Okay, just got to the hotel.

Before I go delving into those lectures and other thing you suggested looking up, can I ask a few questions?

Feynman's lectures are easy reading. The man was a master at explaining complex things in lay terms.

First, is intensity defined implicitly or is there is some other relationship this comes from?

Inensity is defiend as the square of the amplitude.

What exactly is the "potential energy distribution"? Is it the potential energy at a given point x and something q?

Yes. Think of a string on a guitar. You pluck the string and it begins to oscillate with a standing wave. when the string is bowed out to its maximum displacement it stops reverses direction and swings back past the rest positiion until reaches a maximum dispacement on the other side of the rest position and so on. As it pases though the rest position its total kinetic energy is at a maximum, and when its at rest (maximum displacement) its total potential energy. At any given point along the string the potential energy is given by:

D*F where D is the displacement and F is the restoring force.

The real energy of the system would be Ke + Pe, but I would have had to include time as they are out of phase with each other wtih respect to time.

In retrospect, kinetic energy may have been easier to understand.

Which brings me to my second question... what is q? Is q charge? Is it some form of energy (because you said it needs to be quantized)?

q is the mode. Integral values for q give frequenciles that satisfy the equation condition;

When x=0 and x=L, Y(x,q)=0


q=1 there is a half wave on the string

q=2 there is a full wave on the string

q=3 there is 1.5 waves, etc..

Like the guitar string tied at both ends, it can only oscillate a specific frequencies. These are commonly called harmonics.

And yes, it could certainly be viewed as being related to the totaly energy in the system

And lastly, and this is a slightly personal question, but could you elaborate on that last thing you said: why did you stop teaching?

I had a hard time explaining things to my students in way that made it easy for them to understand. Seriously.

 

I wasn't aware that potential energy had an 'amplitude'.

Sure, it can. In this case different points along the string contribute differently to the total potential energy.

Apart from anything else we are back to the observation that potential energy is strictly non-negative whereas the sine function displayed has positive and negative regions.

Yeah, sorry for that. The best I could come up with in the few minutes I had in the airport lounge. I tried to remove the axis, but couldnt firgure out how to do it. I also wasted a lot of time trying to post the document and keep the formatting. This is what you get when you try to simplify things too much and too quickly. Anyway, the idea is there, but in my usual style, not well explained..

 

I wasn't aware that potential energy had an 'amplitude'.

If the potential energy of a system is a periodic function then it has an amplitude (i.e. if the potential energy of a system is a sine function that sine function has a mangitude).

Apart from anything else we are back to the observation that potential energy is strictly non-negative whereas the sine function displayed has positive and negative regions.

Isn't potential energy a vector? Negative and positive only imply direction in that case... Like, if you had a diagram of a planet, and objects orbiting around them, you would draw arrows indicating in what direction each of their potential energies was pointing, no?

Blazed,
How did you get on with my explanation/justification?

All of what you said made perfect sense. I very much appreciate you explaining it to me. But, in my particular case, these were all observations/conclusions I came to a long time ago, before I knew squat about Quantum Mechanics. In class, we were given the equation and constant values and told to determine the probability that a particle was in a given space (this was in high school, the place where teachers try to avoid explaining things to you at any means possible, though I'm obviously exaggerating a bit). I think it was an extra credit assignment, but it was many years ago. I remember asking similarly why is this true, and explaining most of what you told me to the teacher, who nodded her head. But it never went anywhere from there...

I didn't even know it had anything to do with quantum mechanics. Well, that kind of sounded like I was gloating, but I didn't mean it in that way. Again, thank you for the explanation.

Feynman's lectures are easy reading. The man was a master at explaining complex things in lay terms.

Yeah... there are A LOT of lectures/videos on various topics. I will begin watching/reading the ones I can find for free, for now. Thank you for informing me of them.

Yes. Think of a string on a guitar. You pluck the string and it begins to oscillate with a standing wave. when the string is bowed out to its maximum displacement it stops reverses direction and swings back past the rest positiion until reaches a maximum dispacement on the other side of the rest position and so on. As it pases though the rest position its total kinetic energy is at a maximum, and when its at rest (maximum displacement) its total potential energy. At any given point along the string the potential energy is given by:

D*F where D is the displacement and F is the restoring force.

The real energy of the system would be Ke + Pe, but I would have had to include time as they are out of phase with each other wtih respect to time.

In retrospect, kinetic energy may have been easier to understand.

Luckily, I have at least a decent enough understanding to grasp what you've explained. Thank you.

q is the mode. Integral values for q give frequenciles that satisfy the equation condition;

When x=0 and x=L, Y(x,q)=0


q=1 there is a half wave on the string

q=2 there is a full wave on the string

q=3 there is 1.5 waves, etc..

