【讀者分享】 再回應 《奧本海默》：如何用基礎數學公式理解量子力學？ (Patreon)

【按：這位會員網友為了回應《奧本海默》的文章和對談，寫下長長的一篇專業文章，必須坦承作為量子力學白癡的我，實在是門外漢。希望有同好可以和這位有心讀者專業交流，非常感激，也非常佩服。】

I'm a science enthusiast. While I’m not an expert in quantum physics, I would like to share my understanding of elementary quantum mechanics, first by intuition and then through some mathematical equations. I welcome any feedback from experts here who can point out any errors in my concept/interpretation. Please feel free to provide your insights and corrections.

I would say the strangeness of the theory starts off with wave-particle duality: quantum objects have both wave and particle properties. If an electron has a definite single momentum, it has a single frequency, which behaves like a regular wave. This wave continues on in space forever, so the electron doesn't have a definite position. On the other hand, if we set up a wave with a definite position, we have to sum up waves of many different frequencies. So if an electron wave is localized in space, it doesn't have a definite frequency. This is called Heisenberg's uncertainty principle: the more accurately one physical property of a system is measured, the less accurately the other property can be known.

We can illustrate this concept in a simple experiment where the position of an electron is measured. In order to measure the position of an object, a photon must collide with it and return to the measuring device. Since photons hold some finite momentum, a transfer of momentum occurs when the photon collides with the electron, in accordance with the law of conservation of momentum. This transfer of momentum causes the momentum of the electron to increase. Thus, any attempt to measure the position of a particle will increase the uncertainty in the value of its momentum.

Conversely, if we measure the momentum of the electron with high precision, we use a photon with a longer wavelength (lower momentum) in order to minimize the disturbance to the electron’s momentum (the “push” it delivers is minimal). By gaining higher precision in momentum (minimal photon interference), we lose precision in the electron’s position in light of a larger spread of photons.

If we consider this example with a macroscopic object, such as a basketball, it can be understood that Heisenberg's uncertainty principle has a negligible impact on measurements. Although there is still a transfer of momentum from the photons to the basketball, the mass of the photon is significantly smaller than that of the basketball, and therefore, the change of momentum in the basketball can be neglected.

The world of quantum mechanics is very different from classical (Newtonian) mechanics that we're all much more accustomed to. We cannot derive quantum mechanics from classical laws. In fact, quantum mechanics is a more fundamental theory from which classical mechanics emerges. However, there are many close parallels between the equations of quantum and classical mechanics. To understand the origin of the uncertainty principle in mathematical language, we can refer to the canonical commutation relation [x̂, p̂] ≡ x̂p̂ - p̂x̂ = iℏ, where x̂ and p̂ are operators representing the position and momentum, respectively, in a quantum system, ℏ is the reduced Planck constant, and i is the imaginary unit defined by i² = −1. Its counterpart in classical physics is the Poisson bracket {x, p} = 1, where x and p are classical observables, i.e., number-valued quantities. It's worth noting that each observable in classical mechanics corresponds to an operator in quantum mechanics. The eigenvalue of an operator acting on the quantum state corresponds to the measured values of the observable. Examples of observables are position, momentum, kinetic energy, total energy, angular momentum, etc. If x̂ and p̂ are both numbers, they commute so the product does not depend on the order in which x̂ and p̂ are multiplied, x̂p̂ - p̂x̂ = 0, which contradicts the commutation relation. Nevertheless, this contradiction can be interpreted as the limiting case where quantum mechanics becomes classical mechanics when the Planck constant tends to zero. This means that, at the macroscopic level, quantum effects can often be considered negligible, and classical theory can be used instead.

