Sovrapposizione degli stati energetici di un elettrone in un atomo

Dialogo tra Susana Cooper e Daneel Olivar, appassionati studenti di fisica.

Erano le sei di sera, quando, ormai stanco dopo un pomeriggio di studio in biblioteca, mi stavo preparando per rientrare in appartamento. In quel momento entrò nell’aula Susan (Cooper) e mi si avvicinò velocemente; mi guardò dritto negli occhi e sotto voce mi chiese se avevo 10 minuti da dedicarle. Quindi mi condusse attraverso diversi corridoi in un’aula che non avevo mai visto prima e una volta lì mi disse che voleva parlarmi di un problema di fisica che credeva di avere finalmente risolto. Non che questo problema non fosse già stato risolto da moltissimo tempo, ma quando si imbatteva in un argomento che la affascinava, le piaceva provare a venirne a capo da sola.

SUSAN: qualche tempo fa ho seguito una lezione di meccanica quantistica relativamente alla quantizzazione dell’energia di un elettrone in orbita intorno ad un nucleo atomico.

DANEEL: ok Susan. L’ho seguita anch’io e mi era sembrata molto chiara oltre che interessante.

SUSAN: ricordi cosa accadde verso la fine?

DANEEL: non ricordo niente di particolare, ad essere sincero.

SUSAN: ad un certo punto uno studente ha chiesto se l’elettrone potesse trovarsi in una sovrapposizione di stati di energia. E il professore ci ha invitati a provare a dare una risposta al quesito.

Ci fu una brevissima pausa.

DANEEL: capisco. E naturalmente tu, non hai potuto evitare di lanciarti in questa sfida.

SUSAN: ormai mi conosci Daneel.

Ci scambiammo uno sguardo di intesa con un impercettibile sorriso.

DANEEL: ora sono curioso e tutt’orecchi.

SUSAN: non sono sicurissima che la soluzione che ho trovato sia corretta, ho ancora qualche dubbio, per questo vorrei una tua opinione.

DANEEL: allora forza!

SUSAN : per prima cosa, l’equazione di Schrödinger non vieta in alcun modo questa possibilità, essendo una equazione lineare: 2 eigenstate dell’energia corrispondono a 2 onde stazionarie, entrambe soluzioni dell’equazione, quindi una qualunque loro sovrapposizione ne sarà ancora una soluzione.

Il problema riguarda un altro aspetto. In generale, qualunque corpo ha un’energia data dalla famosa equazione

E = \sqrt{p^2 c^2 + {m_0}^2 c^4}

dove m_0 è la massa a riposo del corpo, p la sua quantità di moto relativistica e c la velocità della luce; essa sostituisce la vecchia energia cinetica E_k = \frac{1}{2}m_0v^2. Ora, immaginiamo che il corpo in questione sia un atomo di idrogeno e che esso non sia immerso in alcun campo potenziale, in altri termini che non sia sottoposto ad alcuna forza esterna. Immaginiamo poi che il suo elettrone sia in una sovrapposizione di stati di energia. Siccome l’energia dell’atomo include anche quella dell’elettrone, allora anche l’atomo si trova in una sovrapposizione di stati di energia. Ma tale sovrapposizione può essere attribuita ad una eventuale sovrapposizione di stati legati alla sua quantità di moto?

DANEEL: hmmm…a prima vista direi di no.

SUSAN: anch’io la penso così. L’orbitale o eigenstate energetico assunto dall’elettrone riguarda unicamente la relazione tra nucleo ed esso, nulla ha a che fare con il moto nel suo complesso dell’atomo. Se un atomo è fermo all’istante t e un fotone viene assorbito dal suo elettrone, che quindi si sposta su un orbitale con maggiore energia, questo incremento vale tanto per il sistema di riferimento originario rispetto a cui l’atomo era fermo, quanto per un riferimento rispetto a cui l’atomo continui a risultare fermo! E’ un incremento intrinseco.

DANEEL: capisco.

SUSAN: ma allora tale sovrapposizione dovrebbe essere attribuita alla sua energia a riposo, cioè m_0c^2. In altre parole la sua stessa massa a riposo dovrebbe trovarsi in una sovrapposizione di stati! Ma nell’equazione di Schrödinger la massa è una ed una sola, e lo è anche nella sua versione relativistica per particelle con spin 0, l’equazione di Klein Gordon.

