Energy definition

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How is energy defined in quantum mechanics?is it defined by the De-Broglie relations or from the eigenvalue equation of H operator?Are the two somehow equivalent?
 

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dextercioby
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Interesting question. Actually we have the energy observable (that is what is measured) and the Hamilton operator. If we measure the energy of a quantum system we should get a value from the operator's spectrum.

The measurement problem is really tricky and under debate, but energy is always associted to the hamiltonian.
 
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What is the status of the De-Broglie relations then?Is it a derived result or an axiom?does it always hold?
 
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It is derived. It doesn't always hold, because it really only applies to particles in free space. If there is a non-constant potential present, then you cannot necessarily even assign a single "wavelength" to the eigenstates (only a spectrum of them). For example, the Hydrogen atom's ground state is exponentially decaying with radius.
 
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so the de-broglie relation holds only for constant potential,is that what you are saying?what is the definiton of energy then?
 
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Let me turn this around a bit -- what do you think the definition of energy in classical mechanics is?
 
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ummm...a quantity that remains conserved because the lagrangian is time-independent?
can anyone please give me a sort of outline how i can derive the De-broglie relation for energy ?
 

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