Proof of Schrodinger equation solution persisting in time

[e4c6]

Homework Statement

Proof of Schrodinger equation solution persisting in time

Relevant Equations

Schrodinger equation

I've started reading Introduction to Quantum Mechanics by Griffiths and I encountered this proof that once normalized the solution of Schrödinger equation will always be normalized in future:

And I am not 100% convinced to this proof. In 1.26 he states that $\Psi^{*} \frac{\partial \Psi}{\partial x} - \Psi \frac{\partial \Psi^{*}}{\partial x}$ must go to 0 at infinity as $\Psi$ must vanish in infinity(I've also found very similar reasoning in some youtube video but nothing more precise). However $\Psi$ vanishing in the infinity doesn't imply that its derivative also goes to 0. For example consider:

$$\Psi = \frac{sin(x^3)}{x} - \frac{i}{x}$$

Then:
$$\Psi^{*} \Psi = \left(\frac{sin(x^3)}{x}\right)^2 + \frac{1}{x^2}$$

So integral of probability is finite and then:

$$\Psi^{*} \frac{\partial \Psi}{\partial x} - \Psi \frac{\partial \Psi^{*}}{\partial x} = 6icos(x^3)$$

Which doesn't equal 0. I've only written x - dependent part of the wave function but we can add some constants and time dependent part to get a proper(I think) solution which is a counterexample. What's wrong with my reasoning?

 

[PeroK]

The allowable wavefunctions that repesent a physical system do not include the sort of counterexample you have found. Somewhere else in the book Griffiths says something like "any good maths student could find a counterexample".

The results in QM like this one depend on the wavefunction and all its derivatives vanishing at infinity. Whereas, a function like $\sin(x^3)$ has unbounded derivatives as $x \rightarrow \pm \infty$. This makes it physically impossible for a localised system.

 

[e4c6]

Thank you for your answer. Up to now where I am Griffiths haven't mentioned that. I understand that wavefunction must vanish at infinity as otherwise we couldn't get probability equal to 1. However why there is also condition of vanishing derivative? At a first glance I don't see any reason why it can't be a physical solution. How do we know that there isn't any proper example of such situation?

 

[PeroK]

Thank you for your answer. Up to now where I am Griffiths haven't mentioned that. I understand that wavefunction must vanish at infinity as otherwise we couldn't get probability equal to 1. However why there is also condition of vanishing derivative? At a first glance I don't see any reason why it can't be a physical solution. How do we know that there isn't any proper example of such situation?

It's not just QM. If you have a localised system (gravitational, electromagentic or whatever), then the influence of that system must vanish if you go far enough from that system. Not just in terms of the strength of the field, but in terms of the perturbations.

If that's not the case and you imagine a universe where effects may be significant out to "infinity", then probably you need a different class of physical theories than the ones we have.

In this case, if the wavefunction does not remain normalised, then in some sense the amount of "particle" or system is increasing over time. If that's the case, then QM is not the right theory.

I would turn your question round and say that if you want to include pathological mathematical functions in your physics, then you are not studying the physics we assume to be true in our universe. And, I would suggest you look for experimental evidence of the physical phenomona represented by such functions.

PS and what we really want to, in fact, is not what Griffiths has done, but make the assumption that the wavefunction stays normalised and use that to constrain the class of wavefunctions that we allow.

 

[e4c6]

It's not just QM. If you have a localised system (gravitational, electromagentic or whatever), then the influence of that system must vanish if you go far enough from that system. Not just in terms of the strength of the field, but in terms of the perturbations.

If that's not the case and you imagine a universe where effects may be significant out to "infinity", then probably you need a different class of physical theories than the ones we have.

In this case, if the wavefunction does not remain normalised, then in some sense the amount of "particle" or system is increasing over time. If that's the case, then QM is not the right theory.

I would turn your question round and say that if you want to include pathological mathematical functions in your physics, then you are not studying the physics we assume to be true in our universe. And, I would suggest you look for experimental evidence of the physical phenomona represented by such functions.

PS and what we really want to, in fact, is not what Griffiths has done, but make the assumption that the wavefunction stays normalised and use that to constrain the class of wavefunctions that we allow.

Sorry I didn't know that such a wavefunction is pathological. As I said I'm just beginner and I didn't know that this function doesn't make physical sense. Later in Griffiths there is an example with complex potential and there the overall probability is decreasing over time. I was just curious what could happen here.

 

[PeroK]

Sorry I didn't know that such a wavefunction is pathological. As I said I'm just beginner and I didn't know that this function doesn't make physical sense. Later in Griffiths there is an example with complex potential and there the overall probability is decreasing over time. I was just curious what could happen here.

It was a good question - and it was good to find a counterexample. I think Griffiths ought to say something about the class of wavefunctions being more constrained than just being square integrable. You can assume that all derivatives tend to zero as well. And, in general, faster than any power of $x^n$.

Some textbooks do try to establish the class of allowable wavefunctions more precisely. There was a thread on here a while ago about it, I think. It might be hard to find now, though.