Like the guitar string tied at both ends, it can only oscillate a specific frequencies. These are commonly called harmonics.

And yes, it could certainly be viewed as being related to the totaly energy in the system

Ok, I know what is the mode of a string now that I've brushed up a bit on it. It's been a while since I've dealt with them. Thank you.

I had a hard time explaining things to my students in way that made it easy for them to understand. Seriously.

Thanks for sharing. I sort of want to teach myself... at least one day... so I was curious.

Something is still... missing to me though. I like things to click, mathematically speaking, in my head.

I have another question, though I think based on my reading here and there it's got to be true and based on everything you've been explaining to me up till this point:

This fact, that the integral from 0 to L of the magnitude of the wave function squared = 1 is the pobability of finding a particle on that wave, is true for all waves, NOT just schrodinger's wave equation? Since that is the case, things are starting to make a lot more sense.

Everything does, except the squaring of the modulus. I understand that the probability must be a positive number, hence the modulus, but why square it? I believe the answer lies in the statistical method normalization. Squaring a function and setting it equal to 1... normalizes it? It sounds familiar (I've taken classes on statistics before), but I'll have to ponder this for a bit.

Thank you both of you for your help. I do believe I'm making progress... hopefully.

-blazed

 

Well howdy Montreal,
Bill you surely must have better fish to fry there than AAC, but it's nice to hear all the same.

It is all to easy to mix up the physics and the maths, we should remember that procedures may have mathematical reality and meaning but no counterpart in the physical world.
It is well to recap the physical properties involved in these equations.

Schrodingers Equation deals with energy. Its units are Joules, ergs or whatever.

The Wave Equation deals with displacement and time. Its units are metres, fathoms or whatever and seconds.

I started with a particle-centric view of Schrodingers Equation, for this it is necessary to dispense with time and imagine a standing wave from the wave-centric point of view.

The probability interpretation is appropriate to a particle-centric view, not a wave centric view. Indeed, by symmetry, when the wave function is integrated over ±∞ the result is precisely zero, since the + and - parts cancel. This implies that the probability of finding a single particle anywhere in an infinite universe is zero.

It is much easier to develop the pendulum analogy to explain the link to probability.

Unfortunately I also suffer from inability to explain complicated ideas simply, but I do try.

I did once try with the nature of time and its difference from spatial dimensions but that fell on deaf ears.
Perhaps you would like to review post #11 of this thread

http://forum.allaboutcircuits.com/showthread.php?t=10758

Have fun

 

Yes. I can see how they misunderstood your statements. At this point my understanding of space and time leads me to believe that the only real difference between the dimensions is that we can move back and forth through space, but through time we must always travel in one direction. We can speed up and slow down our travel through time, but we are ALWAYS moving through space-time at the speed of light. Besides that, there really is no difference between space and time. We are constantly traveling through space-time (usually through both).

-blazed

 

This fact, that the integral from 0 to L of the magnitude of the wave function squared = 1 is the pobability of finding a particle on that wave, is true for all waves, NOT just schrodinger's wave equation? Since that is the case, things are starting to make a lot more sense.

When the amplitude of the wave function is squared you get a finite real function that in never negative.. Since this is a real particle, and the probability of finding it must equal 100% a normalization constant is calculated to multiply the squared function with in order to force the total integation area beneath to equal 1.. Same could be done with any function if you needed to do so to match it up with the reality of a physical system..

 

I do, but not until 7.

I see what you're saying in that other thread. However you still have all 4 dimensions. Unless the block extends infinitely into at least two other dimensions, you should be able to go around it. No?

For instance, if I’m walking down the street (along X) and I come to a wall in my way, which extends partially into Y and partially into Z. So it is there in T (same time as me) and partially there in X, Y, and Z. I can still just get over it, or walk around it. All it would take is to displace myself into one of the other dimensions until I get around the blockage.

 

I do, but not until 7.

If you're short of something to do, have you read Roger Penrose - The Road to Reality?

With regard to the other. Yes you are correct, you can still side step in another spatial dimension - move, displace yourself, change your x, y or z coordinate or whatever.

That's the whole point, you can do this because you have three spatial dimensions available.

However, no matter how fancy your footwork is in those three, you can't change your t coordinate by one millisecond.
You would require a second time dimension to achieve this.

Did you get my second point about the cross section? If you managed to find a second time dimension to move past the obstruction in, you would still need to move the whole 'extent' in that dimension, not just a cross section.

