What is the physical significance of having +/- a constant multiplying

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The discussion centers on the physical significance of the normalization constant A in the wave function psi = A*e^(-r/a), where A = +/- sqrt(2/a). This constant indicates that the wave function can represent two distinct states, corresponding to the positive and negative values of A. The multiplication of the wave function by a phase factor of the form e^{iθ} does not alter the physical properties, as demonstrated by the probability density remaining unchanged. Specifically, using θ = π results in a phase shift that effectively changes the sign of the wave function without impacting the overall probability density.

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After normalising a wave equation, let's say psi = A*e^(-r/a), and finding that A = +/- sqrt(2/a), what does this mean in terms of the physical significance of the wave?

Would the wave just be a superposition of two waves with the two different A values?
 
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You can multiply any wave function by a constant factor of the form ##e^{i\theta}## without any physical effect. For example, the probability density gets multiplied by ##e^{i\theta}e^{-i\theta} = 1## which of course has no effect.

As a special case of the above, if you let ##\theta = \pi##, the constant factor becomes ##e^{-i\pi} = -1##.
 

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