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## Main Question or Discussion Point

Here is an interesting piece of paradox.

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?

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?