Ci fu una pausa di alcuni secondi. Quindi le dissi…

DANEEL: se il tuo ragionamento è corretto, allora un elettrone non può trovarsi mai in una sovrapposizione di stati energetici.

SUSAN: esatto!

DANEEL: E che mi dice del caso vi sia più di un elettrone intorno al nucleo?

SUSAN: in tal caso la funzione d’onda della nuvola di elettroni è data dal prodotto tensore delle funzioni d’onda dei singoli elettroni. Quindi basta che un solo elettrone sia contemporaneamente in più di uno stato di energia perché la nuvola di elettroni e quindi l’atomo nella sia interezza si trovi in una sovrapposizione di stati di energia, contraddicendo il fatto che l’energia intrinseca deve essere una ed una sola, come detto prima.

Spacetime Loops in Quantum Eraser experiment

When considering the relationship between time and quantum mechanics, particularly interesting is the quantum eraser proposed by Scully and Drühl. For an outline of the experiment, you can see the figure below.

fig. 1

The photon passing through the lens and going to the “interference screen” is called the signal photon, the twin photon (entangled) is called the idler photon.

Now suppose to move all detectors (A, B, C and D) and associated instruments to a very high distance, e.g. 10 light years. Suppose also that photon gun produces one photon at a time. Which figure do you expect to see on the screen after a lot of photons has been fired by the photon gun?  As well explained by Brian you will see a figure without interference, because half of the idler photons are detected by A or B detectors . But if you consider only the signal photons associated to the idler detected by C and D detectors, you will discover they have formed an interference figure on the screen. The incredible thing is that this happens independently of the distance of detectors from the BBO crystal, or, in other words, independently of the time when the idler photons are detected.

But now suppose that the scientists in the laboratory at 10 light years from photon gun, have the possibility to modify the detectors configuration in one of 2 ways as shown below in fig. 2 and fig. 3:

fig. 2

fig. 3

Now suppose also that before going away from Earth they have established with scientist of laboratory with photon gun that every 6 minutes photon gun fires a beam of photons one at a time, separated by half of a second and that every beam lasts for 5 minutes. Their clocks are perfectly synchronized and we suppose that gravity effects on time are negligible. So in the minute that separates 2 subsequent beams of photons, scientist in the laboratory with detectors can change the detectors configuration in one of the two shown configurations in fig. 2 and 3.

When the configuration is that shown in fig. 2 (detectors A and B), 10 years before no interference figure has formed on the screen in laboratory on the Earth, Viceversa when configuration is that shown in fig. 3 (detectors C and D), 10 years before interference figure must have formed on the screen! 

From now on we will indicate with A the laboratory on Earth and with B that placed at 10 light years with the detectors.

Amazing consequences.

The consequences of all this are incredible!

From the point of view of scientists in laboratory A, they are able to predict the behavior of scientists in laboratory B! If they now observe the formation of an interference figure, it means that in 10 years from now the detectors in laboratory B will be in the configuration of fig. 3, while if they do not observe an interference figure then they will necessarily be in the configuration shown in fig. 2..

Let’s now put ourselves in the shoes of the scientists of laboratory B: they can determine with their current behavior, switching detectors configuration from that of fig. 2 to that of fig. 3, the formation or not of the interference figure in laboratory A, 10 years before!

They could use this mechanism to communicate messages to laboratory A in the past, assigning, e.g. the value 0 in the absence of the interference figure and the value 1 in its presence!

This is really incredible, because it seems to violate the cause-effect relationship that wants the cause precedes its effect!

Paradox.

We modify the scheme of figure 1, so that in laboratory B there is not a photon detector but a system of mirrors positioned in such a way as to send back the idler photons coming from laboratory A. We also modify laboratory A so that the 2 possible detectors configuration (A+B or C+D) are placed in it on the trajectory of idler photons reflected by mirrors in laboratory B. So scientist in laboratory A can choose to observe idler photons through detectors in configuration A+B or C+D.

