Here is an interesting piece of paradox.(adsbygoogle = window.adsbygoogle || []).push({});

We all know Hamiltonian is an energy operator in Quantum Mechanics.The Schrodinger's equation tells that

[tex]\ H\psi=\ i \hbar\frac{\partial}{\partial\ t}\psi[/tex]

Now,we also know that Hamiltoian is Hermitian.But are we sure that

[tex]\ i \hbar\frac{\partial}{\partial\ t}[/tex] is also Hermitian?It is difficult to see as [tex]\frac{\partial}{\partial\ t}[/tex] cannot have any matrix representation(as far as I know---in what basis would we expand the operator to find its matrix elements?).

To check this from the definition of the scalar product-

[tex]\int\psi\ *\ A\phi\ d\ V

=

\int(\ A\psi)\ *\phi\ d\ V[/tex]

the first thing looks reasonable to ask is whether [tex]\ d\ V[/tex] should be position or time.

If it is time,then the LHS can be integrated to see that there will be the desired term (RHS) plus a term like [tex][\psi^\ *\phi]_{\ t_\ 1}^{\ t_\ 2}[/tex]

Then there maight be a solution to the problem.Because,wave functions related by a unitary time evolution operator are seen to satisfy the above.

What you people think about it?

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# Paradox Regarding Hermiticity

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