Room occupancy matrix and quanum mechanics

 

Fig.1: 3 rooms with interconnecting doors

@Home with QM

During the pandemic, people have written poetry, jokes, books… All I could do is this is produce some half-baked ruminations (aka insufficiently chewed c(r)ud).

Stuck at home, I worked out that I spent the maximum time per day in my bedroom, and the least in my study. So I can define a room occupancy factor: the percentage of time spent in a given room.

Generalizing, I can also define a room occupancy matrix, with diagonal elements as the percentage of time spent in a given room and off-diagonal elements as the transition probabilities (fraction of time spent transiting) between rooms. The off-diagonal elements are smaller since less time is spent transiting between rooms.

The bedroom may be defined as the ‘ground state’, and the study as the ‘highest energy state’ (since it has the lowest occupancy).

One other – purely formal - assignment can be made for the diagonal elements of the room occupancy matrix. The room that one spends the most time in is the ground state and the one with the lowest occupancy is the highest energy excited rate. So one can vaguely associate a sort of ‘temperature’ if the diagonal elements could be fitted (in a Procrustean way) to an exponential. And associate an energy level with each room. Having stretched this analogy beyond reasonable limits, it would be better to stick to a Maxwell-Boltzmann distribution…

For example, assume just 3 rooms: bedroom, kitchen and study and you get a 3x3 matrix.

( bb bk bs)

(kb kk ks)

(sb sk ss)

And of course it is a symmetric matrix, unless t_bk ¹ t_kb.

Some rooms are connected to others by doors, some are not connected at all. The latter case, like a selection rule in QM, forbids a direct transition. For example, going from State 1 to State 3 is impossible directly because there is no ‘door’.

The analogy to quantum mechanics is purely formal.

But it looks the same whether the matrix elements are time spent in each room, probability of occupancy (dimensionless)  or transition probability (per unit time).

There are three or four other analogues, which one may briefly consider:

a) Markov chain transition matrices

b) the S-matrix

c) Einstein A & B coefficients

d) density matrices.

a)      Markov chain transition matrices:

 

Interestingly the study of Markov chains involves transition matrices. For example [1], the transition matrix (3x3):

(1/4    1/2   1/4)

(1/3     0     2/3)

(1/2     0    1/2)

Is represented diagrammatically by:

Fig.2

Figure2 shows the state transition diagram for the above Markov chain. In this diagram, there are three possible states 1, 2, and 3, and the arrows from each state to other states show the transition probabilities p_ij. When there is no arrow from state i to state j, it means that p_ij = 0.

This seems to be very similar to the room occupancy matrix suggested above.

A mathematician would ask: is the isomorphism between state diagrams and these ‘rooms’ one-to-one? Does it always work? My answer: I haven’t the slightest idea. I do know that giving just one example of a state diagram with no corresponding arrangement of rooms is enough to wipe out the putative ‘isomorphism’. Can I guarantee it, either way? Sorry: that’s way above my pay grade.

Fig.3: Rooms with corridors

However, let me just add that if we consider corridors between rooms, then the time taken to traverse the corridors corresponds to the off-diagonal elements of the room occupancy matrix. If there is no door or corridor between two given rooms i and j then the corresponding traverse time t_ij is infinite.

b) S-matrix:

Another interesting similarity is the scattering matrix, the S matrix [2].

The S-matrix is closely related to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; the elements (individual numerical entries) in the S-matrix are known as scattering amplitudes. In scattering theory the S-matrix maps the free particle in-state Y_i to the free particle out-state Y_f :

Y_f = S Y_i

And the S matrix, expressed in terms of the time evolution operator U, is symmetric if time-reversal symmetry holds.

U = exp(iHt), where H = H₀ + V,

H₀  is the free-space Hamiltonian, and V is the interaction.

 

For free particles in 1D, with no interaction potential V, the S-matrix becomes:

If V is nonzero, the off-diagonal elements deviate from 1, but remain equal, while the two diagonal elements are unequal and nonzero.

So can I consider myself as a particle in a potential field (that varies with position) being scattered between different rooms?

 

a)    Einstein’s A and B coefficients:

Einstein’s A coefficient corresponds to transitions induced by an external radiation field. So far in the room occupancy picture, nothing external has been considered. But one might imagine something like a sunny room proving attractive in winter, and repulsive in summer, as an example of such an external factor inducing transitions between rooms. The B coefficients refer to spontaneous transitions.

For a two-level system, Einstein’s B coefficients are symmetric if the degeneracy g of the two states is the same [3]:

B₁₂ = B₂₁ if g₁ = g₂.

