Understanding Quantum Mechanics #1: It’s not about discreteness

This must be one of the most common misunderstandings about quantum mechanics, that quantum mechanics is about making things discrete. But is an understandable misunderstanding because the word “quantum” suggests that quantum mechanics is about small amounts of something. Indeed, if you ask Google for the meaning of quantum, it offers the definition “a discrete quantity of energy proportional in magnitude to the frequency of the radiation it represents.” Problem is that just because energy is proportional to frequency does not mean it is discrete. In fact, in general it is not.

[joevignoloru]

Its wonderful how Sabine calls this a "understandable misunderstanding". This is contrary to the typical practice by some on YouTube of calling people "idiots" and "morons" whenever they are wrong about something. Misunderstanding something does not mean someone has low intelligence. It just means that their knowledge base on that particular subject needs some enhancement.

[ComplexVariables]

I did know that there was even a problem with my understanding until watching this; thank you!

[balasubr]

Clarifying this “understandable misunderstanding” is very appropriate when applying quantum mechanics to social science. Even mathematicians and statisticians fail to appreciate this critical distinction. Sabine’s effort in making this video is worth many saved hours of discussions. Thanks Professor.

[KpxUrz]

Hosssenfelder is a superb teacher. Her informative videos are filled with comprehensible facts, and none of the fluff that most videos are filled with. Really outstanding.

[meahoola]

Thank You Sabine, this is exactly the level I need, you fill a gap in the available presentations of science on YouTube.
Most popular presentations way too trivial, most lectures too formal.

[DanielL]

Sabine, thank-you so much for the most 'coherent', non-trivial explanations of a field of science that has been convoluted (especially for consumption by the general public, not to mention students) by the misuse of language. I've been waiting for you to show up for years. Please, keep producing these videos but not discretely.

[philochristos]

That last sentence was what I was wanting to know the whole time--whether quantizing gravity means that space is quantized.

[stefanolacchin]

Thank you Sabine. Today I learned (somewhere else) that particles are not really "particles", then that quantum mechanics doesn't really imply a quantised reality. Every time I feel like my mind is grasping some basic notion about the nature of reality, I discover it's just a gross simplification and I'm back at square one. As a layman lacking the mathematical tools to really wrap my mind around these concepts, I want to thank you for your rigorous yet comprehensible insights. Your channel is priceless.

[noeckel]

Very true! In fact, the operators representing position and momentum in Heisenberg's uncertainty relation have a continuous spectrum themselves. Observables that have a discrete spectrum (more precisely trace-class operators) cannot obey a Heisenberg-type uncertainty relation (with a constant lower bound for the uncertainty product).

[Nickelodeon]

I really appreciate this explanation. As an engineer I often deal with the continuous continuum of mechanics being modelled simply i.e. being broken down into something we can model e.g. Finite Element Analysis. In reality, the discretisation is not real, just a way to handle the information in a practical way. Real life is not FEM, but simply a model.

[maxtabmann]

What I have lerned long time ago is that one should not mix up the terms quantization and discretization. Discretization refers to spatial or temporal coordinates that are not discrete in nature but they may discretized on a computer for calculation purposes. Quantization refers to charge appearing in quantized units. Therefore, momentum and energy, because they are Fourier conjugate to time and position, are naturally not discrete. Only in a bound system, frequencies or k-vectors can be discrete. This is true for both classical and quantum systems, for guitar strings as well as for energy levels, as frequency and energy are connected by Plancks constant. So the only thing really quantized is the charge or the spin.

[raminagrobis]

Yes, and no, Sabine :) Yes, you're absolutely right, quantum mechanics does not preclude a "free" particle from taking momentum values from a continuum (or "scale", as you refer to it) of such values. But the very definition of a system being quantized IS about it taking only discrete values. And this happens whenever a particle is bound to another or others - a pretty common situation, to say the least (!). That's the very essence not only of the word but of the concept as well, and it's what troubled Planck so much that he long thought the introduction of his constant (h) was a kind of "fudge factor" that had no real intrinsic meaning and reflected a lack of theoretical "classical" explanation rather than a new, fundamental and more accurate way to describe reality than Newtonian mechanics.
So, yes, quantum mechanics does not "discretize" (!) energy per se, but quantization of a system truly means constraining energy to acquire only discrete values. But I think you have been saying more or less the same thing. I just wanted to emphasize the important point that it IS about reducing a system to adopt only discrete values of momentum (or of energy, or related parameters) whenever we're dealing with an interaction between particles.

