First order in time=> time cannot go backwards ?

[pantheid]

First order in time=>"time cannot go backwards"?

I have had numerous professors mention, but not explain, the differences between PDEs that are second order and first order in time. For example, in the regular wave equation, they say that "time can go backwards," or something to that effect. In order to avoid this in the Schrödinger equation, it was made first order, but imaginary. Can you guys explain how reducing the order implies that time cannot flow backward (if that is indeed what my professors meant and not a misunderstanding on my part), why this was implemented into the Schrödinger equation but not the wave equation, and how this is supposed to be better than complex solutions?

Thanks in advance.

 

[bhobba]

For example, in the regular wave equation, they say that "time can go backwards," or something to that effect. In order to avoid this in the Schrödinger equation, it was made first order, but imaginary.

I have zero idea what whoever is saying this is trying to get across. It may have something to do with the derivation of the Dirac equation but I am going to assume its what you say it is, the Schrödinger equation.

But a few points:

1. It's a strange but true fact that the Schrödinger equation can be derived from the Hamilton-Jacobi equation by simply going over to complex numbers:
http://arxiv.org/abs/1204.0653

The link explains why physically - its to do with path cancellation in Feynman's Sum Over History approach.

The deep mathematical reason is you need complex numbers so you can have continuous transformations between so called pure states:
http://www.scottaaronson.com/democritus/lec9.html
http://arxiv.org/pdf/quant-ph/0101012.pdf

2. The real reason for the 'why' of the Schrödinger equation lies in symmetry, as you will find in Chapter 9 of Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

Thanks
Bill

 

[haael]

I have had numerous professors mention, but not explain, the differences between PDEs that are second order and first order in time. For example, in the regular wave equation, they say that "time can go backwards," or something to that effect.

Equations that depend only on the second power of time (or only on even powers of time in general) are symmetric with respect to time reversal.
Equations that depend only on the first power of time (or only on odd powers of time) are antisymmetric.
Equations that depend both on odd and even powers of time are not symmetric nor antisymmetric.

The problems with symmetric (second-order) equations is that they don't distinguish the time arrow direction. There's no reason for the time to flow forward. Usually there are 2 solutions to such equations, one of which describes processes going forward and one backward in time. The "backward" solution usually violates casuality, which is a bad thing. So physicists say that only one of the solutions is physical and other simply doesn't exist, but that is ugly and arbitrary.

First-order equations on the other hand behave as we expect them to, yielding only one solution that we can assign certain time direction.

There is also a thing called absorber theory that says that the backward solutions actually exist, but are accidentally canceled in our universe.

 

[The_Duck]

I have had numerous professors mention, but not explain, the differences between PDEs that are second order and first order in time. For example, in the regular wave equation, they say that "time can go backwards," or something to that effect. In order to avoid this in the Schrödinger equation, it was made first order, but imaginary. Can you guys explain how reducing the order implies that time cannot flow backward (if that is indeed what my professors meant and not a misunderstanding on my part), why this was implemented into the Schrödinger equation but not the wave equation, and how this is supposed to be better than complex solutions?

Perhaps what they are getting at is that in the wave equation, if you know the position and velocity of every point on the wave at a given time $t$, not only can you derive the shape of the wave at any future time, you can also derive the shape of the wave at any *past* time. Given what the wave looks like now, it is possible to compute what the wave looked like five minutes ago. You do this in exactly the same way you would compute the future shape, except you change a sign somewhere.

This is not possible with all differential equations. For example, for the heat equation

$\frac{d}{dt} \phi = - \nabla^2 \phi$

it is in general *not* possible to derive the past state of $\phi$ from a given present state because you run into singularities when you try to run the equation backwards.

The conclusion is *not* that any equation that is first order in time can't be run backwards. In fact, for the Schrödinger equation,

$i \frac{d}{dt} \psi = - \nabla^2 \psi$

it *is* possible to solve for the past state of $\psi$ given the present state. It is very much like the wave equation in this respect. So I'm not sure why you think the Schrödinger equation can't be run backwards in time.

 

[Jano L.]

This is not possible with all differential equations. For example, for the heat equation

$\frac{d}{dt} \phi = - \nabla^2 \phi$

it is in general *not* possible to derive the past state of $\phi$ from a given present state because you run into singularities when you try to run the equation backwards.

This sounds interesting. Can you provide some reference illustrating this?

 

[Jano L.]

The heat conduction equation is
$$
\frac{\partial T}{\partial t} = \kappa \Delta T
$$
there is partial derivative with respect to $t$ and there is no minus on the right-hand side.