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Difference between classical wave function and quantum wave function 
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#1
May2712, 05:01 PM

P: 81

We have the wave equation in classical mechanics in one dimension in the following way
[itex]\frac{\partial^2 \psi}{\partial x^2}=c^2\frac{\partial^2 \psi}{\partial t^2}[/itex] on the other hand we have the Schrodinger equation in quantum mechanics in one dimension in the following way [itex]i\hbar\frac{\partial\psi}{\partial t}=\mathbf{H}\psi[/itex] both are called wave functions. Which are the difference between them? 


#2
May2712, 05:18 PM

P: 28

The classical wave equation describes the behaviour of waves, whereas the schrodinger equation describes the behaviour of a quantum particle, A good descriptioon of it can be found here http://www.youtube.com/watch?v=G3NgO...6&feature=plcp



#3
May2812, 02:07 AM

Sci Advisor
P: 5,366

The difference between classical an quantum mechanics is not the wave function itself but the interpretation, e.g. the probabilistic interpretation "to find a particle somewhere" plus the "collapse of the wave function" which is absent in classical theories and which is not subject to the Schrödinger equation i.e. which cannot be described by the linear dynamics of the quantum world (there are other interpretations as well, but I think none of themsolves the "measurement problem").



#4
May2812, 10:37 AM

PF Gold
P: 3,080

Difference between classical wave function and quantum wave function
In addition to the probabilistic and particlelike elements of the quantum wave function, we also have the mathematical difference that the QM wave function uses a i d/dt not a d^{2}/dt^{2}. For wave functions that correspond to particles of definite energy and wave modes of definite frequency, the two work the same (if we deal with complex amplitudes appropriately, according to whichever interpretation we are using), but in the general case (particles with no definite energy, wave modes that superimpose frequencies), these are mathematically different. For one thing, a secondorder in time equation requires two initial conditions, often the wave function and its first time derivative at t=0, whereas a firstorder in time equation requires only one (often the wave function at t=0). That's a pretty important difference classically, we need to know the current state and how it is changing (analogous to position and velocity of the particles), but quantum mechanically, we need only know the current state, and how it is changing in time is prescribed automatically.
Now, since we view quantum mechanics as the more fundamental theory, we would tend to regard it as "closer to the truth" in some sense, so we would tend to think that the universe does not need to be told how it is changing, only what its state is. So then we can ask, not why is QM a d/dt, but rather why is CM a d^{2}/dt^{2}? Why did we think we had to be told the particle velocities and positions, when in fact it is inconsistent to know both at the same time? Apparently it is because our measurements were so imprecise, we could better "pin down the state" by including information about both position and momentum. More precise measurements actually mess with the system in ways that we tend to want to avoid, so we found it is better to combine imprecise measurements and we can still make good predictions in the classical limit without strongly affecting the system. 


#5
May2812, 10:19 PM

P: 81




#6
May2812, 10:27 PM

P: 81

http://www.imamu.edu.sa/Scientific_s...rpretation.pdf Going back to the main subject. Is it only the name of those functions the only thing that they have in common? Or they have some mathematical or physical properties in common? 


#7
May2812, 11:22 PM

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#8
May2912, 05:08 AM

PF Gold
P: 171

@ Ken G:
You wrote: " For one thing, a secondorder in time equation requires two initial conditions, often the wave function and its first time derivative at t=0, whereas a firstorder in time equation requires only one (often the wave function at t=0). That's a pretty important difference classically, we need to know the current state and how it is changing (analogous to position and velocity of the particles), but quantum mechanically, we need only know the current state, and how it is changing in time is prescribed automatically." This is amazing! Suppose I describe a simple symmetric pulse, perhaps a Sine Gaussian wave packet. But I describe it three times at exactly the same location. Only the first time it is moving left, the second time it is moving right and the third time it is a half and half superposition of the first two. If I have no velocity information, how can I tell the difference? TIA. Jim Graber 


#9
May2912, 11:04 AM

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P: 3,080

Then, sometimes we figure out a way to reduce a time derivative, as in going from Newton's laws to the Schroedinger equation. But it comes at the cost of a lot of other complexities, like probability amplitudes and so on. These are necessary to get agreement with experiment, and we do amazingly well it's an odd case of the equations getting more accurate when they in some sense got simpler. But we had to change the kinds of questions we want answers to (probabilities rather than definite outcomes). So I think that's what is really going on as long as we are flexible in what we want to know, we can keep the time derivatives surprisingly low in our equations. I don't know why it works at all. 


#10
May2912, 11:10 AM

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#11
May2912, 11:14 AM

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#12
May3012, 06:50 AM

PF Gold
P: 171

So the second pulse would be the complex conjugate of the first pulse?
I think the most confusing thing about QM, and the biggest difference from CM is not probabilistic versus deterministic, and not continuous versus discrete, but rather all those extra dimensions, complex and in the Hilbert space. TIA Jim Graber 


#13
May3012, 09:24 PM

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