was wondering if someone could explain the basic points of Quantum Mechanics to me.

Not really, it is a rather broad subject. If you have specific questions I would be happy to answer, but I can't really see anyone teaching you how to do quantum mechanics in full via the forum.

dinky earnshaw

I was thinking i might find some help understanding some of the stuff Ive read in brysons a short history of nearly everything. After reading the previous posts I think the question should be Do Sesame street have a programme on qm im totally lost.

waitinforatrain

I also watched a video about the double slit experiment a while back and have some questions:

1. The "observer". From what I understand, the photon begins to behave like a particle when it is being measured. Therefore the "observer" changes the thing observed. But does this observer need to be a conscious entity? Does the photon counter count as an observer? Does the photon counter have a discrete state and value before I (the experimenter) observes it?

If I've totally misconceived and these questions are unanswerable then let me know and I'll go back to the drawing board :pac:

2. How difficult would it be to replicate the double slit experiment for the average joe? This is something I've found surprisingly difficult to find out.

3. How can we be sure that only one photon is firing at a time?

Azelfafage

waitinforatrain wrote: »

Therefore the "observer" changes the thing observed. But does this observer need to be a conscious entity?

Naw.

The Tyrannosarus Rex existed long before the human race exited.

T-Rex did not invent the things he was eating.

.

Professor_Fink

waitinforatrain wrote: »

1. The "observer". From what I understand, the photon begins to behave like a particle when it is being measured. Therefore the "observer" changes the thing observed. But does this observer need to be a conscious entity? Does the photon counter count as an observer? Does the photon counter have a discrete state and value before I (the experimenter) observes it?

No, it's just an effect of open systems (systems which interact with a broader system). If a system is closed, the evolution of the wave function is unitary, but when it is allowed to interact with other systems it causes it to appear as if the wave function collapses.

So the observer is simply another system. It doesn't need to be an animal or person, simply a few electrons will do.

waitinforatrain wrote: »

2. How difficult would it be to replicate the double slit experiment for the average joe? This is something I've found surprisingly difficult to find out.

Well, you can do the Young's slit experiment fairly easily, but doing so with single photons is much harder.

waitinforatrain wrote: »

3. How can we be sure that only one photon is firing at a time?

Usually an attenuated laser is used, but single photons are detected. Another way to do it is to use a single photon source. There are plenty of candidates which work. As it happens, though, we can do far better than simply using photons. The double slit experiment has been done with neutrons, atoms and even molecules.

waitinforatrain

Professor_Fink wrote: »

No, it's just an effect of open systems (systems which interact with a broader system). If a system is closed, the evolution of the wave function is unitary, but when it is allowed to interact with other systems it causes it to appear as if the wave function collapses.

So the observer is simply another system. It doesn't need to be an animal or person, simply a few electrons will do.

Ah thanks, that really clarifies it for me.

A few more quick questions for you if you get the time: What do you mean by 'unitary' (I googled and all I got was "Having a magnitude of one" which doesn't help me to understand it). Are you just saying that the wave function behaves as a wave, but when another system depends on it, it converges to a discrete value.

Is this 'wave' function vibrating in physical space, or is it a representation of the probability of the particle being at a certain point when observed?

You say "causes it to appear", does it not actually collapse or is it just relative to the system that is observing/interacting with it? Could two different systems measure it as collapsing to a different point?

Do you have any book recommendations for this stuff (hopefully that avoids complex maths)?

Professor_Fink

waitinforatrain wrote: »

Ah thanks, that really clarifies it for me.

A few more quick questions for you if you get the time: What do you mean by 'unitary' (I googled and all I got was "Having a magnitude of one" which doesn't help me to understand it). Are you just saying that the wave function behaves as a wave, but when another system depends on it, it converges to a discrete value.

Is this 'wave' function vibrating in physical space, or is it a representation of the probability of the particle being at a certain point when observed?

You say "causes it to appear", does it not actually collapse or is it just relative to the system that is observing/interacting with it? Could two different systems measure it as collapsing to a different point?

It's a little more tricky. Basically quantum systems in pure states are systems in a state such that we can write the state as a superposition of classical states. By a superposition I mean that each classical state psi_i is waited by some complex amplitude A_i, such that |A_i|^2 is the probability of measuring the system in that state, so the 'wave function' which is just a list of these amplitudes has nothing to do with physical vibrations in general. Unitary operations are operations which preserve this structure.

When a quantum system interacts with the environment it undergoes a non-unitary operation. The effect of this is to bring the system into a mixed state. A mixed state is simply a classical probability distribution of quantum states. When the interaction is strong enough, the system will loose the complex phase of A_i (called dephasing), essentially making the system classical.

waitinforatrain wrote: »

Do you have any book recommendations for this stuff (hopefully that avoids complex maths)?

The Feynman Lectures, perhaps.

Azelfafage

Cats Cannot Speak.
Be they living or dead.

So ............Alter the Experiment.

Use a human victim...without harming the human.

Humans can talk.

This experiment is Less lethal to a living being but just as valid:

Put an Irishman in a Box-room with a bottle of whiskey.

Is he drunk or sober before you open the door?

That shows you the sheer absurdity of Schrodinger's "experiment".

.

Professor_Fink

Azelfafage wrote: »

Put an Irishman in a Box-room with a bottle of whiskey.

Is he drunk or sober before you open the door?

That shows you the sheer absurdity of Schrodinger's "experiment".

Look, Azelfafage, at this stage you're comments are degenerating into complete nonsense (and frankly offensive nonsense). If you want to go start denying the existence of quantum mechanics, fine, but if so you may want to stop using pretty much every modern piece of technology, including your computer.

Azelfafage

Schrodinger was merely making an analogy about Quantum Physics.

He was astounded that anybody believed about the cat.

A cat has too many atoms to be relevent to Quantum Physics.

I jokingly reproduced the experiment,replacing the cat with a consious being with a bottle of whiskey.

Schrodinger was joking about the cat.

I was joking about the Whiskey.

.

Professor_Fink

Azelfafage wrote: »

Schrodinger was merely making an analogy about Quantum Physics.

He was astounded that anybody believed about the cat.

A cat has too many atoms to be relevent to Quantum Physics.

That's not correct. The number of atoms has nothing to do with whether quantum effects will be visible or not. Decohrence (and hence the transition to classical physics) is entirely determined by interaction with an environment.

onedoubleo

I recently attended a talk given by as part of a series in from Markus Arndt in the University of Vienna who mentioned that they had managed to get a bucky ball to show interference and he also mentioned that there was evidence of quantum mechanics in processes like photosynthesis.
So I have two questions, is there anywhere else in nature we are seeing quantum effects like tunneling play a major role? And other than verification of theory is there any other reason to get large scale object to show wave phenomenon?

Professor_Fink

onedoubleo wrote: »

So I have two questions, is there anywhere else in nature we are seeing quantum effects like tunneling play a major role? And other than verification of theory is there any other reason to get large scale object to show wave phenomenon?

Q1) Photosynthesis, your sense of smell and bird navigation are all believed to exploit quantum effects.

Q2) To build a quantum computer.

onedoubleo

Sense of smell I can kinda get but how Bird Navigation?