# Is there any Hamiltonian that contains time derivative? [duplicate]

Quantum mechanics is governed by Schrodinger's equation:

$$\hat{H}\psi=i\hbar\partial_t \psi$$

It seems that Hamiltonian acts on wave functions like a time derivative. Just out of curiosity, is there any Hamiltonian that contains time derivatives, either first order or second order?

## marked as duplicate by ACuriousMind♦ quantum-mechanics StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 13 '16 at 10:40

The Hamiltonian operator is a unitary operator that maps state vectors to other state vectors in a given Hilbert Space, regardless of time. Lubos's answer in this thread discusses this distinction very clearly: Why $\displaystyle i\hbar\frac{\partial}{\partial t}$ can not be considered as the Hamiltonian operator?
For any particular interaction, you will have a predetermined Hamiltonian. For example, if a particle is free, then $$\hat H = \hat P^2/2m$$ If a particle is subject to some kind of scalar potential energy V(x), then $$\hat H = \hat P^2/2m + V(x)$$
• Do you mean that time derivatives are not unitary? I think $-i\partial_t$ is unitary. – Chong Wang Jun 13 '16 at 10:21