It is interesting to note that Werner Heisenberg was working on his own quantum mechanics at the same time Schroedinger was coming up with his equation. In fact, together with colleagues Max Born and Pascual Jordan, Heisenberg actually developed and published his theory a short time before Schroedinger.
This alternative theory was based on the mathematics of matrices instead of complex wave functions. This made the whole theory look completely different from what we have just described. At first, Heisenberg’s theory had an easier time explaining certain observations than Schroedinger’s theory did. The so-called matrix mechanics was more focused on explaining the “quantum jumps” between energy states of the atom, and the trio came up with a very successful form of advanced algebra to do just that.
By contrast, Schroedinger developed his wave mechanics with the hope of eliminating the need for quantum jumps. He believed that he’d ultimately show that wave functions could move smoothly and continuously (albeit rapidly) from one state to another. While he succeeded in developing his wave-based theory, he struck out when it came to striking down the discontinuous jumps.
QUANTUM QUOTE
If all these damned quantum jumping were really here to stay, I should be sorry I ever got involved with quantum theory!
—Erwin Schroedinger during a debate with Heisenberg and Bohr
For a short time in 1925 and 1926, Heisenberg’s matrix mechanics and Schroedinger’s wave mechanics were in competition with each other. It appeared that they were both attempting to explain the same quantum phenomena in very different ways. As the theories spread and other scientists began working with them, it appeared that each theory had its own strengths and weaknesses, but both were remarkably successful at predicting the most important physical results.
QUANTUM LEAP
One of the nice features of Heisenberg’s matrix mechanics is that it can, in a very straightforward way, identify conjugate pairs of observables (like position and momentum, or energy and time). That means in the matrix formulation, given one observable quantity, one can easily see what the conjugate partner of that quantity should be. Since, as you recall, conjugate pairs are the ones that obey Heisenberg’s uncertainty principle, this is not too surprising.
Eventually, physicists started to gravitate to the wave theory given its resemblance to classical wave theories that most of them was already comfortable with. A true embrace of matrix mechanics, on the other hand, took a lot more guts.
By the middle of 1926, however, Schroedinger (who had come in second in the race to make a quantum mechanics) showed convincingly that the two theories were really just different mathematical formulations of exactly the same theory. Fundamentally and mathematically, they were identical, and would always make the same prediction if correctly applied.
So why did we spend so much effort presenting Schroedinger’s theory and not matrix mechanics? Part of the reason lies in the details of the math. Differential equations are difficult enough to explain, but matrix operations are at least as tricky, and look even more like “magic” to most people who are not familiar with higher math.
Moreover, the most widespread interpretations of quantum mechanics are based on the Schroedinger version of the theory, and the concept of wave functions. This formalism is a little easier to connect with probability and uncertainty, the ideas that are so important to developing intuition about the quantum world.
We will explore many of the implications of Schroedinger’s wave mechanics in the coming but first let’s pause to appreciate the significance of this breakthrough. Schreodinger’s formalism is consistent with wave/particle duality and all the weirdness of the uncertainty principle. It predicts the quantization conditions, which had previously been imposed as ad hoc assumptions. It can handle stationary states as well as predict how wave packets move and change over time. With regard to atoms, the Schroedinger equation doesn’t just predict the energy levels of hydrogen, it can be applied to all atoms, common and exotic. What’s more, it can easily be put into a form that is consistent with Einstein’s special relativity.
In other words, Schroedinger’s wave theory is the real deal. Now that we have a sense of what it is and what it does, let’s take a look at some real live wave functions.
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