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I have a problem in understanding wave functions

Let [itex]q \mapsto \phi(q)[/itex] a position wave function for a single particle of mass [itex]m[/itex]

The equivalent momentum wave function is said to be computable with using fourier transform:

[tex]\psi : p \mapsto \int_q \phi(q) \cdot e^{-i/\hbar \cdot \langle p, q\rangle} \delta^3 q[/tex]

with [itex]\langle , \rangle[/itex] the duality bracket or inner product (depending if you consider duality or not). But I feel unconfortable because technically (if you consider strictly galilean space-time) [itex]q[/itex] is not a vector but a point in an affine space. Turning it into a vector is equivalent to choosing a origin (= injecting non-physical data into my modelisation). Similarly the set of all possible [itex]p[/itex] for my particle is also a affine space. Turning it into a vector is equivalent to choosing a inertial frame (= which again corresponds to non-physical data in my modelisation).

By making explicit those two origins, I can compute:

[tex]\psi_0 : p \mapsto \int_q \phi(q) \cdot e^{-i/\hbar \cdot \langle p - p_0, q - q_0\rangle} \delta^3 q[/tex]

[tex]\psi_1 : p \mapsto \int_q \phi(q) \cdot e^{-i/\hbar \cdot \langle p - p_1, q - q_1\rangle} \delta^3 q[/tex]

And of course [itex]\psi_0 \neq \psi_1[/itex].

But by defining the equivalence relation :

[tex]\psi_A \sim \psi_B \Leftrightarrow \left( \exists p_*, \exists \vec p, \exists \vec q: \psi_A(p) = e^{-i/\hbar \cdot \langle p - p_*, \vec q\rangle} \cdot \psi_B(p + \vec p) \right)[/tex]

we have [itex]\psi_0 \sim \psi_1[/itex].

It seems that [itex]\sim[/itex]-equivalence on (position or momentum) wave functions is compatible with vector space structure and with Fourier transform (though I've not checked for the hermitian product).

So my question is: am I right to consider that two equivalent (position or momentum) wave functions are physically equivalent and that one should not consider the classical Hilbert spaces of position and momentum wave functions but rather their respective quotient spaces by relation [itex]\sim[/itex] ? If true, should I consider an even weaker equivalence relation ?

Thanks

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# Equivalence of wave functions

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