In science, today’s breakthroughs quickly become tomorrow’s monotony. Many of us use quantum mechanics everyday, but we don’t always think about its paradigm-shifting consequences and its remaining unanswered questions.

There are many online lists stating the most remarkable facts of quantum mechanics, but they often don’t adequately distinguish between the *formalism* and the *interpretation* of quantum mechanics. In my opinion, it is somewhat disingenuous to present interpretations of quantum mechanics as being part of the formalism, though this line is not always clear. The many-worlds view of quantum mechanics is a prime example that often gets media coverage. Not only is this only an interpretation of quantum mechanics, but it does not even maintain a consensus within the scientific community.

Here is a list that attempts to discuss some of what I find to be some of the most remarkable consequences of quantum mechanics. Some of these items do require some interpretation, but for these points they are at least consensus viewpoints.

#### 1. The wavefunction is not a measurable quantity

Unlike in most other realms of physics (such as classical mechanics and general relativity), in quantum mechanics, one of the main quantities that physicists attempt to calculate cannot be directly measured. This is not because we don’t have adequate tools to do so. This is because it is not *possible* to do so. The wavefunction is a complex quantity, and as such, cannot be observed.

#### 2. Quantum mechanics makes probabilistic, not deterministic, predictions

Within the realm of quantum theory, it is only possible to predict the probability of an outcome. In that sense, one does not know the trajectory of a single particle in a Young’s two-slit setup for instance, but one can predict the statistical distribution of many particles on the screen behind the slits.

#### 3. Heisenberg uncertainty relations

This well-known theorem states that, for instance, one cannot measure the angular momentum of a particle in the *x-* and *y-* directions simultaneously without some inherent inaccuracy.

#### 4. Identical particles, spin-statistics theorem and quantum statistics

All electrons are made the same. All photons (of same frequency) are made the same. It turns out that there are two categories of identical particles in three dimensions, *bosons*, which are of integer spin and *fermions*, which are of half-integer spin. The properties of bosons allow for superlative low-temperature phenomena like Bose-Einstein condensation. The existence of fermions and the Pauli exclusion principle, on the other hand, make sure that your hand doesn’t go through a table when you put your hand on it!

#### 5. Non-locality, Entanglement, Bell’s Theorem

I’ve written about this on several occasions, but I will just say that quantum theory is inherently non-local. Einstein spent a pain-staking 15 years trying to make the theory of gravity local, only to see quantum mechanics pull out the rug from under is feet. See here for further details.

#### 6. The scalar and vector potentials have measurable consequences and Berry phases

In classical electrodynamics, the quantities that have measurable experimental consequences are the electric and magnetic fields. In quantum mechanics, the Aharonov-Bohm effect demonstrates that a change in the vector potential can have experimental consequences. It is important to note that *differences* in potentials are gauge-invariant.

If you think I’ve left anything off, please let me know!