Undergrad Is the wavefunction always an analytic signal?

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The wavefunction is not always an analytic signal; it can be any square integrable function. The imaginary part of the wavefunction is not necessarily the Hilbert transform of the real part. This distinction highlights the flexibility of wavefunctions in quantum mechanics. The relationship between the real and imaginary components is not strictly defined as a Hilbert transform. Understanding these properties is crucial for deeper insights into quantum theory.
olgerm
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Is wavefunction always analytic signal?
Is imagnigray part of wavefunction hilbert transform of real part of wavefunction?
 
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olgerm said:
Is the wavefunction always an analytic signal?
Is the imaginay part of the wavefunction a hilbert transform of the real part of the wavefunction?
No. It can be any square integrable function.
 
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Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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