The Problem with Schrödinger’s Wave Equation
The biggest question still plaguing Schrödinger’s wave equation was the role of the wavefunction. Sure, mathematically it’s clear: it’s the solution to Schrödinger’s wave equation and the “all-powerful function” as a result. However, physically it was still a big mystery to everyone, including Schrödinger himself.
Originally he had interpreted it as being connected to “some vibration process in the atom.” Later, for a system of electrons, Schrödinger became more exacting and interpreted the absolute square of the wavefunction (the absolute value of the wavefunction multiplied by itself) as a type of “weight-function” related to the charge density (or “density of electricity,” as he put it) at a particular region in space. Therefore, for a single electron, Schrödinger actually envisioned it to be spread out over the entire space. In other words, he literally imagined the electron not as a particle at a particular point in space, but as a wave spread throughout it.
Born’s Quantum Probability
Schrödinger wasn’t the only one to be thinking about the physical meaning of the wavefunction. Several people were beginning to conclude that the wavefunction was really associated with a kind of quantum probability very different from the probability of classical mechanics. Among them were Paul Dirac (1902–1984), Eugene Wigner (1902–1995), and, most notably, Max Born. Max Born’s work clearly defined the physical meaning of the wavefunction and the nature of quantum probability. He says, “The motion of particles follows probability laws ….”
In other words, the motion of quantum particles, such as electrons, isn’t governed by deterministic equations like classical particles (or objects). As a result, unlike a classical particle, a quantum particle doesn’t move along a well-defined physical path with well-defined values for its key properties, such as position, momentum, energy, and the like, at every instant in time. Rather, according to Born, these physical quantities (and many others) are determined entirely by a quantum probability, which is proportional to the absolute square of the wavefunction. As it was for Schrödinger, so was it for Born that the absolute square of the wavefunction was the secret to revealing the true physical meaning of the wavefunction. However, his perception of it was altogether different.
Born also notes that “the probability itself propagates according to the law of causality.” So, although the motion of a quantum particle isn’t deterministic, the quantum probability governing the final outcome is, and it’s given by Schrödinger’s wave equation (since this determines the wavefunction, and therefore the absolute square of it). This is somewhat reminiscent of when we talked about the Boltzmann probability. Recall that the Boltzmann probability gives the probability of a certain microstate occurring from the many that are available to a system of particles. However, there’s a crucial difference.
The Boltzmann probability was a mathematical convenience for handling a system containing an extremely large number of particles. For such a system, it’s simply impossible to use Newton’s equation to determine the physical path of each and every particle. That doesn’t mean that their paths and the respective positions, momentum, energy, and the like don’t exist. Surely, in classical mechanics, they do exist. It just means that solving this mathematical problem is unwieldy. Thus, we appeal to using the Boltzmann probability, which greatly simplifies the original problem by allowing us to calculate average quantities of the collective system of particles.
It’s a totally different scenario with the probability associated with quantum mechanics. Here, the quantum probability isn’t merely a way of simplifying some complicated mathematics into a more tractable problem. In the quantum world, the probabilistic nature is the physical reality. Therefore, the only thing one can know about a quantum particle is the probability that you will find it in a given quantum (micro) state. So whereas in classical mechanics one talks of a particle evolving, according to Newton’s equations, along a given path, in quantum mechanics it’s the probability of the particle that evolves, according to Schrödinger’s equation, leading it from one quantum state to another.
Born’s interpretation reconciled a major challenge of Schrödinger’s wave equation. Now, it was clear how the continuous wavefunction could give rise to the discontinuous energy states available to an electron: the wavefunction moves in a probability space – not a physical space – that “guides” the electron from one quantum state to another. So, the probabilistic nature of quantum, which Heisenberg had explicitly included in his theory, is also present in Schrödinger’s theory. The electron jumping in Bohr’s atom had also been reaffirmed by Born’s interpretation; now these electronic transitions had been endowed with a mathematical probability of their occurrence. However, the electronic orbits (a purely classical notion), which had been falling out of favor for some time, were finally gone once and for all. For Schrödinger this was all too much.
One would have thought that Schrödinger would have been a big fan of Born’s statistical interpretation. After all, this was the guy who, in his 1922 inaugural address at the University of Zürich, said, “It is quite possible that the laws of nature without exception have a statistical character.” Evidently, quantum probability was not what he was thinking when he made this statement:
“But 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.”
Schrödinger wasn’t the only one to have trouble with quantum probability; Einstein did as well.
Einstein’s Break with Quantum Mechanics
After leading the way in quantum theory for almost twenty years and being the first to introduce transition probabilities into it (in 1916), Einstein had had enough. In 1917, he noted that according to quantum theory, the direction of momentum transfer in the spontaneous emission of a photon from an atom was apparently governed by “chance,” and it troubled him ever since. By 1926, he had become completely unforgiving of quantum probability, and in response to a letter Born had written to him, Einstein wrote:
“Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.”
Einstein’s denouncement came as a “hard blow” to Born. It was official: Einstein was turning his back on quantum mechanics and so it would be the rest of his life. In 1944, he would reaffirm his original statement to Born: “The great initial success of the quantum theory cannot convert me to believe in that fundamental game of dice.”
While Born and Einstein were disagreeing on quantum probability, Bohr and Schrödinger were disagreeing on – well, pretty much everything.