The Schrödinger Equation1. The Schrōdinger Equation. The fundamental equation of motion in quantum mechanics is Erwin Schrōdinger's famous wave equation that describes the evolution in time of his wave function ψ.
iℏ δψ / δt = H ψ (1)Max Born interpreted the square of the absolute value of Schrōdinger's wave function |ψn |2 (or < ψn | ψn > in Dirac notation) as providing the probability of finding a quantum system in a particular state n. As long as this absolute value (in Dirac bra-ket notation) is finite,
< ψn | ψn > ≡ ∫ ψ* (q) ψ (q) dq < ∞, (2)then ψ can be normalized, so that the probability of finding a particle somewhere < ψ | ψ > = 1, which is necessary for its interpretation as a probability. The normalized wave function can then be used to calculate "observables" like the energy, momentum, etc. For example, the probable or expectation value for the position r of the system, in configuration space q, is
< ψ | r | ψ > = ∫ ψ* (q) r ψ (q) dq. (3)2. The Principle of Superposition. The Schrōdinger equation (1) is a linear equation. It has no quadratic or higher power terms, and this introduces a profound - and for many scientists and philosophers a disturbing - feature of quantum mechanics, one that is impossible in classical physics, namely the principle of superposition of quantum states. If ψa and ψb are both solutions of equation (1), then an arbitrary linear combination of these,
| ψ > = ca | ψa > + cb | ψb >, (4)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.
H ψn = En ψn, (5)The discrete spectrum energy eigenvalues En limit interactions (for example, with photons) to specific energy differences En - Em. In the old quantum theory, 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. 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 (6)where the "delta function"
δnm = 1, if n = m, and = 0, if n ≠ m. (7)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 >. (8)The expansion coefficients are
cn = < ψn | φ >. (9)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. 2.1 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. 3. The Axiom of Measurement. The axiom of measurement depends on the idea of "observables," physical quantities that can be measured in experiments. A physical observable is represented as an operator A that is "Hermitean" (one that is "self-adjoint" - equal to its complex conjugate, A* = A). The diagonal n, n elements of the operator's matrix,
< ψn | A | ψn > = ∫ ∫ ψ* (q) A (q) ψ (q) dq, (11)are interpreted as giving the expectation value for An (when we make a measurement). The off-diagonal n, m elements describe the uniquely quantum property of interference between wave functions and provide a measure of the probabilities for transitions between states n and m. It is the intrinsic quantum probabilities that provide the ultimate source of indeterminism, and consequently of irreducible irreversibility, as we shall see. Transitions between states are irreducibly random, like the decay of a radioactive nucleus (discovered by Rutherford in 1901) or the emission of a photon by an electron transitioning to a lower energy level in an atom (explained by Einstein in 1916). The axiom of measurement is the formalization of Bohr's 1913 postulate that atomic electrons will be found in stationary states with energies En. In 1913, Bohr visualized them as orbiting the nucleus. Later, he said they could not be visualized, but chemists routinely visualize them as clouds of probability amplitude with easily calculated shapes that correctly predict chemical bonding. The off-diagonal transition probabilities are the formalism of Bohr's "quantum jumps" between his stationary states, emitting or absorbing energy hν = En - Em. Einstein explained clearly in 1916 that the jumps are accompanied by his discrete light quanta (photons), but Bohr continued to insist that the radiation was classical for another ten years, deliberately ignoring Einstein's foundational efforts in what Bohr might have felt was his area of expertise (quantum mechanics). The axiom of measurement asserts that a large number of measurements of the observable A, known to have eigenvalues An, will result in the number of measurements with value An that is proportional to the probability of finding the system in eigenstate ψn. Quantum mechanics is a probabilistic and statistical theory. The probabilities are theories about what experiments will show. Experiments provide the statistics (the frequency of outcomes) that confirm the predictions of quantum theory - with the highest accuracy of any theory ever discovered! 4. The Projection Postulate. The third novel idea of quantum theory is often considered the most radical. It has certainly produced some of the most radical ideas ever to appear in physics, in attempts by various "interpretations" to deny it. The projection postulate is actually very simple, and arguably intuitive as well. It says that when a measurement is made, the system of interest will be found in (will instantly "collapse" into) one of the possible eigenstates of the measured observable. We have several possibilities for eigenvalues. We can calculate the probabilities for each eigenvalue. Measurement simply makes one of these actual, and it does so, said Max Born, in proportion to the absolute square of the probability amplitude wave function ψn. In this way, ontological chance enters physics, and it is partly this fact of quantum randomness that bothered Einstein ("God does not play dice") and Schrōdinger (whose equation of motion for the probability-amplitude wave function is deterministic). The projection postulate, or collapse of the wave function, is the element of quantum mechanics most often denied by various "interpretations." The sudden discrete and discontinuous "quantum jumps" are considered so non-intuitive that interpreters have replaced them with the most outlandish (literally) alternatives. The famous "many-worlds interpretation" substitutes a "splitting" of the entire universe into two equally large universes, massively violating the most fundamental conservation principles of physics, rather than allow a diagonal photon arriving at a polarizer to suddenly "collapse" into a horizontal or vertical state. 4.1 An example of projection. Given a quantum system in an initial state | φ >, we can expand it in a linear combination of the eigenstates of our measurement apparatus, the | ψn >.
| φ > = ∑ n = 0 n = ∞ cn | ψn >. (8)In the case of Dirac's polarized photons, the diagonal state | d > is a linear combination of the horizontal and vertical states of the measurement apparatus, | v > and | h >. When we square the (1/√2) coefficients, we see there is a 50% chance of measuring the photon as either horizontal or vertically polarized.
| d > = ( 1/√2) | v > + ( 1/√2) | h > (10)4.2 Visualizing projection. When a photon is prepared in a vertically polarized state | v >, its interaction with a vertical polarizer is easy to visualize. We can picture the state vector of the whole photon simply passing through the polarizer unchanged. The same is true of a photon prepared in a horizontally polarized state | h > going through a horizontal polarizer. And the interaction of a horizontal photon with a vertical polarizer is easy to understand. The vertical polarizer will absorb the horizontal photon completely. The diagonally polarized photon | d >, however, fully reveals the non-intuitive nature of quantum physics. We can visualize quantum indeterminacy, its statistical nature, and we can dramatically visualize the process of collapse, as a state vector aligned in one direction must rotate instantaneously into another vector direction.