Schrödinger’s Cat is a thought experiment used to illustrate the weirdness of quantum mechanics. Namely, unlike classical Newtonian mechanics where a particle can only be in one state at a given time, quantum theory says that it can be in two or more states ‘at the same time’. The latter is often referred to as ‘superposition’ of states.
If D refers to the state that the cat is dead, and if A is the state that the cat is alive, then classically we can only have either A or D at any given time — we cannot have both states or neither.
In quantum theory, however, we can not only have states A and D but can also have many more states, such a for example this combination or superposition state:
ψ = 0.8A + 0.6D.
Notice that the coefficient numbers 0.8 and 0.6 here have their squares adding up to exactly 1:
(0.8)2 + (0.6)2 = 0.64 + 0.36 = 1.
This is because these squares, (0.8)2 and (0.6)2, refer to the probabilities associated to states A and D, respectively. Interpretation: there is a 64% chance that this mixed state ψ will collapse to the state A (cat is alive) and a 36% chance it will collapse to D (cat is dead) — if one proceeds to measure or find out the status of the cat were it to be in a quantum mechanical state described by ψ.
Realistically, it is hard to comprehend that a cat can be described by such a state. Well, maybe it isn’t that hard if we had some information about the probability of decay of the radioactive substance inside the box.
Nevertheless, I thought to share another related experiment where one could better ‘see’ and appreciate superposition states like ψ above. The great physicist Richard Feynman did a great job illustrating this with his use of the Stern-Gerlach experiment (which I will tell you about). (See chapters 5 and 6 of Volume III of the Feynman Lectures on Physics.)
In this experiment we have a magnetic field with north/south poles as shown. Then you have a beam of spin-half particles coming out of a furnace heading toward the magnetic field. The result is that the beam of particles will split into two beams. What essentially happened is that the magnetic field made a ‘measurement’ of the spins and some of them turned into spin-up particles (the upper half of the split beam) and the others into spin-down (the bottom beam). So the incoming beam from the furnace is like the cat being in the superposition state ψ and magnetic field is the agent that determined — measured! — whether the cat is alive (upper beam) or dead (lower beam). (Often in physics books they use the Dirac notation |↑〉for spin-up state and |↓〉 for spin-down.)
In a way, you can now see the initial beam emanating from the furnace as being in a superposition state.
Ok, so the superposition state of the initial beam has now collapsed the state of each particle into two specific states: spin-up state (upper beam) and the spin-down state (lower beam). Does this mean that these states are no longer superposition states?
Yes and No! They are no longer in superposition if the split beams enter another magnetic field that points in the same direction as the original one. If you pass the upper beam into a second identical magnetic field, it will remain an upper beam — and the same with the lower beam. The magnetic field ‘made a decision’ and it’s going to stick with it!
That is why we call these states (upper and lower beams) ‘eigenstates’ of the original magnetic field. They are no longer mixed superposition states — the cat is either dead or alive as far as this field is concerned and not in any ‘in between fuzzy’ states.
Ok, that addresses the “Yes” part of the answer. Now for the “No” part.
Let’s suppose we have a different magnetic field, one just like the original one but perpendicular in direction to it. (So it’s like you’ve rotated the original field by 90 degrees; you can rotate by a different angle as well.)
In this case if you pass the original upper beam (that was split by the first magnetic field) into the second perpendicular field, this upper beam will split into two beams! So with respect to the second field the upper beam is now in a superposition state!
Essential Principle: the notion of superposition (in quantum theory) is always with respect to a variable that is being measured. In our case, that variable is the magnetic field. (And here we have two magnetic fields, hence we have two different variables — non-commuting variables as we say in quantum theory.)
Therefore, what was an eigenstate (collapsed state) for the first field (the upper beam) is no longer an eigenstate (no longer ‘collapsed’) for the second (perpendicular) field. (So if a wavefunction is collapsed, it can collapse again and again!)
The Schrodinger Cat experiment could possibly be better understood not as one single experiment, but as a stream of many many boxes with cats in them. This view might better relate to the fact that we have beam of particles each of which is being ‘measured’ by the field to determine its spin status (as being up or down).
Best wishes, Sam
Reference. R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. III (Quantum Mechanics), chapters 5 and 6.
