Undergrad Fourier Transform of the Wave function

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The discussion centers on the relationship between the wave function in momentum space and its conjugate in relation to the Fourier transform. It questions whether the conjugate of the wave function in momentum space is equivalent to the conjugate of the entire transformation integral. A key point raised is the assertion that the Fourier transform of the conjugate wave function equals the conjugate of the Fourier transform itself. The equation F(ψ*) = F(ψ)* is proposed for proof or disproof. This highlights the mathematical properties of Fourier transforms in quantum mechanics.
Haynes Kwon
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Given that the wave function represented in momentum space is a Fourier transform of the wave function in configuration space, is the conjugate of the wave function in p-space is the conjugate of the whole transformation integral?
 
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Haynes Kwon said:
Given that the wave function represented in momentum space is a Fourier transform of the wave function in configuration space, is the conjugate of the wave function in p-space is the conjugate of the whole transformation integral?

Can you prove or disprove that:

##F(\psi^*) = F(\psi)^*##

Where ##F## is the Fourier Transform.

Which I think is what you are asking?
 
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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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