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In a equation in quantum mechanics for example

$$\mathrm i\hbar \frac{d}{dt}|\phi\rangle=H|\phi\rangle \tag{1}.$$

Would it also be a hermitian operator?

Qmechanic
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TKFT
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3 Answers3

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The Hamiltonian corresponds to the energy of the system. The equation you have written is the Schrodinger equation and it tells you that the Hamiltonian is a special observable operator that dictates time-evolution in quantum mechanics.

Also yes, it is Hermitian, as all operators corresponding to observables are.

Charlie
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Taking a step back, in classical mechanics, the system energy can be expressed as the sum of the kinetic and potential energies.

For quantum mechanics, the elements of this energy expression are transformed into the corresponding quantum mechanical operators. The Hamiltonian contains the operations associated with the kinetic and potential energies, ordinarily one writes $$H=T+V$$

Where $T$ is the Kinetic Energy, and $V$ is the potential energy. More symbolically $$H= \frac{- \hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)$$

Which is the Hamiltonian for a particle in one dimension.

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The Hamiltonian was originally defined in a classical context. If you know how to express the Hamiltonian in terms of generalized coordinates and their conjugate momenta, then you can find the time evolution of the system. Not all systems are Hamiltonian systems. Systems in which energy is dissipated into heat cannot be represented in the Hamiltonian formalism. The time evolution of a Hamiltonian system is always reversible, which is clearly not the case if you consider an example like a book sliding across a table and coming to a stop.

The Hamiltonian is often equal to the energy of the system, but not always. See When is the Hamiltonian of a system not equal to its total energy?

In quantum mechanics, as in classical mechanics, all of the following are true: (1) the Hamiltonian is the thing that tells you how the system will evolve over time; (2) the time evolution is reversible (and probability is conserved, which is analogous to Liouville's theorem); and (3) the Hamiltonian is often, but not always, equal to the energy.

The Hamiltonian has to be hermitian because otherwise the time evolution would be nonunitary (probability would not be conserved). It follows from this that the Hamiltonian can be considered as an observable. In cases where it equals the energy, it is the observable for energy.

An answer by Kevin says:

Taking a step back, in classical mechanics, the system energy can be expressed as the sum of the kinetic and potential energies.

This is wrong. See the link above.

An answer by Charlie says:

The Hamiltonian corresponds to the energy of the system.

This is the same mistake.

Considering that there are so many incorrect answers, it's ironic that people are voting to close the question for being too elementary.

user282813
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    While what you are saying is correct, it is generally unhelpful to throw technicalities and exceptions at someone who is new enough to the subject to be asking what the Hamiltonian means. If OP wants a rigorous and careful explanation of the exact physical interpretation of the Hamiltonian along with lots of examples/counterexamples they can find that in a textbook, not an answer on a Q&A site. If OP is new to quantum mechanics the place to start is not with fringe cases. The question is being close voted because the first paragraph of Wikipedia could have answer this for them. – Charlie Dec 16 '20 at 15:17