Is dψ/dx zero when x is infinite in QM?

[dextercioby]

It is correct, but it is a non-physical one because it's not square integrable, not to mention that it's not in the self-adjointness domain of the Hamiltonian.

 

[houlahound]

Interesting, the first solution of SWE every physics student must learn is not even physical.

 

[bhobba]

Oh dear.

To the OP put such issues to one side at the moment, its full solution requires what are called Rigged Hilbert Spaces a very advanced area.

To start with read the first few chapters of Ballentine which will give you an overview. Beyond that its Phd Level:
https://arxiv.org/abs/quant-ph/0502053

I spent far too much time sorting this out - wait until you really understsand the physics then I can give you some reccomendations. Its certainly interesting, beautifull and elegant, but more of a side issue.

Thanks
Bill

 

[bhobba]

Forgive my ignorance but the wave function for a free particle is non zero for x-> infinity...and so is its derivative.

Or not?

It is. ts not physically relizeable and part of the RHS formalism of QM. Its done for mathematical convenience, but its full elucidation is a very advanced area of functional analysis.

Thanks
Bill

 

[vanhees71]

While it's again beyond any doubt that the unitary evolution applies in principle to any wavefunction, because any unitary operator is bounded, Stone's theorem which advocates the existence of a self-adjoint Hamiltonian will not render this Hamiltonian necessarily bounded, thus as per Hellinger-Toeplitz theorem, the Hamiltonian's domain will be smaller than the domain of its exponential. One more time: e^(itH) is unitary, if H is self-adjoint. I cannot find any self-adjoint Hamiltonian which has that wavefunction in its domain. Unitary time evolution in Quantum Mechanics makes sense only for wavefunctions in the self-adjointness domain of the Hamiltonian.

You can unitarily extend the operator $\exp(\mathrm{i} t \hat{H})$ beyond the domain of $\hat{H}$. So also the somewhat funny wave function discussed above has a unitary time-evolution given that function as initial condition.

 

[DrDu]

In principle, you can shift any wavefunction to anywhere, also to infinity So if $\psi'(x)|_{x=0}\neq 0$ then $\lim_{a \to \infty} ( d/dx \exp(ipa) \psi(x)|_{x=a}) \neq 0$.

 

[Nugatory]

Interesting, the first solution of SWE every physics student must learn is not even physical.

It's the infinite plane wave solution for a particle that is not localized and never has been, has equal probability of being found anywhere in an infinite and completely featureless universe. So yes, it's well and thoroughly unphysical... But then again, so are the ideal point masses with zero size and infinite density that are the first examples you'll see of Newton's law of gravity.

We teach the infinite plane wave solutions first for two reasons:
1) Like the unphysical point masses in high school treaments of gravity, they're mathematically easy to work with. Schrödinger's equation with $V(x)=0$ is one of the simpler differential equations around.
2) Although they are unphysical, you need them to build the superpositions that describe physically realizable states.

 

[Vanadium 50]

And 3) For many problems they are a good approximation.

Physics, as it is practiced, is about the search for ever-better approximations and not about the search for Absolute Truth.

 

[dextercioby]

Nonetheless, the absolute truth is most frequently sought here, on Physicsforums: https://www.physicsforums.com/threads/the-typical-and-the-exceptional-in-physics.885480/. But what do we all practice, then? I am in accounting/financial services. :)

 

[vanhees71]

It's the infinite plane wave solution for a particle that is not localized and never has been, has equal probability of being found anywhere in an infinite and completely featureless universe. So yes, it's well and thoroughly unphysical... But then again, so are the ideal point masses with zero size and infinite density that are the first examples you'll see of Newton's law of gravity.

We teach the infinite plane wave solutions first for two reasons:
1) Like the unphysical point masses in high school treaments of gravity, they're mathematically easy to work with. Schrödinger's equation with $V(x)=0$ is one of the simpler differential equations around.
2) Although they are unphysical, you need them to build the superpositions that describe physically realizable states.

As in all of physics, classical field theory and quantum theory, the plane-wave solutions are generalized functions (distributions in the sense of functional analysis). They are of eminent importance not as physical solutions but for building such physical solutions in terms of Fourier series or Fourier integrals. In quantum theory the Fourier transformation of the position wave function is the momentum wave function and vice versa. In quantum field theory it's the mode decomposition of the free-field solutions with respect to the momentum-spin/helicity eigenbasis in terms of annihilation and creation operators and Fock states constructed from the corresponding single-particle basis as (anti-)symmetrized products of defined particle number(s).

 

[houlahound]

^^ Plane waves are like building blocks for theories??

 

[vanhees71]

They are an important way to build solutions of linear partial differential equations. BTW Fourier developed the theory of the corresponding series named after him when investigating the heat equation.

 

[houlahound]

Praise be to Euler.

That would make a good T-shirt, bumper sticker or coffee mug slogan. You heard it here first folks.

 

[Mentz114]

^^ Plane waves are like building blocks for theories??

You might find rigged Hilbert spaces relevant. Here's a quote from a paper,

As well, in order to gain further insight into the physical meaning of bras and kets, we shall present the analogy between classical plane waves and the bras and kets.

The paper is here http://arxiv.org/abs/quant-ph/0502053v1