 

Blazed and Triggernum

I think you should have another look at Bill's equation in post 18.

He observes that the (wave) function that satisfies the differential equation (remember that the solutions to diff equations are functions not just numbers) must have a zero at 0 and L.

He does not say, and it is not true, that the function is zero outside this range.

Indeed this property is the whole basis of the quantum theory and the reason transistors work at all.

There are other points on the axis where the wave function is non zero, therefore (assuming the link to probability) there are other points where there is a possibility of finding the particle, however small.

The logical implication of this is the integration has to be performed between the limits of - ∞ and +∞

These other non zero points are the basis of my original post as well.

Incidentally my understanding of a phonon is that it is a P (longitudinal) wave. Bill's model used an S (transverse) wave. Once again there is a distinction between the Maths and the Physics.

 

With regard to the other. Yes you are correct, you can still side step in another spatial dimension - move, displace yourself, change your x, y or z coordinate or whatever.

That's the whole point, you can do this because you have three spatial dimensions available.

However, no matter how fancy your footwork is in those three, you can't change your t coordinate by one millisecond.
You would require a second time dimension to achieve this.

Did you get my second point about the cross section? If you managed to find a second time dimension to move past the obstruction in, you would still need to move the whole 'extent' in that dimension, not just a cross section.

Now that makes a lot of sense!

-blazed

Edit:

Blazed and Triggernum

I think you should have another look at Bill's equation in post 18.

He observes that the (wave) function that satisfies the differential equation (remember that the solutions to diff equations are functions not just numbers) must have a zero at 0 and L.

He does not say, and it is not true, that the function is zero outside this range.

Indeed this property is the whole basis of the quantum theory and the reason transistors work at all.

There are other points on the axis where the wave function is non zero, therefore (assuming the link to probability) there are other points where there is a possibility of finding the particle, however small.

The logical implication of this is the integration has to be performed between the limits of - ∞ and +∞

These other non zero points are the basis of my original post as well.

Incidentally my understanding of a phonon is that it is a P (longitudinal) wave. Bill's model used an S (transverse) wave. Once again there is a distinction between the Maths and the Physics.

Yes, I see. I was trying to point out to myself mostly that the entire idea of why that is in fact the probability is not something only relevant in the context of quantum mechanics. By making it applicable to everything, things seem to make a lot more sense. For some reason or other, while I hoped I was past all that nonsense, the subject of quantum mechanics can be intimidating...

-blazed

 

the subject of quantum mechanics can be intimidating

Take heart, perhaps you would like to know that Einstein worked out the Physics of relativity, long before the Mathematics. He worked out (his great insight) , by logical physical reasoning, an explanation to a physically observed phenomenon - the Michelson - Morley experiment.

On a lesser plane

No form of energy is a vector under any definition of the word. Negative energy does not exist.

It is shown in high school physics courses that the energy in a wave is proportional to the square of its amplitude.
So if we regard the solution to the wave or Schrodinger's equation as a plot of energy (density) alonmg the axis, it is perhaps reasonable to associate high probability with high energy density. However this does regard the a particle as a quantum wave smeared out along the axis.

 

Take heart, perhaps you would like to know that Einstein worked out the Physics of relativity, long before the Mathematics. He worked out (his great insight) , by logical physical reasoning, an explanation to a physically observed phenomenon - the Michelson - Morley experiment.

On a lesser plane

No form of energy is a vector under any definition of the word. Negative energy does not exist.

It is shown in high school physics courses that the energy in a wave is proportional to the square of its amplitude.
So if we regard the solution to the wave or Schrodinger's equation as a plot of energy (density) alonmg the axis, it is perhaps reasonable to associate high probability with high energy density. However this does regard the a particle as a quantum wave smeared out along the axis.

Not to mention Einstein also made mistakes along the way (the aether)...

-blazed

 

Everything does, except the squaring of the modulus. I understand that the probability must be a positive number, hence the modulus, but why square it? I believe the answer lies in the statistical method normalization. Squaring a function and setting it equal to 1... normalizes it? It sounds familiar (I've taken classes on statistics before), but I'll have to ponder this for a bit.

The modulus operator is for complex systems where there is a real part and an imaginary part. Quite often the solutions to non-linear systems of differential equations are complex functions. The wave DE and the Schrodinger DE are almost identical and both can produce complex wave equations as their solution.

So before you square a complex wave function to find its intensity you must take its modulus to isolate the real component, otherwise you end up with negative intensity! Ugly, real ugly.