Now scientists from laboratory A can cause the interference pattern to form or not for each beam of photons emitted by the gun, by placing the associated detectors configuration 20 years later (see fig. 4)! But this puts them in a position to violate the consequences of quantum mechanics: in fact they can always decide to position the detector in the opposite way to what is required. For example, if at a given moment they observe an interference pattern, 20 years later they can decide to choose detectors configuration as shown in fig. 2, giving rise to a paradox (the figure they should have observed 20 years earlier, had to be without the typical pattern of interference).

fig. 4

There is another situation that causes the paradox. Look at figure 5 below.

fig. 5

In A1 the laboratory A produces a beam of signal photons and associated idler photon sent to laboratory B and received in B1, where the idler photons can be detected as shown in fig. 2 (A+B) or can be detected as shown in fig. 3 (C+D) through the quantum eraser configuration. The same thing happens between B2 and A2, with reversed parts.

When laboratory A is in A2, it knows which kind of figure formed in its interference screen (interference pattern or not) and so through A2-B2 connection it can “cause” the opposite figure in laboratory B at point B2. And so laboratory B can cause that figure in A1 through B2, where it detects the idler photons arriving from A1. So we can have again the contradiction!

An attempt at resolution.

In this chapter I suggest a solution to the paradox just seen, supposing we can avoid to consider gravity effects, and so treat only inertial reference systems related to each other through special relativity transformation laws (Lorentz transformations).

I want to consider 2 important phenomena of Physics: simultaneity and entanglement. We know that entanglement happens between 2 particles, started from the same origin in a certain instant and observed at the same time. So entanglement requires that we measure the same physical quantity of the 2 particles at the same instant!

But we know that simultaneity of 2 events is not absolute.

Consider the figure below, where the trajectories of 2 entangled particles emitted from a source positioned at the origin of the reference system shown, are represented in the Minkowsky space time.

fig. 6

Event A, in the trajectory of particle 2, can be in a simultaneity relationship with all events between B and C belonging to particle 1 trajectory, because for any event inside the interval BC (we can call it H), an opportune reference system can exist from whose point of view event A and H are simultaneous. 

In the case H is one of the 2 extremes (B and C), the reference system from whose point of view the 2 considered events are simultaneous, is that attached to the photon going from B to A or from A to C respectively.

If we measure particle 2 in A (obviously we are measuring a specific observable of particle 2) what happens to particle 1? Where does its wave function collapse between B and C? There is only one possibility: it must collapse in B and it cannot change from B to C, otherwise there would exist at least one reference system from whose point of view the 2 particles wouldn’t be entangled. We’ll come back on this later.

A new type of simultaneity

After these considerations we can introduce a new type of simultaneity, which I’ll call absolute simultaneity (abbreviated as a-simultaneity), defined as follow: 2 events are in an a-simultaneity relationship if an opportune inertial reference system can be imagined, with respect to which they are simultaneous.

With this definition, we infer that all events belonging to the trajectories of 2 photons starting from the same origin at the same time, are in an a-simultaneity relationship.

For simplicity, from now on, we’ll say that 2 events are a-simultaneous meaning that they are in a-simultaneity relationship.

Direct Paths

Now I want to investigate which are the simplest paths through which information can travel.

First we need to introduce the concept of “direct path” between 2 events A and B: it is the shortest path from A to B, extending the classic Euclidean distance of a 3 dimensional space, to the 4 dimensional space time, in which we define the 4th coordinate as T=ct (where t is the reference system time and c the speed of light). With this definition we can introduce 5 types of “direct path”:

  1. a path from past to future, abbreviated to PF
  2. a path from future to past, abbreviated to FP
  3. a path along the geodesic trajectory of a photon from past to future: abbreviated to PHPF
  4. a path along the geodesic  trajectory of a photon from future to past: abbreviated to PHFP
  5. a path connecting 2 a-simultaneous events, abbreviated to AS

Direct paths PHPF and PHFP are special cases of AS path, but we need to specify them as you will understand in a while. Moreover we use the past-to-future and future-to-past concepts to define them because, although they are a-simultaneous path, for any reference system excluded that of an hypothetical photon, they “move” from past to future and in the opposite direction respectively.

Below a representation of all possible direct paths.

fig. 7

Some of these are the basic types of path through which information can travel, others are forbidden paths.

