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If you are asking about the possibility for the electron to be found in energy between two energy levels, the answer is no. If you are asking about the possibility that the electron is found in space between two orbitals corresponding to two energy levels than the answer is yes. An explanation follows.
Electron is a quantum particle and as such, it will be represented by a wave function `Psi(x,t)`. The square of the absolute value of the wave function `|Psi(x,t)``|^2` represents the probability that the electron is present in that infinitesimal space element `dx` and at that given time `t` . This behavior comes from the Heisenberg incertitude principle that says to us that we never can know with 100% certitude at the same time the position and the speed of a quantum particle. Otherwise said, the electron might be there in space more probable than in other part, you just never know until you made a measurement on it.
Having the above said, the wave function that represents the electron position need to satisfy the Schrodinger equation
`hatH*Psi(x,t) = E*Psi(x,t)` or equivalent
`[hatp^2/(2m) +V(x,t)]Psi(x,t) = E*psi(x,t)`
`[-(ih*grad)/(4pim) +V(x,t)] =E*Psi(x,t)`
This equation means that there are only certain fixed energy levels `E` (the eigenvalues of `hatH` ) that the electron can take, and never can an electron have an energy intermediate between two eigenvalues of the above equation.
Now, about the electron presence in space. After solving the Schrodinger equation, the electron wave function will be known. Thus the form of the trajectories of the electrons for different electron energy levels will be known to a certain approximation. It is more correct to say orbitals instead of trajectories, to account for the uncertainty of the electron presence in space (there are 1 up to 4 orbitals for each energy level of the electron). Because of this uncertainty, electrons from two different energy levels (and thus two different orbitals) will have a probability different of zero to be found in the same spatial element `dx` which means they both can occupy the same space in an atom (in other words orbitals from different energy levels in atom can overlap). Regardless of this, they will never have an intermediate energy between two allowed energies in the atom.
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