Are Superposition, the Uncertainty Principle, and Duality the same?

[eow]

Are quantum superposition, Heisenberg uncertainty principle, wave/particle duality the same thing? Do they imply each other?

They all seem the same to me like the wave-like nature of photons -> superposition, and superposition means there's not definite position -> uncertainty principle.

Sorry if it's a dumb question. I have no formal physics education (trying to learn on my own).

 

[DrClaude]

No, they are different things.

First, there is no wave-particle duality. That's an outdated concept fro the beginnings of quantum mechanics. Quantum objects have properties that can't be reduced to simple classical concepts such as wave and particle.

Second, quantum superpositions come about because the the Schrödinger equation is linear: any linear combination of solutions is itself a solution to the Schrödinger equation. Therefore, quantum systems can be in superpositions of states, where is state is a solution to the time-independent Schrödinger equation. Note that there is nothing special about superpositions, since they correspond to different point of views. If a photon is left-circularly polarized, that's a single state, but also a superposition of linear polarizations along x and y. Likewise, a photon linearly polarized along x is in a superposition of left and right circular polarizations.

The Heisenberg uncertainty principle comes from the fact that different observables correspond to operators that do not commute. This means that both attributes can't be defined at the same time. For example, if an electron is in a spin state pointing along z, you can't say anything about its spin along x or y.

 

[lynn_esther16]

I'm a bit confused by the first part. Are you saying that the wave particle duality no longer exists? Because I thought that all things propogate like a wave and exchange energy like a particle? Are you saying that this is no longer the case and if so, what are you proposing is?

 

[PeroK]

Are you saying that this is no longer the case and if so, what are you proposing is?

He's proposing Quantum Mechanics, which resolved the issue of wave-particle duality about 100 years ago.

I have two undergraduate textbooks on QM. The first has one reference to wave-particle duality as a historical footnote. The second doesn't mention it at all.

This is in contrast to popular science sources, where wave-particle duality is a central theme.

 

[lynn_esther16]

Ohh I see,

Thank you.

Where can I get those textbooks?

 

[PeroK]

The first was is Griffiths:

https://www.abebooks.co.uk/servlet/...4LDnk0TXnkySQMp9AN5mcDlWMVVzc4uYaAvAIEALw_wcB

You can get the second edition for not much now the third edition is out.

 

[PeroK]

Ohh I see,

Thank you.

Where can I get those textbooks?

There are also these notes online. They are both excellent, but the second one is perhaps the most accessible introduction to QM:

http://www.damtp.cam.ac.uk/user/tong/quantum.html

http://physics.mq.edu.au/~jcresser/Phys304/Handouts/QuantumPhysicsNotes.pdf

 

[Sargon38]

Actually, there is a link between the 3 concepts that were asked by the TS almost 8 years ago.

The three notions ("wave particle duality", "quantum superposition" and "Heisenbergs' uncertainty relationships" are in fact tightly related to the FUNDAMENTAL ASSUMPTION of quantum theory that there are different incompatible sets of observations of nature. THAT is the fundamental idea behind quantum theory, that there are "incompatible ways of looking at nature".
As pointed out, it is only when quantum theory got a firmer ground that the different concepts fell together as different aspects of the same thing.

Essentially, one postulates in quantum theory that there are different "complete observations" of whatever is the subject of the quantum theory at hand, say M and N, and that to each complete set of observations correspond an exhaustive list of all possible outcomes: the possible outcomes of M are {m1, m2, ....} and the possible outcomes of N are {n1, n2, ....}. The fundamental hypothesis is that if one does a complete observation of the quantum system at hand M, and one has an outcome m_i, then it is impossible to know what would be the outcome if one did N. And vice versa: if one did a complete observation N, and had outcome n_j, then it is impossible to know the outcome if one does M. In other words, the fundamental hypothesis of quantum theory is that there is a way of observing the system, M, that doesn't allow us to know what would be the outcome if one observed the system with observations N. It is impossible to know "entirely" a system's state, that would give us the outcomes of observations M, and observations N. THAT is the core idea of quantum theory.

Now, mathematically, this is represented by having a Hilbert space of which a basis corresponds to each possible outcome of M, so to the entire set of possible outcomes {m1, m2....} corresponds a basis of Hilbert space, each basis vector representing a different outcome ; and by having a DIFFERENT basis in that same Hilbert space that corresponds to the possible outcomes of N. So to {n1, n2, ...} correspond also basis vectors of the Hilbert space, but they are "rotated" from those of M, in such a way that each basis vector corresponding to {n1, ...} has components of all basis vectors corresponding to {m1, ...}.

A typical notation of these basis vectors is $\{ |n_1> , |n_2> .... \}$ for the basis vectors corresponding to the observation N with possible results n1, n2 ,... respectively. We have another set of basis vectors $\{ |m_1> , |m_2> ... \}$ for the observation M with possible results m1, m2, ...

This is the "superposition principle" namely, that the basis vectors corresponding to the outcomes of N are linear superpositions of the basis vectors of the outcomes of M. The basis of N maps to the basis of M through a unitary operator.

One can construct "measurement operators" out of the basis vectors. One can construct a set of Hermitean operators corresponding to the complete measurement M (all the "numeric quantities" that make up an observation m1 for instance). They are simply diagonal operators in the corresponding basis (here, the basis corresponding to M), and have the numerical outcomes on their diagonals.
So to measurement M, correspond a set of Hermitean operators, say $H_M1, H_M2, ... H_M13$.

When we apply such an operator to one of the $| m_i >$ vectors, they turn out the same vector, with a coefficient, corresponding to the real-valued measurement value of that particular outcome and operator.

