Consider a particle in a 1D-Box.

The box ranges from x=0 to x=l in the first case, and from x=-l/2 to x=l/2 in the second case.

The only difference I see is that the origin is shifted.

On solving the Time Independent Schroedinger's Equation, we get different wave functions in both the cases.

Why is it that the choice of origin affects the wave function?

Do observables too depend on the choice of origin? As surely the particle "knows" that it has a region of L length to move about.

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    $\begingroup$ Please note that it is not $\psi$ but $|\psi|^2$ that has a physical interpretation!! $\endgroup$ – Amey Joshi Oct 5 '19 at 5:09
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    $\begingroup$ Do you realize that sines and cosines are just shifted versions of each other? $\endgroup$ – G. Smith Oct 5 '19 at 5:16
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    $\begingroup$ @Hrsht you are absolutely correct. Both wavefunctions completely specify the system. Although they differ, their physical content is identical. $\endgroup$ – Amey Joshi Oct 5 '19 at 6:47
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    $\begingroup$ @Hrsht If you're getting different energy eigenvalues then you made an error somewhere. $\endgroup$ – BioPhysicist Oct 5 '19 at 11:30
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    $\begingroup$ I got different quantisation condtitions and not just simply shifted wave functions. If you don’t show any of your work, we can’t tell you what you might be doing wrong. $\endgroup$ – G. Smith Oct 5 '19 at 15:53

When plotted both cases are indistinguishable. The plots only differ by what you specify on the x-axis.

Physics is invariant under space and time translation. This property is closely related to energy and momentum conservation via Noether's theorem.

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  • $\begingroup$ Physics is invariant under space and time translation. I would consider qualifying this statement. You make it sound like momentum is conserved in every system. $\endgroup$ – BioPhysicist Oct 5 '19 at 11:46
  • $\begingroup$ @AaronStevens In every isolated system momentum is conserved. Perhaps not in general relativity if you'd like to comment on that. I consider it off topic here. $\endgroup$ – my2cts Oct 5 '19 at 12:14
  • $\begingroup$ I didn't say anything about GR, and your answer doesn't say anything about isolated systems, which is a kind of qualifier I was mentioning in my first comment. $\endgroup$ – BioPhysicist Oct 5 '19 at 12:39
  • $\begingroup$ @AaronStevens The statement "Physics is invariant under space and time translation." is self contained. Hence I had to guess after the direction of your thinking. $\endgroup$ – my2cts Oct 5 '19 at 14:40

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