Did you get a chance to Google the Copenhagen Interpretation? It’s really simple to understand. Basically a bunch of physicist of notable fame (most of the modern greats were there) got together in Copenhagen to decide on the universally accepted interpretation of the quantum wave equation. It was decided at that conference to use the square of the modulus of the wave function as a predictor of the probability of the event that is the subject of the wave function.

Back in the day (I’m talking when personal computers were Commodore PETs and such) we scoured the literature on probability theory and could find nothing that related continuous, three dimensional complex wave functions to probability in this way. As far as I can tell the acceptance of this comes from the understanding that intensity is related to proximity. The higher the intensity the closer you are likely to be.

 

Incidentally my understanding of a phonon is that it is a P (longitudinal) wave. Bill's model used an S (transverse) wave. Once again there is a distinction between the Maths and the Physics.

Yeah, another simplification, again sorry. In a real system like this vibratiing string, there is also a longitudinal wave related to the tension on the string, so the one-dimensional phonon exists here. Remember though, my postulate was that the observer did not know anything about the system beyond the equation. That was an attempt to make this like a QM discussion.

However, if my recollection serves me, and it may not as I have been drinking for more than 2 hours, in 4 dimensions phonons are allowed to take on 'packet' or particle like attributes and would necessarily have transverse components.

 

Did you get my second point about the cross section? If you managed to find a second time dimension to move past the obstruction in, you would still need to move the whole 'extent' in that dimension, not just a cross section.

I thought I did. I'm just not sure why the side step needs to be into another 'time' dimension.

Any obstruction needs to coexist in all 4 dimensions in order to be 'real' in our universe. Take a mountain. It has extent in x, t, y, z. But it can be bypassed. Assuming I can't drill through it, I can certainly walk around it, over it or wait for the wind and rain to reduce it to a mere bump.

Maybe you could help by describing an obstruction in real space-time that would require another (5th) dimension, whether it is a 'time' or a 'space' dimension, to circumvent?

 

In the middle of a working day here. Just installed a new computer system for a client, so I'll be quick 'testing' the system.

I'm just not sure why the side step needs to be into another 'time' dimension.

The point is not to bypass the obstruction. The point is that there is an obstruction.
Remember the other thread was about time travel and its implications.
I was discussing what would be necessary for time travel to happen and what might be the practical implications.

In mathematics we have four axes, ostensibly identical, with numbers along their grids.

In Physics one of these is different.

Travel is defined as assigning new coordinates at will to any of these axes.

We can do this for all four axes in Maths but only three in Physics.

As you say we could mathematically overcome obstructions by side-stepping in any of these axes, but we can't side step in the fourth physically.

 

To expand a bit further: we can going through a mountain by moving far enough left of it so that moving in the original direction no longer crosses into the mountain. Then, when we're far enough past it, we can move back right to align with our old path. We "sidestep" the mountain.

But that only works because we actually HAVE another space (or two) dimension to "sidestep" into. But there is only ONE time dimension. We can't avoid an obstacle by sidestepping through time, because there's no where to go besides forwards or backwards.

That makes a lot of sense. But what makes time so special we can't move freely through it? We are always moving in the same direction. Granted, we can speed up and slow down our movement through it, but we can never change our direction.

Or do you think this is a result of time being only one dimension again somehow?

-blazed

 

We can sort of move through time in an impeded fashion.. c limits the rate at which we can move through space, and the rate at which we move through space determines how we move through time.. The situation we are in is analogous to a space-craft that can approach c stuck at/within the event horizon of a black hole.. No matter how hard it guns its engine, it won't be able to progress forward, merely slow the rate at which it is pulled backwards, just like we can't exceed c to go backwards in time like the notion of tachyons implies..
Time dilation though could be regarded as forward time travel wrt the traveller's initial frame, and its accomplished via energy just like spatial movement..

 

Okay, I could set up a conference call to discuss this, but really my problem is envisioning anything that could obstruct time travel (mathematically first) that could not be circumvented unless it had infinite extent in at least 3 of the dimensions. In this case t and 2 spatial dimensions.

Now I know the ‘real’ nature of t and that we cannot arbitrarily assign values to it as we can the spatial dimensions, but the whole idea of time travel is to find a way to do just that.

Besides, limited time travel has been demonstrated within the accessible Minowski cone, but its always into the future and only at a slower than normal rate.

However, neutrinos are known to travel into the past, relativisticly speaking, by somehow traversing space faster than photons.

So let me know if you want to do that call.

 

I don't know if my phone system will support a conference call, but I would be pleased to have a chat. I can call any standard landline number in North America on my package at no charge, if you would like to pm a number and a time.
I have had some interesting chats this way in other forums.