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Erwin Schrödinger
Erwin Schrödinger is perhaps the most complex figure in twentieth-century discussions of quantum mechanical uncertainty, ontological chance, indeterminism, and the
statistical interpretation of quantum mechanics.
In his early career, Schrödinger was a great exponent of fundamental chance in the universe. He followed his teacher Franz S. Exner, who was himself a colleague of the great Ludwig Boltzmann at the University of Vienna. Boltzmann used intrinsic randomness in molecular collisions (molecular chaos) to derive the increasing entropy of the Second Law of Thermodynamics. The macroscopic
irreversibility of entropy increase depends on Boltzmann's molecular chaos which depends on the randomness in microscopic irreversibility.
Before the twentieth century, most physicists, mathematicians, and philosophers believed that the chance described by the calculus of probabilities was actually completely determined. The "bell curve" or "normal distribution" of random outcomes was itself so consistent that they argued for underlying deterministic laws governing individual events. They thought that we simply lack the knowledge necessary to make exact predictions for these individual events. Pierre-Simon Laplace was first to see in his "calculus of probabilities" a universal law that determined the motions of everything from the largest astronomical objects to the smallest particles. In a Laplacian world, there is only one possible future.
On the other hand, in his inaugural lecture at Zurich in 1922, Schrödinger argued that evidence did not justify our assumptions that physical laws were deterministic and strictly causal. His inaugural lecture was modeled on that of Franz Serafin Exner in Vienna in 1908.
"Exner's assertion amounts to this: It is quite possible that Nature's laws are of thoroughly statistical character. The demand for an absolute law in the background of the statistical law — a demand which at the present day almost everybody considers imperative — goes
Several years later, Schrödinger presented a paper on "Indeterminism in Physics" to the June, 1931
"Fifty years ago it was simply a matter of taste or philosophic prejudice whether the preference was given to determinism or indeterminism. The former was favored by ancient custom, or possibly by an a priori belief. In favor of the latter it could be urged that this ancient habit demonstrably rested on the actual laws which we observe functioning in our surroundings. As soon, however, as the great majority or possibly all of these laws are seen to be of a statistical nature, they cease to provide a rational argument for the retention of determinism.
Despite these strong arguments against determinism, just after he completed the wave mechanical formulation of quantum mechanics in June 1926 (the year Exner died), Schrödinger began to side with the determinists, including especially Max Planck and Albert Einstein (who in 1916 had discovered that ontological chance is involved in the emission of radiation).
Schrödinger's wave equation is a
continuous function that evolves smoothly in time, in sharp contrast to the discrete, discontinuous, and indeterministic "quantum jumps" of the Born-Heisenberg matrix mechanics. His wave equation seemed to Schrödinger to restore the continuous and deterministic nature of classical mechanics and dynamics. And it suggests that we may visualize particles as wave packets moving in spacetime, which was very important to Schrödinger. By contrast, Bohr and Heisenberg and their Copenhagen Interpretation of quantum mechanics insisted that visualization of quantum events is simply not possible. Einstein agreed with Schrödinger that visualization (Anschaulichkeit) should be the goal of describing reality.
Max Born, Werner Heisenberg's mentor and the senior partner in the team that created matrix mechanics, shocked Schrödinger with the interpretation of the wave function as a "probability amplitude."
The motions of particles are indeterministic and probabilistic, even if the equation of motion for the probability is deterministic.
It is true, said Born, that the wave function itself evolves deterministically, but its significance is that it predicts only the probability of finding an atomic particle somewhere. When and where particles would appear - to an observer or to an observing system like a photographic plate - was completely and irreducibly random, he said. Born credited Einstein for the idea that the relationship between waves and particles is that the waves give us the probability of finding a particle, but this "statistical interpretation" of the wave function came to be known as "Born's Rule.".