The basic types of direct path through which Information can travel are:

  • PF: this is obvious;
  • PHPF: this is also obvious because we use electromagnetic signals to transmit information;
  • PHFP: this is the way information travels in the case of entanglement;

In all the other cases information cannot travel directly. 

In the case of AS path, it can travel through the composition of:

  • a PHFP with a PF
  • or a PHFP with a PHPF

The FP direct path for information is not admitted, but information can travel through the composition (for example) of 2 PHFP!

Now we introduce a new rule of Nature which is about loops of information, where a loop is a trajectory where information starts from a point (belonging to it) and through a sequence of basic direct paths returns to the starting point. 

It says: “No information loops in spacetime, formed by basic admitted direct paths, are possible”.

And now, with this new rule we can try to solve the 2 paradoxes introduced before and we can try to have a deeper view on entanglement.

Solving Paradox of fig. 4

Now we can also try to solve the paradox described through fig. 4, which is reproduced below:

First we need to understand where information starts. There only 2 possibilities: A or C. But we know that the figure formed on the screen in A, is a consequence of the detectors configuration in C: so C is where information starts! And now the question: does it travel through a loop made with allowed direct paths? Yes! The loop is formed by:

  1. C-B: which is a PHFP path
  2. B-A: which is a PHFP path too
  3. A-C: which is a PF path

But this violates the rule postulated about information loops. So the Paradox is not possible if we accept this rule.

Solving Paradox of fig. 5

For clarity the paradox is reproduced below (same as fig.5):

Where information starts? As for the previous case, it starts in A2, where scientists of laboratory A decide detectors configuration. And as before we see there is a loop:

  1. A2-B2: PHFP path
  2. B2-B1: PF path
  3. B1-A1: PHFP path
  4. A1-A2: PF path

And as before, if we accept the new rule about information loops, it cannot happen!

How Nature avoid loops?

What happens if there’s a loop as in the case of the 2 paradoxes? I think that simply Nature avoids the loop, breaking the connection between the idler photons and the associated signal photons, so that these always form an interference pattern on the screen. No information will go through the PHFP paths.

A more general case

We now consider the case illustrated in the figure below, where B-M and M-A are PHFP direct paths and A-B is a PF path which can be used by laboratory A to send information to laboratory B. 

fig. 8

Here the problem is that laboratory A could send information about the pattern formed on its screen by signal photons, or could not. This case includes the previous two, considering that laboratory A could forget information about the pattern formed by signal photons! Sending information in those cases corresponds to keep it.

How can Nature know if laboratory A sends information or not after having seen the pattern formed by the signal photons? Remember that the type of pattern formed by signal photons happens before the information is sent to B.

I think the only possibility is to relax the rule about information loops in this way: “No information loops in spacetime, formed by possible basic admitted direct paths, are possible”.

Possible means that it is really possible that information travels on that path. If the instruments of lab. A sending information are broken, Nature “knows” that there is no possibility that information reaches the lab. B on time. Quantum mechanics has taught us that possibilities are real. All possible evolutions of a system are real, even if only one is observed, When laboratory A sends the idler photons, Nature says: “is its state compatible with the possibility that it sends information to laboratory B?”. And if the possibility of that is not 0 then it must break the loop, always in the same way, breaking the correlation between idler and signal photons: only interference patterns will be observed on the screen in lab. A.

To be clearer we consider some examples.

Suppose that the device used in laboratory A to decide whether sending or not information to laboratory B about the pattern formed by the signal photons, is based on quantum mechanics: for example and head or tales quantum coin, practically a quantum bit with state 1/sqrt(2)|0> + 1/sqrt(2)|1>. If the observation is head then information will be sent, otherwise it will not. Nature cannot rules out the possibility that head will be the result, so it has to break the loop as explained before.

Suppose now that the device used for deciding whether sending or not information to laboratory B, is a classical stochastic head or tails coin (in practice a real coin). Because it “does not obey quantum laws”, in the sense that its behaviour is practically deterministic, Nature knows with precision which face of the coin will be the result, so only in the case that the head will be the result it will break the loop.

And what will happen if the “device” used to decide whether sending or not information is a human being? Because a human being can always decide to make fun of Nature (and its laws) the loop will be always broken.

