How is energy defined in quantum mechanics?

[xboy]

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?

 

[dextercioby]

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.

 

[xboy]

What is the status of the De-Broglie relations then?Is it a derived result or an axiom?does it always hold?

 

[Manchot]

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.

 

[xboy]

so the de-broglie relation holds only for constant potential,is that what you are saying?what is the definition of energy then?

 

[StatMechGuy]

Let me turn this around a bit -- what do you think the definition of energy in classical mechanics is?

 

[xboy]

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 ?