What's the different between wave equation and Schrodinger's eq?

[Lh0907]

Homework Statement

As I know wave equation has d^2/dt^2,but Schrödinger's equation has only d/dt

(Time-dependent).

Why these eq has different thing(d/dt, d^2/dt^2)?

I assume if Schrödinger's equation has d^2/dt^2(not d/dt),

eigenfunction of Schrödinger's equation is not stable along with time.

And continuity equation is also not used.

But I don't know I wrong or not.

Homework Equations

Gasiorowicz 3rd edtion(chapter 2)

But this is only said analysis of probability is also changed.

That's all. (I'm not sure in English edition.My book is not an English edition)

The Attempt at a Solution

I'm not sure.

 

[mathfeel]

Homework Statement

As I know wave equation has d^2/dt^2,but Schrödinger's equation has only d/dt

(Time-dependent).

Why these eq has different thing(d/dt, d^2/dt^2)?

I assume if Schrödinger's equation has d^2/dt^2(not d/dt),

eigenfunction of Schrödinger's equation is not stable along with time.

First, it's $i\frac{\partial}{\partial t}$, not just $\frac{\partial}{\partial t}$. If you replace $\frac{\partial^2}{\partial t^2}$ in wave equation with $\frac{\partial}{\partial t}$, you get the diffusion equation. The solution decays in time.

One way you can see that Schrödinger's equation should be first power in $i\frac{\partial}{\partial t}$ is that time is conjugates to energy, so that $i\frac{\partial}{\partial t}$ is the energy operator. Just like $i\frac{\partial}{\partial x}$ is the momentum opreator.

So that Schrondinger's equation, with its first power in $i\frac{\partial}{\partial t}$, is simply saying that $E = p^2/2m$, which is just the non-relativistic definition of energy.

On the other hand, the equation $-\frac{\partial^2}{\partial t^2} \psi = -\nabla^2 \psi$ corresponds to something like $E^2 = p^2$, which is obvious not correct for a non-relativistic particle. This actually is the Klein-Gordon equation for a massless particle.

 

[DavidAlan]

The wave equation, as derived for a very long string, makes reference to the instantaneous forces between elements in the continuum. In quantum mechanics you cannot take this approach for obvious reasons.

If you differentiate the Schroedinger equation then you get the RHS involving a second-time derivative and the LHS applying the Hamiltonian (which does not change with time) to the time-evolution of the state.

The wave equation can be factored into operators quite neatly, but each term only involves a first order derivative in position.

The Schroedinger equation is very similar to the diffusion equation, however, and I believe this is by virtue of the restrictions you put on the probability amplitude. Consider the continuity equation.

Try analyzing the differences between the diffusion and wave equations; I think this is an acceptable analogy.

For instance, is the wave equation well posed for t positive? Yes. But the diffusion equation is only well posed for bounded solutions, sound familiar?
Is the wave equation well posed for t negative? Yes. The diffusion equation, however, is NOT well posed for t<0, sound familiar?

You can also ask yourself interesting questions like; "is information lost?"

 

[Lh0907]

Thank you for reply.

But how about change of probability?

This question is really I want to to get.

And how will be changed flux of probability & density of probability if Schrödinger's eq has d^2/dt^2(not i*d/dt)?

 

[DavidAlan]

Thank you for reply.

But how about change of probability?

This question is really I want to to get.

And how will be changed flux of probability & density of probability if Schrödinger's eq has d^2/dt^2(not i*d/dt)?

Compute it; it's very easy. Differentiate the probability amplitude and make substitutions using your "new" Schroedinger equation. The probability current will still be defined in the same manner.

 

[Lh0907]

Thank you David.

I've some works these days, so I made very late reply about your advice.

Thanks.

 

[dextercioby]

I think we can safely call <temporal conservation of probilities/statistics> as a regularity condition.

In other words, if a valid state vector is finite norm, hence rescaled to unit norm, we are forced to require that letting time 'pass' over this vector, its norm won't change, it will still be 1. Thus all the information encoded about the system at an arbitrary t_1 will change in a controlled manner over time.

Getting a time evolution equation of 1st order in time derivative is easily proved, if norm conservation is assumed/postulated.
An analysis is usually performed when the Dirac equation is introduced.

 

[moj20062001]

excuse me,
i have a question.
can u help me please?
i want to know the difference between the wave equation and the maxwell equations.
i mean what is the different uses of these two?
thank u.

 

[dextercioby]

Maxwell equations lead to wave equations for the electromagnetic field.