Postscript. It occurred to me to add a uniquely quantum mechanical feature that is contrary to classical physics thinking. The Stern-Gerlach experiment is a good place to see this feature.
We noted that when the spin-half particles emerge from the furnace and into the magnetic field, they split into upper and lower beams. Classically, one might think that before entering the field the particles already had their spins either up or down before a measurement takes place (i.e., before entering the magnetic field) — just as one might say that the earth has a certain velocity as it moves around the sun before we measure it. Quantum theory does not see it that way. In the predominant Copenhagen Interpretation of quantum theory, one cannot say that the particle spins were already a mix of up or down spins before entering the field. Reason we cannot say this is that if we had rotated the field at an angle (say at right angles to the original), the beams would still split into two, but not the same two beams as before! So we cannot say that the particles were already in a mix of those that had spins in one direction or the other. That is one of the strange features of quantum theory, but wonderfully illustrated by Stern-Gerlack.
Mathematics. In vector space theory there is a neat way to illustrate this quantum phenomenon by means of `bases’. For example, the vectors (1,0) and (0,1) form a basis for 2D Euclidean space R2. So any vector (x,y) can be expressed (uniquely!) as a superposition of them; thus,
(x,y) = x(1,0) + y(0,1).
So this would be, to make an analogy, like how the beam of particles, described by (x,y), can be split into to beams — described by the basis vectors (1,0) and (0,1).
However, there are a myriad of other bases. For example, (2,1) and (1,1) also form a basis for R2. A general vector (x,y) can still be expressed in terms of a superposition of these two:
(x,y) = a(2,1) + b(1,1)
for some constants a and b (which are easy to solve in terms of x, y). So this other basis could, by analogy, represent a magnetic field that is at an angle with respect to the original — and its associated beams (2,1) and (1,1) (it’s eigenstates!) would be different because of their different directions. As a matter of fact, we can see here that these eigenstates (collapsed states), represented by (2,1) and (1,1), are actual (uncollapsed) superpositions of the former two, namely (1,0) and (0,1). And vice versa!
Analogy: let’s suppose the vector (5,3) represents the particle states coming out of the furnace. Let’s think of the basis vectors (1,0) and (0,1) represent the spin-up and spin-down beams, respectively, as they enter the first magnetic field, and let the other basis vectors (2,1) and (1,1) represent the spin-up and spin-down beams when they enter the second rotated (maybe perpendicular) magnetic field. Then the particle state (5,3) is a superposition in each of these bases!
(5,3) = 5(1,0) + 3(0,1),
(5,3) = 2(2,1) + 1(1,1).
So it is now quite conceivable that the initial mixed state of particles as they exit the furnace can in fact split in any number of ways as they enter any magnetic field! I.e., it’s not as though they were initially all either (1,0),(0,1) or (2,1),(1,1), but (5,3) could be a simultaneous combination of each — and in fact (5,3) can be combination (superposition) in an infinite number of bases.
Indeed, it now looks like this strange feature of quantum theory can be described naturally from a mathematical perspective! Vector Space bases furnish a great example!
One beautiful summer morning I spent a couple hours in a park reflecting on what I know about quantum mechanics and thought to sketch it out from memory. (A good brain exercise to recapture things you learned and admire.) This note is an edited summary of my handwritten draft (without too much math).
Being a big subject, I will stick to some basic ideas (or principles) of quantum theory that may be worth noting.
Two key concepts are that of a `state’ and that of an ‘observable’.
The former describes the state of the system under study. The observable is a thing we measure. So for example, an electron can be in the ground state of an atom – which means that it is in `orbital’ of lowest energy. Then we have other states that it can be in at higher energies.
The observable is a quantity distinct from a state and one that we measure. Such as for example measuring a particle’s energy, its mass, position, momentum, velocity, charge, spin, angular momentum, etc.
QM gives us principles / interpretations by how states and observables can be mathematically described and how they relate to one another. So here is the first principle.
Principle 1. The state of a system is described by a function (or vector) ψ. The probability density associated with it is given by |ψ|².
This vector is usually a mathematical function of space, time (sometime momentum) variables.
For example, f(x) = exp(-x^2) is one such example. You can also have wave examples such as g(x) = exp(-x^2) sin(x) which looks like a localized wave (a packet) that captures both being a particle (localized) and a wave (due to the wave nature of sin(x)). This wave nature of the function allows it to interfere constructively or destructively with other similar functions — so you can have interference! In actual QM these wavefunctions involve more variables that one x variable, but I used one variable to illustrate.
