Energy - Difference between Continuous & Discreet

[Neha Sanghvi]

Hey, can anybody please help me with this question? What is exactly the difference between continuous and discreet values? I mean, I know what the terms mean, but if we compare them with analog and digital, we can say that continuous values comprise of very very small discreet values assumed to be continuous ( maybe due to inefficient accuracy ). I am not sure on this point, so any help would be grateful.

Thanks.

 

[Gokul43201]

To get an idea of the numerical values, consider, for example, the difference between the energy levels of a piece of solid silicon.

The energy difference between bands is of the order of an electron volt (~ 1 eV); but there are a macroscopic (~10²³) number of levels within each band (that is less than an eV wide). So, these levels are separated by less than 10^-23 eV.

 

[Neha Sanghvi]

To get an idea of the numerical values, consider, for example, the difference between the energy levels of a piece of solid silicon.

The energy difference between bands is of the order of an electron volt (~ 1 eV); but there are a macroscopic (~10²³) number of levels within each band (that is less than an eV wide). So, these levels are separated by less than 10^-23 eV.

I got your point. But doesn't this support my argument of continuous values being very very small discreet values? And could you please tell me the difference between digital and analog?

Another question ( not related to this topic ) I wanted to ask was how are position and velocity independent of each other? I can explain this philosophically, but had some problems explaining it physically.

 

[Claude Bile]

There are theories that quantities we take to be continuous such as space and time are actually quantised.

http://www.physlink.com/Education/AskExperts/ae281.cfm

The difference between digital and analogue is simple. Digital signals can only occupy certain pre-determined values, while analogue signals can have any value. Digital signals are advantageous as they are very robust in terms of noise and they are readily integrated with digital processors.

For more info: http://www.answerbag.com/q_view.php/1799

Another question ( not related to this topic ) I wanted to ask was how are position and velocity independent of each other? I can explain this philosophically, but had some problems explaining it physically.

You should have problems explaining this physically since Position and velocity are not independant of one another since;

$$v=\frac{ds}{dt}$$

where v = velocity, s = displacement and t = time.

 

[Neha Sanghvi]

You should have problems explaining this physically since Position and velocity are not independant of one another since;

$$v=\frac{ds}{dt}$$

where v = velocity, s = displacement and t = time.

Hey, I know this, but if you point to the same logic for the HUP, then it doesn't work. Because if they are dependent, then you can know them simultaneously by the same formula, isn't it so? So, they will have to be independent for HUP to be true. And I stand by HUP firmly bcoz it explains many more other theories.

As for the philosophical explanation of them being independent, it is the basic fact that everything is independent of each other. It's like the natural tendency of humans to make thought out decisions and therefore, when things go against their thinking, their decisions go for probability reasoning. Oh, leave this out, I just don't want to get philosophical right now. So, please just explain the HUP question I asked you above. Thanks.

And seriously, for me, it was your idea. So, hats up to you!

 

[Neha Sanghvi]

Hey Claude,
By that idea, I meant that one explaining insufficient energy absorption by an electron ( the one you posted in the other thread ).

 

[inha]

In QM the expectation value of velocity is the derivative of the expectation value of position with respect to time.
Position and momentum operators do not commute but that doesn't remove the connection between velocity and position.

 

[Neha Sanghvi]

In QM the expectation value of velocity is the derivative of the expectation value of position with respect to time.
Position and momentum operators do not commute but that doesn't remove the connection between velocity and position.

Listen, we can determine the position of the electron accurately ( by using a laser beam, or I suppose you must have done that experiment of single slit ), so according to your statement, as the position and time is known, we can very well get the velocity. Whereas it is not so, that's what HUP states. Actually HUP does not mean ( though it states so ) the fact that you can't find the position and velocity of a particle simultaneously. It rather means that the position and velocity ( or call it momentum ) have to be independent of each other for one to be able to be indetermined even if the other is perfectly defined.
And as you would know, QP is applicable for all cases of CP. Therefore, position and velocity should be independent for CP as well.

 

[Antiphon]

Hey, can anybody please help me with this question? What is exactly the difference between continuous and discreet values? I mean, I know what the terms mean, but if we compare them with analog and digital, we can say that continuous values comprise of very very small discreet values assumed to be continuous ( maybe due to inefficient accuracy ). I am not sure on this point, so any help would be grateful.

Thanks.

Think of the difference between integers and real numbers.

Analog and digital are the wrong ideas. You start with continuous systems
most of the time. You then confine the system and it will take on a discrete
charachter.

For example, I have an infinite steel wire with a certain tension. If I pluck
it it will send a "pop" down the wire but it will not sound a musical note.

In order to sound a musical note, I have to tie down two point of the wire
and pluck it in between. Now the wire has a discrete frequency. Before,
the wire would ring however I plucked it, not at a specific frequency.

It is the confinement of continuous systems that makes them have discreet
features. This is how quantum systems become discrete.

 

[Claude Bile]

Listen, we can determine the position of the electron accurately ( by using a laser beam, or I suppose you must have done that experiment of single slit ), so according to your statement, as the position and time is known, we can very well get the velocity. Whereas it is not so, that's what HUP states. Actually HUP does not mean ( though it states so ) the fact that you can't find the position and velocity of a particle simultaneously. It rather means that the position and velocity ( or call it momentum ) have to be independent of each other for one to be able to be indetermined even if the other is perfectly defined.
And as you would know, QP is applicable for all cases of CP. Therefore, position and velocity should be independent for CP as well.

Velocity is defined as the first time derivative of displacement.

I'm not sure you are applying the HUP correctly in this case. Take the single slit diffraction experiment. There is an uncertainty in position, dx defined by the slit width which results in an uncertainty in momentum, dp which can be related to the angular spread of the diffraction pattern. I can detect the final position of the photon using a fluroescent screen and hence calculate its momentum. What I cannot do is predict, in advance, where each photon is going to land (and thus predict what each of their momenta will be), and this is what the HUP prohibits.

Claude.

 

[Neha Sanghvi]

Velocity is defined as the first time derivative of displacement.

I'm not sure you are applying the HUP correctly in this case. Take the single slit diffraction experiment. There is an uncertainty in position, dx defined by the slit width which results in an uncertainty in momentum, dp which can be related to the angular spread of the diffraction pattern. I can detect the final position of the photon using a fluroescent screen and hence calculate its momentum. What I cannot do is predict, in advance, where each photon is going to land (and thus predict what each of their momenta will be), and this is what the HUP prohibits.

Claude.

Hey, I am not actually sure abt this, but isn't this velocity derived from Newton's Laws? Coz if it is so, then you aren't considering the non-inertial frame of reference.

 

[inha]

Velocity isn't derived from anything. It's just defined as Claude stated.