How is the energy operator in quantum mechanics derived?

The latter formula is the emissivity of unit frequency interval, which is derived from Planck's blackbody radiation formula, so the order you say should actually be reversed.

When microscopic particles are in a certain state, their mechanical quantities (such as coordinates, momentum, angular momentum, energy, etc. Generally, there are a series of possible values, and each possible value appears with a certain probability.

When the state of the particle is determined, the probability that the mechanical quantity has a possible value is completely determined. This is the uncertainty relation obtained by Heisenberg in 1927. At the same time, Bohr put forward the principle of union, which further explained quantum mechanics.

The operation of measuring the physical quantity of a physical system in a certain state corresponds to the effect of the operator representing the quantity on its state function; The possible value of measurement is determined by the eigenequation of the operator, and the expected value of measurement is calculated by the integral equation containing the operator. Generally speaking, quantum mechanics cannot predict a definite result for an observation.