# Proof of Schrodinger equation solution persisting in time

• e4c6
In summary: So, my question still stands. How do we know that there isn't any proper example of such situation? 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...normalize. So, my question still stands. How do we know that there isn't any proper example of such situation?
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 Schrodinger 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 probablity 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.

PhDeezNutz
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?

e4c6 said:
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.

PhDeezNutz
PeroK said:
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.

PhDeezNutz
e4c6 said:
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.

PhDeezNutz

## 1. What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior and evolution of a quantum system over time. It is named after Austrian physicist Erwin Schrodinger and is used to calculate the probability of finding a particle at a certain position and time.

## 2. How is the Schrodinger equation solved?

The Schrodinger equation is solved using mathematical techniques such as separation of variables, perturbation theory, and numerical methods. The solution provides information about the state of a quantum system at any given time, including the probability of finding a particle in a particular state.

## 3. Why is it important for the Schrodinger equation solution to persist in time?

The persistence of the Schrodinger equation solution in time is important because it allows us to accurately predict the behavior and evolution of quantum systems. This is essential for understanding and manipulating the behavior of particles at the microscopic level, which has numerous applications in fields such as chemistry, physics, and technology.

## 4. How does the Schrodinger equation relate to the concept of superposition?

The Schrodinger equation is closely related to the concept of superposition, which states that a quantum system can exist in multiple states simultaneously. The solution of the Schrodinger equation provides information about the probability of finding a particle in a particular state, which can be a combination of multiple states due to superposition.

## 5. Are there any limitations to the Schrodinger equation solution persisting in time?

While the Schrodinger equation is a fundamental and powerful tool in quantum mechanics, it does have some limitations. For example, it does not take into account the effects of relativity and gravity, and it only applies to non-relativistic particles. Additionally, it cannot fully explain certain phenomena such as quantum tunneling and entanglement, which require more advanced theories.

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