Difference between classical wave function and quantum wave function

[Casco]

We have the wave equation in classical mechanics in one dimension in the following way

$\frac{\partial^2 \psi}{\partial x^2}=c^2\frac{\partial^2 \psi}{\partial t^2}$

on the other hand we have the Schrödinger equation in quantum mechanics in one dimension in the following way

$i\hbar\frac{\partial\psi}{\partial t}=\mathbf{H}\psi$

both are called wave functions. Which are the difference between them?

 

[juzzy]

The classical wave equation describes the behaviour of waves, whereas the Schrödinger equation describes the behaviour of a quantum particle, A good descriptioon of it can be found here http://www.youtube.com/watch?v=G3NgOHwxhMg&list=UU_fHG6JygMd7oIvQ5S_cSIg&index=6&feature=plcp

 

[tom.stoer]

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").

 

[Ken G]

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 d²/dt². 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 second-order in time equation requires two initial conditions, often the wave function and its first time derivative at t=0, whereas a first-order 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 d²/dt²? 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.

 

[Casco]

So then we can ask, not why is QM a d/dt, but rather why is CM a d²/dt²?.

This reminds me a crazy thought that always has been in my mind. Why almost all, or at least a big number of differential equations in physics are of second order? It is very weird to find a differential equation of third order in physics. I believe that nature wants to tell us something whith that. For me this is even mysterious. What do you think about that?

 

[Casco]

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").

Anyone can give a different interpretation. I find this paper very illustrative:

http://www.imamu.edu.sa/Scientific_selections/abstracts/Physics/Quantum%20Theory%20Needs%20No%20Interpretation.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?

 

[lugita15]

It is very weird to find a differential equation of third order in physics. I

Yes, it is weird to find a third order differential equation in physics, and when you do see such a beast, be on the lookout for something really weird. See the Abraham-Lorentz equation.

 

[jimgraber]

@ Ken G:
You wrote: " For one thing, a second-order in time equation requires two initial conditions, often the wave function and its first time derivative at t=0, whereas a first-order 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

 

[Ken G]

This reminds me a crazy thought that always has been in my mind. Why almost all, or at least a big number of differential equations in physics are of second order? It is very weird to find a differential equation of third order in physics. I believe that nature wants to tell us something whith that. For me this is even mysterious. What do you think about that?

I think it's a curious point, for sure, but I'm not sure it comes from nature, I think it comes from us. We decide what kinds of questions we want answers to, and we are always seeking simplicity. So we start with equations that have no time dependence, but we don't get dynamics from them. So we bring in a first time derivative, and a wide array of phenomena open up to us, but still much remains hidden. So we bring in a second derivative, and a host of new phenomena become understandable to us, and those that don't we can just label "too complicated" and move on. I think we make a mistake if we think nature herself is determined by simple laws, instead I think that we lay simple laws next to nature, like templates, and amaze ourselves by how well she fits them, but they are still coming from us.

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.

 

[Ken G]

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?

The key physical property they have in common is the importance of interference. A "wave equation" describes superimposable waves, so each wave can be solved independently, but the ultimate behavior involves superimposing them, which allows them to interfere with each other. So a wave that can get somewhere in two different ways can arrive with zero amplitude, which is less than either wave would have by itself. I think that is the fundamental similarity of these "waves," along with being unlocalized. The "wave" concept is used even more generally, like with solitons in nonlinear systems, but in most cases one is talking about a linear superimposable signal when one talks about "waves."

 

[Ken G]

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?

You can tell because the pulses you are desribing are not real-valued, they are complex-valued. That means it is not just their magnitude that changes over the pulse, but also the phase. Phase information is how you encode the direction of propagation. Granted, this is something of a cheat-- we still need two pieces of information (magnitude and phase), rather than position and velocity. But all the same, there is no need to know how anything is changing with time, the initial condition can be just a snapshot (if you could "see complex"!).

 

[jimgraber]

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

 

[Ken G]

So the second pulse would be the complex conjugate of the first pulse?

Exactly.

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.

I agree with you, though I think I would have said it that the connection between those things is what is the most confusing element of quantum mechanics.