Einstein had seen clearly for many years that quantum transitions involve chance, that quantum jumps are random, but he did not want to believe it. Although the Schrödinger equation of motion is itself continuous and deterministic, it is impossible to restore continuous deterministic behavior to material particles and return physics to strict causality. Even more than Einstein, Schrödinger hated this idea and never accepted it, despite the great success of quantum mechanics, which today uses Schrödinger's wave functions to calculate Heisenberg's matrix elements for atomic transition probabilities and all atomic properties. Discouraged, Schrödinger wrote to his friend Willie Wien in August 1926 "[That discontinuous quantum jumps]...offer the greatest conceptual difficulty for the achievement of a classical theory is gradually becoming even more evident to me."...[yet] today I no longer like to assume with Born that an individual process of this kind is "absolutely random." i.e., completely undetermined. I no longer believe today that this conception (which I championed so enthusiastically four years ago) accomplishes much. From an offprint of Born's work in the
Why did Schrödinger not simply welcome Born's absolute chance? It provides strong evidence that Boltzmann's assumption of chance in atomic collisions (molecular disorder) was completely justified. Boltzmann's idea that entropy is statistically irreversible depends on microscopic irreversibility. Exner thought chance is absolute, but did not live to see how fundamental it was to physics. And the early Epicurean idea that atoms Could it be that senior scientists like Max Planck and Einstein were so delighted with Schrödinger's work that it turned his head? Planck, universally revered as the elder statesman of physics, invited Schrödinger to Berlin to take Planck's chair as the most important lecturer in physics at a German university. And Schrödinger shared Einstein's goal to develop a unified (continuous and deterministic) field theory. Schrödinger won the Nobel prize in 1933. But how different our thinking about absolute chance would be if perhaps the greatest theoretician of quantum mechanics had accepted random quantum jumps in 1926? In his vigorous debates with Neils Bohr and Werner Heisenberg, Schrödinger attacked the probabilistic Copenhagen interpretation of his wave function with a famous thought experiment (which was actually based on another Einstein suggestion) called Schrödinger's Cat.Schrödinger was very pleased to read the Einstein-Podolsky-Rosen paper in 1935. He immediately wrote to Einstein in support of an attack on Bohr, Born, and Heisenberg and their "dogmatic" quantum mechanics. "I was very happy that in the paper just published in P.R. you have evidently caught dogmatic q.m. by the coat-tails...My interpretation is that we do not have a q.m. that is consistent with relativity theory, i.e., with a finite transmission speed of all influences. We have only the analogy of the old absolute mechanics . . . The separation process is not at all encompassed by the orthodox scheme.' Einstein had said in 1927 at the Solvay conference that nonlocality (faster-than-light signaling between particles in a space-like separation) seemed to violate relativity in the case of a single-particle wave function with non-zero probabilities of finding the particle at more than one place. What instantaneous "action-at-a-distance" prevents particles from appearing at more than one place, Einstein oddly asked.
In his 1935 EPR paper, Einstein cleverly introduced
Here we must explain the
Schrödinger challenged Einstein's idea that two systems that had previously interacted can be treated as ψ and _{1}ψ.
_{2}
Einstein called this his "separability principle ( When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call thatIn the following year, Schrödinger looked more carefully at Einstein's assumption that the entangled system could be separated enough to be regarded as two systems with independent wave functions: Years ago I pointed out that when two systems separate far enough to make it possible to experiment on one of them without interfering with the other, they are bound to pass, during the process of separation, through stages which were beyond the range of quantum mechanics as it stood then. For it seems hard to-imagine a complete separation, whilst the systems are still so close to each other, that, from the classical point of view, their interaction could still be described as an unretarded Schrödinger described the puzzle of entanglement in terms of what one can answer to questions about the two entangled particles, which set an unfortunate precedent of explaining entanglement in terms of knowledge (epistemology) about the entangled particles rather than what may "really" be going on (ontology). He wrote:
the result of measuring p Schrödinger says that the entangled system may become disentangled (Einstein's separation) and yet some perfect correlations between later measurements might remain. Note that the entangled system could simply decohere as a result of interactions with the environment, as proposed by decoherence theorists. The perfectly correlated results of Bell-inequality experiments might nevertheless be preserved, depending on the interaction. And of course they will be separated by a measurement of either particle, for example, by Alice or Bob in the case of Bell's Theorem.
Bell's Theorem
Following David Bohm's version of EPR, John Bell considered two spin-1/2 particles in an entangled state with total spin zero. We can rewrite Schrödinger's separation equation above as
| ψ > = (1/√2) | + - > - (1/√2) | - + >
This is a superposition of two states, either of which conserves total spin zero. The minus sign ensures the state is anti-symmetric, changing sign under interchange of identical electrons.
Schrödinger does not mention
Let's assume that Alice makes a measurement of a spin component of one particle, say the x-component. First, her measurement projects the entangled system into either the | + - > or
Whenever Alice measures spin up, Bob measures spin down, but that is
Bell's inequality was a study of how these perfect correlations fall off as a function of the angle between measurements by Alice and Bob. Bell predicted local hidden variables would produce a
Bell wrote that "Since we can predict in advance the result of measuring any chosen component of
But note that these values were not determined (they did not even exist according to the Copenhagen Interpretation) before Alice's measurement. According to Werner Heisenberg and Pascual Jordan, the spin components are
Quantum Jumping
In 1952, Schrödinger wrote two influential articles in the British Journal for the Philosophy of Science denying quantum jumping. These papers greatly influenced generations of quantum collapse deniers, including John Bell, John Wheeler, Wojciech Zurek, and H. Dieter Zeh.
On Determinism and Free Will
In Schrödinger's mystical epilogue to his essay What Is Life? (1944), he "proves God and immortality at a stroke" but leaves us in the dark about free will.
As a reward for the serious trouble I have taken to expound the purely scientific aspects of our problem Order, Disorder, and Entropy
Chapter 6 of What Is Life?
Normal | Teacher | ScholarNec corpus mentem ad cogitandum, nec mens corpus ad motum, neque ad quietem, nec ad aliquid (si quid est) aliud determinare potent.' SPINOZA, Ethics, Pt III, Prop.2 |