However, it is not obvious that the transition elements are always going to be symmetric in general. The proof for symmetry involves the principle of detailed balance (which is, by definition, above my pay grade).

d) Density Matrices:

In the density matrix r_mn, the diagonal elements r_nn refer to the probabilities of occupying a quantum state n, so they are also called populations [4]. The off-diagonal elements are complex, and are called ‘coherences’ because they have a time-dependent phase factor that describes the evolution of coherent superpositions (i.e. transitions). Decay of off-diagonal elements is referred to as ‘dephasing’, while decay of diagonal elements is ‘relaxation’ (best visualized on the Bloch sphere).

The diagonal elements follow Boltzmann’s law:

r_nn = p_n = exp(- bE_n)/Z

Where   b = 1/(kT) and  Z is the partition function.

The density matrix is Hermitian i.e. (r_mn)* = r_nm.

If we assume the elements were all real, this would just mean that the density matrix is symmetric. It is also satisfies the condition: Tr (r) = 1 i.e. the sum of all probabilities (diagonal elements) is one.

Ok so we have briefly considered 4 possible analogs. In one sense, the first, the Markov chain, seems to be closest. It is true that both the scattering matrix and the density matrix are symmetric, as is the Einstein B-coefficient. But, in general, all these matrix elements are complex and have to be, because they involve the phase factor. The room occupancy matrix seems to be stuck with real numbers. Not quite the same.

Another worry is the off-diagonal element of the room occupancy matrix. Does it refer to the time required to go from room 1 to room 2 – or the probability of going from room 1 to room 2? Or are they the same? I'm a bit hazy about the units, but the Markov chain matrix elements seem to be dimensionless.

If we go back to the starting point of actual physical rooms, then the time spent in both rooms and corridors is well-defined over the course of a day or some shorter or longer time period. This does not translate easily into a transition probability from one room to another. One might associate the most occupied room as the ground state and the least occupied one as the highest energy level – but nothing guarantees that it will be exponential. First, to even define a distribution you would need some variable like ‘energy’ and it is not clear that it could be defined in the room occupancy picture. Second, you would need to fit the probability of occupation to an exponential.

However, one can show that the room occupancy diagram is equivalent to the normal energy level diagram, taken from Schroeder [5] of a 1D infinite square well:

Fig.4: energy level diagram [5] is topologically equivalent to rooms  

The left-hand part is from Schroeder [5], and I added in the schematic ‘rooms’ on the right hand side. The arrows are intended not to indicate transitions, but to indicate that the ‘rooms’ can be squeezed (vertically) to become (thick) lines which are topologically equivalent to the energy levels from Ref.[5]. So the two diagrams convey the same information. Does room occupancy convey anything more? Probably not. Here I'm forced to leave out the corridors, and assume that I can 'teleport' from one room to another. I agree: not satisfactory.

The only thing lacking is an energy variable on the right-hand side!

There probably is a way to define the ‘energy variable’ as the eigenvalues of a transition matrix – only it is beyond my ken. Marcus Woo [6] in Quanta magazine discussed the solution of the 15-puzzle by mathematicians Yang Chu and Robert Hough [7]. The 15 puzzle is a game consisting of a 4x4 grid with 15 tiles and one empty space. The tiles can be moved around so that the empty space ‘moves’. “The goal is to slide the tiles around and put them in numerical order or, in some versions, arrange them to form an image” [6]. The moves can be described as a Markov chain and one can study the transition from order to randomness. Chu and Hough define transition matrices in the configuration space and solve them to find the eigenvalues. More specifically, their aim was to find the number of steps needed to reach randomness (which they define in two ways).

However, as I mentioned earlier, when I see theorems and lemmas, there are no dilemmas: I just head for the hills! And there I shall let this rather half-cooked, useless pandemic-induced recipe rest in peace.

 

References:

1.     1.  https://www.probabilitycourse.com/chapter11/11_2_2_state_transition_matrix_and_diagram.php

2.      2.  https://en.wikipedia.org/wiki/S-matrix

3.      3.  https://en.wikipedia.org/wiki/Einstein_coefficients

4.       4. http://ocw.mit.edu      MIT Open Course Ware: 5.74 Introductory Quantum Mechanics II (Spring 2009)

5.       5. Daniel V.Schroeder, “Notes on Quantum Mechanics,”(Weber State University, January 2020)

6.       6. Marcus Woo https://www.quantamagazine.org/mathematicians-calculate-how-randomness-creeps-in-20191112/

7.       7.Yang Chu & Robert Hough, “Solution of the 15-puzzle problem” arXiv 19Aug.2019 1908.07106v1