[davidschneide]

Even the textbook definition of "quantum" is confusing. How appropriate.

[Andrey.Balandin]

Sabine, my life has been a lie until now.

[robmorgan]

Wonderful content as usual!

I like to tackle the aporoach to understanding quantum from the historical perspective of thermodynamic and statistical mech: ie Plank black body and Einstein AB coefficients maybe the Gibbs paradox, then add in the statistical/algebraic constraints to recover the effects of commutation and ultimately the idea of Lie algebra and spinnors and wave mechanics to motivate and "derrive" the schrodenger/dirac eq's.

People usually don't "understand" QM because it's an example of a theory that places statistics and algebra/topology on equal footing with geometry.

This combined with the problem of integrating a conserved quantity into a theory (such as basis independent total probability aka the Bohrn rule) without an associated charge per Noether's theorem is difficult to comprehend... for me it was this realization, namely that the born rule places something akin to a Goldstone boson (aka something required for mathematical consistency but not DIRECTLY observable in reality: aka complex probabilistic wave functions that permit superposition... until you look for phase effects ala Aharanov Bohm and the EPR predictions and other entanglement effects) straight into the heart of the theory for no reason other than the fact that probability must be conserved (ie sum to 1).

Now it's possible to liberate one's self from the confusion by reflecting on the consequences that might result in a statistical theory from introducing a charge and it's associated field from an implicit rather than explicitly conserved quantity into the partition function (like the total probability or the "unfreezing" of degrees of freedom in your state space in the grand canonical partition function). This like any other poorly formulated constraint is essentially swept up into the entropy of the grand canonical partition function and the conjugate temperature/chemical potential that creates the psudo forces coupling "the system" to its own phase space and it's environment in unusual/unexpected or "unnatural" ways. You only notice the problem when your experiment reveals a heat capacity that exceeds your predictions. I'm sympathetic to the guys studying extra dimensions since this is very much analogous but the difference here is profound.

After all it was the ultra violet catastrophe, Gibbs parodox and other problems in thermodynamics that illuminated the path forward for the development of quantum mechanics. The string theorists and phenomenologists were trying to figure out WHERE to look next instead of searching for and attacking problems living in experimental data head on.

As you point out in your book, It's the "unnaturalness" of this statistical approach that tends to offend most physicists who prefer to work fully in the geometric picture...which mostly makes sense if you like Lagrangian dynamics, GR, or E&M.

The irony is statmech is very good at allowing you to basically set up your geometry a priori via the real constraints your experimental observations place on your partition fn (as pointed out by Jaynes)...and hence your Lagrangian... oh well. I've wasted years tilting at this particular windmill. Looking forward to seeing what's next!

[ddimin]

I found out that studying quantum informatics is a clearer and much faster way to understand what "quantum" means (compared to studying quantum mechanics). That's because everything is in finite dimensions, operators are just matrices, you can calculate everything by hand and thinking about bits and qubits is easier than thinking about particles. You still have all the quantum stuff: entanglement, teleportation, measurement, non-commuting operators, etc.

[arijitbose]

Outstanding explanation❤️❤️❤️❤️❤️❤️...No one can explain the basic concept of quantum theory like this method in long academic lectures.

[rv]

For me quantum mechanics is:
1) Linearity: can (essentially) sum states, that is "quantum superposition";
2) Probability theory done with complex numbers instead of positive reals, so there can be "destructive interference";
3) Noncommutativity of the algebra of observables, which forces the "uncertainty principle".

[davidwright]

Thanks Sabine! Clear and helpful, as always. The way one is taught physics - even in grad school! - is poor at understanding other people's misunderstanding of the course content! One is taught to solve problems, not think about them. Your series provides a very good counterbalance. I also enjoy your musical numbers. Lotte Lehmann, eat yer heart out! Likewise Edith Piaf.

[luudest]

Thank you for this video!

I always wondered why Quantum Mechanics is associated with discrete values.