Time derivative of schrodinger equation

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The time-dependent Schrödinger equation (TDSE) is a first-order time derivative because it ensures that wave functions remain complex, which is essential for accurately describing quantum states. This first-order requirement is linked to Noether's theorem, where energy conservation arises from time invariance. If the TDSE were a second-order time derivative, it could lead to non-physical solutions, such as instabilities or negative probabilities. The necessity for complex wave functions allows for interference and superposition, fundamental aspects of quantum mechanics. Understanding these principles is crucial for grasping the foundational postulates of quantum theory.
geet89
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Why is the TDSE first derivative in time. Now I know that it is required so that the wave functions are complex... but is there any physical interpretation for this requirment??
 
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Ultimately, it's a postulate of QM. But you can rationalize it in various ways, for instance by Noether's theorem, energy is the conserved quantity under time translations.
 
Can u give me some examples of what would happen if it were a second order derivative in time... and why should the wave functions always be complex?
 
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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