The electromagnet spectrum looks to be continuous, yet energy is discrete. Shouldn't one see evidence of the electron's "quantum leap"? What's going on?
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Answer
Particles in bound states exhibit quantized energy levels. This is a result of boundary conditions on the wave function. For instance, an electron in an infinite well must have 0 probability of being found outside the well or at the boundaries (to keep the wave function continuous). Solutions to the Shrodinger equation will look like this (as you can see, only a discrete set of energy values are allowed):
As another example, an electron orbiting a hydrogen nucleus should have 0 probability of being found infinitely far from the nucleus, so that the wave function is normalizable. The math for this one is more complicated, but you still get a discrete set of solutions to the Shrodinger equation, with discrete energies. Thus the electron can only be observed in these discrete states, and if you excite hydrogen gas, you will observe a discrete emission spectrum: 
Free particles however, have no boundary restrictions on their wave functions, and may be observed to have any energy whatsoever. This is why you see a continuous spectrum from natural light.
(Note that it is technically still discrete, because each photon has a definite energy and there is a finite number of photons, so you cannot actually obtain every possible real number energy from them)
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