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

[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. Schrodinger'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. Schrodinger'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