# Why is quantum mechanical momentum the derivative of the wave function with respect to the position? [duplicate]

In classical mechanics the momentum is defined as mass times the time-derivative of position.

In quantum mechanics, however, the time-derivative of the wave function is the hamiltonian, while the momentum is defined as $i\hbar \frac{\partial}{\partial x} \Psi(x)$, which is a space-derivative and not a time-derivative.

Note that I understand why momentum is an operator on the wave function (it's a measurable quantity, so it's an operator as per a postulate of QM). I understand the derivation from spatial translation, but I don't understand why it's an equivalent of the classical momentum as it's a space derivative and not a time derivative.

## marked as duplicate by Chris♦, Rococo, Qmechanic♦ 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(); } ); }); }); Feb 6 '18 at 7:04

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## 1 Answer

This interlinkage between classical and quantum mechanical momentum can be thought of as a consequence of the De Broglie's law - that the velocity of the wave depends on it's wavelength. Though the Hamilton operator is obtained from the same law, looking things at this perspective could help in realizing that this relationship is fundamental in nature, and is not derived from anywhere else.