Maybe this is proof that the human brain works (not only but also) as a quantum system!

On Entanglement

In the case of 2 entangled particles, which are measured at the same time with respect to an appropriate reference system, if we choose the observation of one of them as the starting point of information (although this information is not usable to send messages to anyone), we see that it travels on a loop. This seems to violate the rule about information loops in spacetime, but it’s not the case, because there is a symmetry (a reflection symmetry) between the 2 particles; there isn’t one source of information, both the observations of the 2 particles cause information travelling. There is not a loop of information but 2 symmetric paths starting from 2 symmetric origins.

About the speed of light.

Another interesting consideration concerns the speed of photons. We know that in special relativity the fundamental postulate is that the speed of light is constant and does not vary passing from one inertial reference to another. 

Let’s imagine for a moment that the scientists of the 2 laboratories know nothing of this postulate and suppose that the scientists of laboratory A wonder if it is possible to warn of the result of a sequence of photons the scientists of laboratory B before this is reached by the sequence of idler photons. If this were possible, the scientists of laboratory B could decide to position the detectors in the opposite configuration with respect to the figure that emerged on the screen of laboratory A, creating an evident contradiction in quantum mechanics. They know that quantum mechanics is correct and therefore deduce that it is not possible to transmit information that travels faster than the photons themselves. but still being tied to the classical vision of space and time, they imagine a stratagem: a spaceship at very high speed will pass in correspondence with laboratory A just when the sequence of photons formed the figure on the screen, and recovered this information it will transmit it to the laboratory B through an electromagnetic signal (photons); as they consider the Galilean transformations valid, they deduce that the photons emitted by the spaceship will have higher speeds than the idler ones coming from the laboratory, therefore laboratory B will be informed on the figure appeared on the screen before receiving the photons idler themselves. As before then the scientists of the lab. B could change detectors configuration in such a way to contradict what is foreseen by quantum mechanics. Once again they know that this is not possible, so they deduce that the photons emitted by the spacecraft cannot travel faster than those coming out of the lab. A. And since one would not understand why they should travel slower, they deduce that the speed of photons does not depend on the speed of their source. At the end they come to the conclusion that light must have the same speed for any observer!

It therefore seems possible to deduce the postulate on the constancy of the speed of light for any inertial reference, starting from quantum mechanics.

Massless Particles: why they must move at the speed of light

Relativistic-Quantum approach.

From the very beginning of my courses of Physics as an hobbyist, I was wondering why a particle of mass = 0 has to move to the speed of light. Why cannot it move at a lower speed?
Finally after reviewing Professor Barton Zwiebach’s first lessons of his course in the 2016 about Quantum Mechanics (available on OpenCourseWare site) and being inspired by it, I think I have found the answer

First I started from the relativistic energy equation:

and if m0= 0, then E = pc.

For simplicity I will limit this reasoning to the case of a particle moving in one dimension. For any particle moving freely (and this is the case, because no force can act on a massless particle) the wave function associated can be expressed as the sum of momentum eigenstates, each of type:

where I ignore the complex amplitude for simplicity and k and ω are associated with momentum and energy respectively by the well known relationships:

Which is the speed of this component? It’s as follows:

This is true independently of k and ω, so the entire wave packet associated to the particle moves with speed c.

Also reasoning in terms of group velocity, we come obviously to the same conclusions:

Pure relativistic approach

This is the most simple approach. The energy of such a particle would be:
E = pc
If we suppose that it has a speed v < c, then we could choose a reference frame in which the particle has speed zero. So in that frame its momentum would be null and consequently its energy too (because E=pc). In that reference frame that particle would not exist! But how can a particle not exist in a frame and exist instead in another? It’s not possible.
Equivalently if a particle exists for an observer it exists for any other.

An intuitive approach.

Suppose that a particle with a very very small mass appears in the space at an instant; the force needed to accelerate so mutch to take it at a velocity next to speed of light must not be so great. Obviously there is the limit of speed of light and, as its speed approaches that of light, its relativistic mass increases so that it cannot overcome it.
But if we choose a mass gradually lower and approaching zero, the force necessary to bring it ever closer to the speed of light, allowing it to maintain a relativistic mass however small, is gradually less and less.
The same considerations can be applied to the interval of time during which the force is present.
So, if the mass of the particle approaches zero, the force required to push it at speeds close to that of light and the time interval in which this force must be present, go down and approach zero themselves.

