- #1
amjad-sh
- 246
- 13
Take a wavefunction ##\psi## and let this wavefunction be a solution of Schroedinger equation,such that:
##i \hbar \frac{\partial \psi}{\partial t}=H\psi##
The complex conjugate of this wavefunction will satisfy the "wrong-sign Schrodinger equation" and not the schrodinger equation,such that ##i \hbar \frac{\partial \psi^*}{\partial t}=-H\psi^*##.
This means that the complex conjugate of any wavefunction that satisfy the Schroedinger equation is not a physical wavefunction and it does not belong to the Hilbert space.
But we know also that the complex conjugate of that function is square integrable since it satisfies ##\int _{-∞}^{+∞}\psi(x)\psi^{*}(x)dx=1.##
So can we conclude that if the function is square integrable it does not necessarily belong to hilbert space?
##i \hbar \frac{\partial \psi}{\partial t}=H\psi##
The complex conjugate of this wavefunction will satisfy the "wrong-sign Schrodinger equation" and not the schrodinger equation,such that ##i \hbar \frac{\partial \psi^*}{\partial t}=-H\psi^*##.
This means that the complex conjugate of any wavefunction that satisfy the Schroedinger equation is not a physical wavefunction and it does not belong to the Hilbert space.
But we know also that the complex conjugate of that function is square integrable since it satisfies ##\int _{-∞}^{+∞}\psi(x)\psi^{*}(x)dx=1.##
So can we conclude that if the function is square integrable it does not necessarily belong to hilbert space?