We can do the same of course for the observation N, and we will also have a set of Hermitean operators $H_N1 ... H_N5$

These operators will be diagonal in the basis corresponding to N.

But of course, the N-operators will not be diagonal in the basis of M, and the M-operators will not be diagonal in the basis of N. They will NOT COMMUTE.

$H_M1$ commutes of course with $H_M3$ because they are diagonal in the same basis. And $H_N3$ commutes with $H_N4$ because they too, are diagonal in the same basis. But $H_M1$ doesn't commute with $H_N3$, because they are diagonal in different bases.

That's the non-commutativity between incompatible observations. It was BUILT IN. The whole of quantum theory was on purpose constructed to obtain this.

One can show that the "amount" of superposed basis vectors of the M basis, needed to construct a basis vector of the N basis, is related to this non-commutativity. That is Heisenberg's uncertainty relationship: you need "many possible outcomes" of M in order to make a "certain outcome" of N.

Finally, the "wave particle" duality is an application of all this to the quantum mechanics of a SINGLE PARTICLE.

One "complete observation" is the position in space of the particle which we will note by the observation X. The postulated incompatible observation is the momentum of the particle, which we will denote by the observation P.

In 3-dim space, X can be represented by 3 quantities: x, y and z. We have 3 Hermitean operators corresponding to that, $H_x, H_y, H_z$.

We also have observation P to be represented by 3 quantities, px, py and pz. We also have 3 Hermitean operators corresponding to that $H_px, H_py, H_pz$

It turns out that a basis vector corresponding to a position $| x, y, z >$ can be written as a superposition of all the momentum basis vectors $| px, py, pz >$ and vice versa.

So a specific basis vector of momentum will have contributions of all position basis vectors.

Now, there's a way to express the coefficients of each of these basis vectors, and that will be a coefficient per position basis vector. But that's nothing else but a coefficient corresponding to every position (x, y, z).

We can write that as a "function of x, y, z" $\Psi(x,y,z)$. But that looks like a complex function "in space".

That was the "wave" in "wave mechanics", and the "wave particle duality" came about because according to the observations we did on a particle, we had it as a "position" (which we called "particle") or as a "momentum" which corresponded to a superposition of "positions", also called "a wave".

The "wave particle" duality was nothing else but the postulated incompatibility of the observation of momentum and of position, applied to a system of a single particle.

If you had measured the position, you had no idea about the momentum ; if you had measured the momentum, you had no idea about the position. That's Heisenberg's uncertainty.

Of course, once you have two complete sets of incompatible measurements, M and N (or X and P in our single particle case), you can find still other incompatible sets of measurements. They have still other basis vectors in the same Hilbert space, which are superpositions of the other basis vectors.

 

[DaveE]

Wave-Particle Duality is really just a crude description of similarity to simpler classical physics concepts. It's not a real thing. Wave-Particle Duality is like describing a lizard as having Snake-Mouse duality; no, it's a lizard.

 

[Sargon38]

Wave-Particle Duality is really just a crude description of similarity to simpler classical physics concepts. It's not a real thing. Wave-Particle Duality is like describing a lizard as having Snake-Mouse duality; no, it's a lizard.

To me, what was called the wave-particle duality was simply the single-particle system, where position states were "particles" and momentum or other non-position states were "waves", that is to say, superpositions of position states.
It was the expression of a different basis (say, the momentum basis) in the position basis. This becomes a "wave" (that is to say, a complex scalar function in space) if one has a single particle system. $\psi(x,y,z)$. Of course this breaks down if one has more than one particle in the system.

After all, and with some caveats, what's called the classical EM field can be seen as the single photon wavefunction (I'll get beaten up by some hard core theorists here). It is in the same spirit that one can think of the "wave" in the wave-particle duality.

 

[weirdoguy]

To me, what was called the wave-particle duality was simply the single-particle system, where position states were "particles" and momentum or other non-position states were "waves"

Can you give any reference for that?

 

[Sargon38]

Can you give any reference for that?

Well, this is what you can understand, for instance, if you look at p 59 of the first volume of Messiah, section 8: "universal Character of the Wave-Corpuscle Duality".
I cite:
"they appear under two apparently irreconcilable aspects, the wave aspect on the one hand, exhibiting the superposition property characteristic of waves, and the corpuscular aspect on the other hand, namely LOCALIZED (emphasis mine) grains of energy and momentum".

The next part, which is on that same page, is then "II The Schroedinger equation" where the author goes on introducing the single-particle Schroedinger equation, as a "wave equation".

Of course, I know that historically, the "wave particle duality" was much more shrouded in "mystery" than it is once you have a firmer theoretical basis which was given by the Hilbert space formulation of quantum theory, but in the ground, this is what it came down to, and is what inspired Schroedinger to write a "wave equation", which was later understood in more algebraic terms, the fact that the "wave" of a single-particle system was just a superposition of position states. So that helps you to "look back" and to understand that what was seemingly a dualistic aspect of existence was nothing else but quantum states seen in different bases.

I'm not saying that it was immediately UNDERSTOOD this way. I'm saying that this is how one can, a posteriori, understand what was in fact the "wave particle duality".

 

[gentzen]

is what inspired Schroedinger to write a "wave equation", which was later understood in more algebraic terms

The official version how that happened is related to de Broglie and somebody (Debye?) asking for a corresponding wave equation. When you study the history related to Sommerfeld, you learn that Wolfgang Pauli had a close correspondence with Schrödinger, and kept him up-to-date of Pauli‘s own progress. (And hence also of Heisenberg‘s progress…) Without that knowledge, it would be hard to understand why Schrödinger didn‘t publish the Klein-Gordon equation instead of the Schrödinger equation.