Paradox Regarding Hermiticity

[neelakash]

Here is an interesting piece of paradox.

We all know Hamiltonian is an energy operator in Quantum Mechanics.The Schrodinger's equation tells that
$$\ H\psi=\ i \hbar\frac{\partial}{\partial\ t}\psi$$

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

$$\ i \hbar\frac{\partial}{\partial\ t}$$ is also Hermitian?It is difficult to see as $$\frac{\partial}{\partial\ t}$$ 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-

$$\int\psi\ *\ A\phi\ d\ V = \int(\ A\psi)\ *\phi\ d\ V$$

the first thing looks reasonable to ask is whether $$\ d\ V$$ 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 $$[\psi^\ *\phi]_{\ t_\ 1}^{\ t_\ 2}$$

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?

 

[neelakash]

Let me tell you what I understood regarding this problem:Let me use the formula:

$$\int\psi^\ *\ A\phi\ d\ V =\int(\ A\psi)^\ *\phi\ d\ V$$

$$\int\psi\ *[\ i\hbar\frac{\partial}{\partial\ t}\phi]\ dt =\ i\hbar\int\psi\ *[\frac{\partial}{\partial\ t}]\phi\ dt =\ i\hbar[\int\( -\frac{\partial}{\partial\ t}\psi\ *)\phi\ dt\ +\[\psi*\phi]_{\ t_\ 1}^{\ t_\ 2}] =\int[\ i\hbar\frac{\partial}{\partial\ t}\psi]\ *\phi\ dt\ +\ i\hbar[\psi*\phi]_{\ t_\ 1}^{\ t_\ 2}$$

Now we can clearly see that the operator would be hermitian if the boundary term $$\psi(\ t_2)\ *\phi(\ t_2)-\psi(\ t_1)\ *\phi(\ t_1)=0$$

Note that till now,we have used pure mathematics.I have integrated over time,instead over the space (this is my assumption for my method to work).

Now,I search for the functions that have the property.Presumably,the solutions of Schrodinger's equation possibly has this property. if $$\psi$$ and $$\phi$$ are the solutions of the Schrodinger's equation they satisfy:

$$\psi(\ t_2)= exp[\frac{\ -i\int\ H\ dt}{\hbar}]\psi(\ t_1)$$ where the exponential is the unitary time evolution operator.

Similarly,$$\phi(\ t_2)= exp[\frac{\ -i\int\ H\ dt}{\hbar}]\phi(\ t_1)$$

And $$\psi(\ t_2)\ *= exp[\frac{\ i\int\ H\ dt}{\hbar}]\psi(\ t_1)\ *$$

Thus, clearly the solutions of Schrodinger's equation that possesses the above property smoothly fit into the condition

$$\psi(\ t_2)\ *\phi(\ t_2)-\psi(\ t_1)\ *\phi(\ t_1)=0$$

So,my conclusion is not for all functions $$\ i\hbar\frac{\partial}{\partial\ t}$$ is a hermitian operator.However,if the functions are the solutions of Schrodinger's equation.so that they may be relaterd by the unitary time evolution operator,only in that case $$\ i\hbar\frac{\partial}{\partial\ t}$$ is identical with the Hamiltonian operator.And they are hermitian.

However,this method works for I integrated over time.I am not sure if the result works as well if I work by integrating the space.Basically,this is the point of my confusuion.There is no time representation in quantum mechanics,analogous to position representation or momentum representation.So,is the integration over time valid?

 

[Entropia]

I would say that this is indeed an interesting paradox. The Schrodinger's equation tells us that the Hamiltonian operator is Hermitian, meaning that it satisfies the property of Hermiticity which states that the operator is equal to its own conjugate transpose. However, when we look at the time derivative term in the equation, it is not immediately clear if it is also Hermitian.

One possible solution to this paradox is to consider the time derivative term as a generator of time translations. In this case, it can be shown that the time derivative operator is indeed Hermitian. Another approach is to consider the time derivative term as a part of the Hamiltonian operator itself, in which case it would also inherit the Hermitian property.

It is also worth noting that the concept of Hermiticity is closely related to the concept of observables in quantum mechanics. Therefore, the question of whether the time derivative term is Hermitian also raises the question of whether it is a physically meaningful observable.

Ultimately, this paradox highlights the complexities and nuances of quantum mechanics, and serves as a reminder that there is always more to learn and understand in the realm of science.