From Wave_interference#Quantum_interference https://en.wikipedia.org/wiki/Wave_interference#Quantum_inte... :
> Here is a list of some of the differences between classical wave interference and quantum interference:
> - In classical interference, two different waves interfere; In quantum interference, the wavefunction interferes with itself.
> Classical interference is obtained simply by adding the displacements from equilibrium (or amplitudes) of the two waves; In quantum interference, the effect occurs for the probability function associated with the wavefunction and therefore the absolute value of the wavefunction squared.
> The interference involves different types of mathematical functions: A classical wave is a real function representing the displacement from an equilibrium position; a quantum wavefunction is a complex function. A classical wave at any point can be positive or negative; the quantum probability function is non-negative.
> In classical optical interference the energy conservation principle is violated as it requires quanta to cancel. In quantum interference energy conservation is not violated, the quanta merely assume paths per the path integral. All quanta for example terminate in bright areas of the pattern.
From Conservation_of_energy https://en.wikipedia.org/wiki/Conservation_of_energy :
> Classically, conservation of energy was distinct from conservation of mass. However, special relativity showed that mass is related to energy and vice versa by E = mc2, and science now takes the view that mass-energy as a whole is conserved. Theoretically, this implies that any object with mass [e.g. photons, and other massful particles] can itself be converted to pure energy, and vice versa. However this is believed to be possible only under the most extreme of physical conditions, such as likely existed in the universe very shortly after the Big Bang or when black holes emit Hawking radiation.
From Conservation_of_energy#Quantum_theory https://en.wikipedia.org/wiki/Conservation_of_energy#Quantum... :
> In quantum mechanics, energy of a quantum system is described by a self-adjoint (or Hermitian) operator called the Hamiltonian, which acts on the Hilbert space (or a space of wave functions) of the system. If the Hamiltonian is a time-independent operator, emergence probability of the measurement result does not change in time over the evolution of the system. Thus the expectation value of energy is also time independent. The local energy conservation in quantum field theory is ensured by the quantum Noether's theorem for energy-momentum tensor operator. Due to the lack of the (universal) time operator in quantum theory, the uncertainty relations for time and energy are not fundamental in contrast to the position-momentum uncertainty principle, and merely holds in specific cases (see Uncertainty principle). Energy at each fixed time can in principle be exactly measured without any trade-off in precision forced by the time-energy uncertainty relations. Thus the conservation of energy in time is a well defined concept even in quantum mechanics.