The Principle of SuperpositionThe Schrōdinger equation is
H ψn = Enψn, (1)where H is the Hamiltonian operator, ψn is the wave function for state n, and the En are the energy eigenvalues for the states. The discrete spectrum energy eigenvalues En limit interactions (for example, with photons) to specific energy differences En - Em. In 1913, Niels Bohr postulated that electrons in atoms would be in "stationary states" of energy En, and that energy differences would be of the form En - Em = hν, where ν is the frequency of the observed spectral line. Albert Einstein, in 1916, derived these two Bohr postulates from basic physical principles in his paper on the emission and absorption processes of atoms. What for Bohr were assumptions, Einstein grounded in quantum physics, though virtually no one appreciated his foundational work at the time, and few appreciate it today, his work eclipsed by the Copenhagen physicists. The eigenfunctions ψn are orthogonal to each other
< ψn | ψm > = δnm (2)where the "delta function"
δnm = 1, if n = m, and = 0, if n ≠ m. (3)Once they are normalized, the ψn form an orthonormal set of functions (or vectors) which can serve as a basis for the expansion of an arbitrary wave function φ
| φ > = ∑ n = 0 n = ∞ cn | ψn >. (4)The expansion coefficients are
cn = < ψn | φ >. (5)In the abstract Hilbert space, < ψn | φ > is the "projection" of the vector φ onto the orthogonal axes ψn of the ψn "basis" vector set. The Schrōdinger equation is a linear equation. It has no quadratic or higher power terms, and this introduces the principle of superposition of quantum states a profound - and for many scientists and philosophers a disturbing - feature of quantum mechanics, one that is impossible in classical physics. If ψa and ψb are both solutions of equation (1), then an arbitrary linear combination of these,
| ψ > = ca | ψa > + cb | ψb >, (6)with complex coefficients ca and cb, is also a solution. Together with Born's probabilistic (statistical) interpretation of the wave function, the principle of superposition accounts for the major mysteries of quantum theory, some of which we hope to resolve, or at least reduce, with an objective (observer-independent) explanation of irreversible information creation during quantum processes. Observable information is critically necessary for measurements, though observers can come along anytime after the information comes into existence as a consequence of the interaction of a quantum system and a measuring apparatus. The quantum (discrete) nature of physical systems results from there generally being a large number of solutions ψn (called eigenfunctions) of equation (1) in its time independent form, with energy eigenvalues En.
An example of superposition.Dirac tells us that a diagonally polarized photon can be represented as a superposition of vertical and horizontal states, with complex number coefficients that represent "probability amplitudes." Horizontal and vertical polarization eigenstates are the only "possibilities," if the measurement apparatus is designed to measure for horizontal or vertical polarization. Thus,
| d > = ( 1/√2) | v > + ( 1/√2) | h > (10)The vectors (wave functions) v and h are the appropriate choice of basis vectors, the vector lengths are normalized to unity, and the sum of the squares of the probability amplitudes is also unity. This is the orthonormality condition needed to interpret the (squares of the) wave functions as probabilities. When these (in general complex) number coefficients (1/√2) are squared (actually when they are multiplied by their complex conjugates to produce positive real numbers), the numbers (1/2) represent the probabilities of finding the photon in one or the other state, should a measurement be made on an initial state that is diagonally polarized. Note that if the initial state of the photon had been vertical, its projection along the vertical basis vector would be unity, its projection along the horizontal vector would be zero. Our probability predictions then would be - vertical = 1 (certainty), and horizontal = 0 (also certainty). Quantum physics is not always uncertain, despite its reputation.