Quantum mechanics – Heisenberg and Schrödinger

Planck’s quantum law

In 1900 Max Planck (1858-1947) discovered that electromagnetic energy is transmitted in discrete packets – quanta. With this assumption he obtained a theoretical explanation for the emitted light of a glowing black body – such as the filament of a light bulb.

Planck’s radiation law reads: E = h.f.

E is the energy of an emitted and absorbed energy package, f stands for the frequency. The value of h (Planck’s constant): 6,626 ×10-34 Joules per second – is now engraved in his tombstone. With his quantized energy assumption he offered an explanation for why we only get a tan from UV light – high frequency, lots of energy – but not from infrared light – low frequency so too little energy to damage our skin cells.

Of course this raises the question what the physical meaning is of the frequency of a such an energy package.

In 1905 Albert Einstein (1879-1955) explained the photoelectric effect by applying Planck’s energy quanta. He named them light quanta. Later they became better known as photons. This provided Planck’s energy quanta with real particle status and we had suddenly two incompatible descriptions of light: particles and waves.

The quantum atom of Bohr and De Broglie

In 1912, Niels Bohr (1885-1962) arrived at an explanation for the spectral lines of hydrogen. This was a problem because glowing hydrogen does not emit light in all possible frequencies as with an incandescent lamp, but the transmitted frequencies only have certain discrete values. Those broadcasted frequencies have a very specific pattern that even shows a certain mathematically expressible coherence.

Hydrogen emission spectrum. © byjus.com

The atomic model at that time – the beginning of the twentieth century – was a description of negative electrons circled at lightning speed around the positive core, like a mini solar system. But that description was not able to explain those spectral lines and their mathematical expressible behavior. Niels Bohr suggested that Planck’s quantum law – E = h.f – should be applied here. An electron moving around the nucleus could then only be in very specific energy states. Bohr called those energy states orbits. He numbered those orbits in ascending energy state with numbers: n = 1,2,3, … etc. If the electron “jumped” from a higher to a lower energy state, the energy difference was sent away as an Einstein photon.

Bohr also stated that the electron jump did not follow a physical path, in other words it was instantaneous, because otherwise when a physical path was traveled, energy loss had to occur according to Maxwell’s laws for electromagnetic radiation.

Bohr’s quantum model of the hydrogen atom. Only specific electron orbits are allowed.

Bohr could not explain however why only those orbits for electrons were allowed. Luckily Louis Victor de Broglie (1892-1987) came up with a highly imaginative explanation by assuming that if (light) waves could also behave like particles (photons), then particles (electrons) would probably also behave like waves. According to De Broglie, those electron orbits from Bohr were formed by standing electron waves, such as happens with standing waves along a vibrating string. Only electron orbits with a natural number of wavelengths (n = 1,2,3, .. etc) would fit exactly and could exist as standing waves.

The De Broglie electron orbits forming standing waves of 1 to 4 wavelengths.

The theory of De Broglie was confirmed in 1927 by interference behavior observed in experiments with electrons. Interference is an accepted indicator of wave behavior. But the Bohr-De Broglie hydrogen atom model was still as flat as a dime. There was no explanation yet for the the spherical appearance of the atom.

Standing circular De Broglie 8λ wave. By Yuta Aoki

Werner Heisenberg – the first quantum mechanic

Until 1925 there was no real physical coherent quantum theory with one solid mathematical formal basis. Until then quantum physics was still a loose collection of assumptions. In that year, Werner Heisenberg (1901-1976) developed the first version of quantum matrix mechanics with which the location and intensity of the spectral lines of glowing hydrogen gas could be formally calculated. Heisenberg’s matrix mechanics proved to be a difficult formalism, provided no insight into the underlying mechanisms and only produced discrete outcomes. That, of course, was perfectly consistent with the discrete values observed for the wavelengths in the hydrogen spectrum.

Erwin Schrödinger – the wave mechanic

In 1926, Erwin Schrödinger published what is now known as the Schrödinger equation for quantum mechanics. Schrödinger sought and found his famous equation because he realized that the electron wave model of De Broglie for a hydrogen with a single electron could not explain why the hydrogen atom wasn’t flat but a sphere. So he was looking for a solution for a 3-dimensional standing wave that spread spherically around the core, which he found. He thought the solution to his equation described the scattered cloud-like charge of the electron.

In 1929, Heisenberg and Wolfgang Pauli published the foundations of relativistic quantum theory. With that feat, a solid theoretical basis was finally laid for quantum mechanics. They even had now two different ways to make quantum mechanical predictions.

The matrix mechanics of Heisenberg produces concrete outcomes, the solution to the Schrödinger equation describes a wave that can assume all possible values ​​at any point in space and time. So they seemed incompatible theories. Just like particles and waves are not compatible descriptions of reality. But in the end it has been shown that both approaches to quantum mechanics are mathematically equivalent.

Heisenberg’s uncertainty principle: Δx.Δp ≥ ħ/2

In 1927, Heisenberg formulated his famous uncertainty principle that says there is a fundamental inverse relationship between the accuracy with which the momentum p (mass x speed) can be measured and the location of the particle. That it is fundamental means that his uncertainty is not the result of imperfections in our measuring instruments. It is a limit that nature imposes on our measurements of physical objects. The uncertainty in the place is Δx, the uncertainty in the pulse (p) is Δp, the crossed-out ħ (with dash through the upper leg) is the so-called Dirac constant and is equal to h / 2π. That is a notation convention that simplifies the readability of formulas and equations for the physicist, but is of no importance to us here. The value of ħ is so incredibly small – 1,05 ×10−34 – that we only have to deal with it when measuring objects with atomic dimensions.

The probability wave of Max Born

Max Born (1882-1970) finally gave in 1927 the, by most physicist accepted, physical interpretation of the quantum state wave that represents the solution of the Schrödinger equation. The square of the absolute value of the state wave at a certain place and time appears to be the measure of the chance of finding the quantum object – the electron, photon, …, etc. – at that particular place and time.

With such an interpretation, the quantum wave has literally become something elusive, because you can’t pick up probabilities. Probabilities are actually more like thoughts, they are our expectations about something and are therefore more existing in our minds than in the physical tangible reality. Seen in that light the question of how that non-physical probability wave changes into the physical particle that we find in our measuring instrument becomes important, which is the so-called measuring problem.

Schrödinger’s cat

Schrödinger’s cat

Schrödinger was – like many physicists – not happy with the strange implications of quantum physics. To demonstrate the absurdity of quantum physics – the wave of probability in which all possibilities are decided but which is only physically realized upon measurement – he came up with his notorious thought experiment with a cat in a closed box. The cat, due to the quantum state of the box’s contents, would be in a state of simultaneously living and dead from which the cat only ‘escapes’ when an observer opens the box. A lot of – often somewhat macabre – jokes have been made about that cat.

Joking between quantum physicists

Heisenberg and Schrödinger get pulled over for speeding.
The cop asks Heisenberg “Do you know how fast you were going?
Heisenberg replies, “No, but we know exactly where we are!
The officer looks at him confused and says “You were going 108 miles per hour!
Heisenberg throws his arms up and cries, “Great! Now we’re lost!
The officer looks over the car and asks Schrödinger if the two men have anything in the trunk.
A cat,” Schrödinger replies.
The cop opens the trunk and yells “Hey! This cat is dead.
Schrödinger angrily replies, “Well, now he is.”

Next subject: The Quantum Measurement Problem. What is a measurement?