We know that in the vacuum, as predicted by Quantum Mechanics, particles and antiparticles appear continuously, so they would be what is needed to boost our particle with an infinitesimal mass!

The objection to this reasoning is that a particle of zero mass could not be pushed by any force. My answer is: probably Nature works in a continuous way…

Momentum Eigenstates

For simplicity I consider only the one-dimensional case.
Why are momentum eigenstates as follows? 

fig. 1

When we ask the same question about position eigenstates |x> we have no doubts:

fig. 2

but it’s not so obvious why momentum eigenstates are in the form of fig.1.
Leaving out the constant portion of them (fig. 1), we want to understand why they must be in the form:

fig. 3

So we have got a momentum eigenvalue p and we want to deduce its associated eigenstate form in the |x> base (as a function of x).

First of all: can its magnitude depend on position? No! Because otherwise, it would mean that, chosen p, some positions are more special than others. But momentum is logically independent of position.

So it must have constant magnitude over all positions, from – infinity to + infinity.

Could it be real, or equivalently, could its phase be constant? If so, we wouldn’t have any way to distinguish momentum eigenstates!
Can phase be a function of x different from that expressed in fig. 3 (phase = kx)?

First of all, the eigenstate function must be periodic, otherwise we wouldn’t have any information (independent of position ) to associate to momentum p: phase periodicity is the only possible information that is not dependent on position.

The simplest function with this characteristic is that of fig.3.

But you could answer back: why not a different periodic function, not regular within its period as that of fig. 3? in fact it seems the only requirement should be that the opposite momentum eigenstate phase should satisfy relation (to be specular):

fig. 4

The reason is, it is the only function that is position independent also locally: 2 observers close to each other, must see the same function always around and near them (net of the initial phase, which has no information). 

So if we ask for strong independence from the position (both on global and local scale) momentum eigenstate must be in the form expressed in fig. 3!

In conclusion, we have not only understood why momentum eigenstates are in the form of fig. 3, but we have also an intuitive explanation of why they must be complex! It’s the simplest way momentum eigenstates can be independent of position.

And if they can be complex, because momentum is not more special than any other observable, any other physical observable eigenstates can be complex functions.

On Hermitian Operators

The second postulate of quantum mechanics states: “To every observable in classical mechanics corresponds a linear and Hermitian operator in quantum mechanics“. The third then continues entering the details of the relationship between this operator and the associated observable, and finally on how the original wave function changes following the measurement of the observable,

The first time I came across this relationship, I had the feeling of being in front of something magical. How could observables be associated only with hermitian operators?


Certainly the Hermitian operators have only real eigenvalues ​​and this is necessary if we want these eigenvalues ​​to be the measurable values ​​for the observable. But how can you be sure that this operator exists for any observable?
Things became clearer to me by reading the first chapters of the book “Principles of Quantum Mechanics” by professor R. Shankar, in which the author deals with the case of a particle that can only move along an x ​​axis. At each position x we ​​naturally associate a ket |x> which (being in the continuous case) can be imagined as a Dirac pulse in x itself.

More precisely, the following relationships apply:

Any wave function can be expressed as a function of x and we know that the probability of finding the particle in the interval [x, x + dx] is equal to

We also know that

can be expressed as “superposition” of the | x>.
Is there a way to relate the measurable position x with the relative state | x> via a linear operator? The answer is yes and it is through an X operator such that

where X is such that

It is a clearly Hermitian operator whose eigenvalues ​​and associated eigenvectors are x and | x> very simply. Whatever the orthonormal basis of the Hilbert space in which we decide to express X and | x>, the relationship just seen will always be worth:

But what we have seen for position x can be repeated for any observable. No observable is “better” then the others!

The presence therefore of a Hermitian linear operator associated with an observable (whose eigenvalues ​​and eigenstates have the well-known meaning) is therefore not so “magic”.

The postulate that the wave function is the same for all observables and that associated eigenstates bases belong to the same Hilbert space, this is the real magic of Quantum Mechanics!