Yet in the quantum world, x̂ and p̂ are linear operators and they cannot be expressed into numbers simultaneously. Again, the commutator [x̂, p̂] = x̂p̂ - p̂x̂ can be used to clarify this concept. The commutator itself is an operator. Let x and p be the observables, i.e., the eigenvalues of x̂ and p̂ respectively. We then have x̂ ψ = x ψ and p̂ ψ = p ψ. Acting the commutator (an operator indeed) on a quantum state (which can be thought of as measuring an observable in a classical sense), say ψ, gives x̂p̂ ψ - p̂x̂ ψ = x(p ψ) - p(x ψ) = 0. This implies the measurement of x and p doesn’t shift the state of the system, i.e., initial and final states are the same. However, the canonical commutator relation tells us that if we measure x, then measure p, the result for p would differ from directly measuring p. This’s because the first measurement for x has caused the system to “collapse.”

In other words, positions and momentum do not actually have values in the conventional sense simultaneously. The more we confine one observable to be “almost a number” (by performing a measurement), the less the other observable will behave like a number, and vice versa. This does not mean that the quantum system cannot have exact positions and momentum at all times. In fact, they do exist, but they cannot be measured simultaneously in precise numbers. Interpreting this through a probabilistic view, the particle indeed exists in many locations at once.

We can use our knowledge to predict the probabilities of where and when the particle will finally interact with a classical apparatus. It's worth noting that in classical mechanics, both position and momentum observables can be precisely measured simultaneously because they are represented by numbers. We can describe this phenomenon in a neat way using mathematics. The state of a quantum system is given by a wavefunction ψ(x, t), which is a complex function. The probability of finding the particle located between x and x+dx is given by P(x, t)dx ≡ |ψ(x, t)|² dx, where | • | represents the complex modulus used to define the probability density P(•, t). This means that if you start with a cloud of identical particles in the system, each with the same wavefunction ψ(x, t), and you measure the position of each particle at time t, you will obtain different results for different particles in the cloud.

ψ(x, t) can be seen as the wavefunction in the position (x) space, while its Fourier transform Φ(k, t) is in the momentum (k) space. In a similar fashion, the probability of finding the particle with momentum between k and k+dk is given by Q(k, t)dx ≡ |Φ(k, t)|² dk. ψ(x, t) and Φ(k, t) just two representations of the wavefunction, i.e., the state of the same quantum system, in different spaces. We can recover the certainty principle mathematically through the connection of these two spaces: when ψ becomes a very narrow spike in the position space (indicating a high likelihood of finding a particle around the spike), Φ becomes inversely stretched out, almost flat, in the momentum space (indicating a very low but equal probability of having a momentum between k and k+dk along the momentum space).

Suppose we conduct an experiment repeatedly, using the same set of initial conditions, and tally the occurrences of different outcomes. If the outcome is always the same, it means that the system is deterministic, and the initial conditions fully describe it. If the outcome is not always the same, it indicates that the system is not deterministic in nature, or the set of initial conditions are insufficient to fully describe the system. Let’s say we toss a coin. We should be able to predict the outcome fairly accurately if we had a detailed description of the coin, its initial position, and motion, i.e., a full description of the setup in the experiment. However, under normal circumstances, we don't have access to that information, so our description of the system's initial conditions is incomplete. Quantum physics appears to be a “probabilistic” science because initial conditions cannot be purely given in terms of "classical observables," which are quantities that can be measured. As a result, we’re unable to predict the outcome of an experiment with certainty.

Schrödinger's equation is a linear PDE that governs the wavefunction of a quantum system. It is often regarded as the quantum counterpart to Newton's second law of motion, F = ma. In my viewpoint, it is a deterministic equation per se, much like a system without any built-in state simulator. However, certain properties of the system are not fully accessible through observation. Therefore, the probabilistic nature of quantum physics arises from the relationship between initial conditions and outcomes when repeatedly observing a large number of times.

▶️ 陳志宏博士對談《奧本海默》（一）量子力學的顛覆性，究竟在於哪裏？ https://www.youtube.com/watch?v=aMPMKrNEC2I

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https://www.youtube.com/watch?v=Qfpeawzzbkc