Principle 2. Each measurable quantity (called an ‘observable’) in an experiment is represented by a matrix A. (A Hermitian matrix or operator.)
For example, energy is represented by the Hamiltonian matrix H, which gives the energy of a system under study. The system could be the hydrogen atom. In many or most situations, the Hamiltonian is the sum of the kinetic energy plus the potential energy (H = K.E. + V).
For simplicity, I will treat a measurable quantity and its associated matrix on equal footing.
From matrix algebra, a matrix is a rectangular array of numbers – like, say, a square array of 3 by 3 numbers, like this one I grabbed from the net:
Turns out you can multiply such things and do some algebra with them.
Two basic facts about these matrices is:
(1) they generally do not have the commutative property (so AB and BA aren’t always equal), unlike real or complex numbers,
(2) each matrix A comes with `magic’ numbers associated to it called eigenvalues of A.
For example the matrix
(called diagonal because it has all zeros above and below the diagonal) has eigenvalues 1, 4, -3. (When a matrix is not diagonal we have a procedure for finding them. Study Matrix or Linear Algebra!)
Principle 3. The possible measurements of a quantity will be its eigenvalues.
For example, the possible energy levels of an electron in the hydrogen atom are eigenvalues of the associated Hamiltonian matrix!
Principle 4. When you measure a quantity A when the system is in the state ψ the system `collapses’ into an eigenstate f of the matrix A.
Therefore the system makes a transition from state ψ to state f (when A is measured).
So mathematically we write
Af = af
which means that f is an eignstate (or eigenvector) of A with eigenvalue a.
So if A represents energy then `a’ would be energy measurement when the system is in state f.
For a general state ψ we cannot say that Aψ = aψ. This eigenvalue equation is only true for eigenstates, not general states.
Principle 5. Each state ψ of the system can be expressed as a superposition sum of the eigenstates of the measurable quantity (or matrix) A.
So if f, g, h, … are the eigenstates of A, then any other state ψ of the system can be expressed as a superposition (or linear combination) of them:
ψ = bf + cg + dh + …
where b, c, d, … are (complex) numbers. Further, |c|^2 = probability ψ will `collapse’ into the eigenstate g when measurement of A is performed.
These principles illustrate the indeterministic nature of quantum theory, because when measurement of A is made, the system can collapse into any one of its many eigenstates (of the matrix A) with various probabilities. So even if you had the ‘exact same’ setup initially there is no guarantee that you would see your system state change into the same state each time. That’s non-causality! (Quite unlike Newtonian mechanics.)
Principle 6. (Follow-up to Principles 4 and 5.) When measurement of A in the state ψ is performed, the probability that the system will collapse into the eigenstate vector φ is the dot product of Aψ and φ.
The latter dot product is usually written using the Dirac notation as <φ|A|ψ>. In the notation above, this would be same as |c|^2.
Next to the basic eigenvalues of A, there’s also it’s `average’ value or expectation value in a given state. That’s like taking the weighted average of tests in a class – with weights assigned to each eigenstate based on the superposition (as in the weights b, c, d, … in the above superposition for ψ). So we have:
Principle 7. The expected or average value of quantity A in the state described by ψ is <ψ|A|ψ>.
In our notation above where ψ = bf + cg + dh + …, this expected value is
<ψ|A|ψ> = |b|^2 times (eigenvalue of f) + |c|^2 times (eigenvalue of g) + …
which you can see it being the weighed average of the possible outcomes of A, namely from the eigenvalues, each being weighted according to its corresponding probabilities |b|^2, |c|^2, … .
In other words if you carry out measurements of A many many times and calculate the average of the values you get, you get this value.
Principle 8. There are some key complementary measurement observables. (Classic example: Heisenberg relation QP – PQ = ih.)
This means that if you have two quantities P and Q that you could measure, if you measure P first and then Q, you will not get the same result as when you do Q first and then P. (In Newton’s mechanics, you could at least in theory measure P and Q simultaneously to any precision in any order.)
Example, position and momentum are complementary in this respect — which is what leads to the Heisenberg Uncertainty Principle, that you cannot measure both the position and momentum of a subatomic particle with arbitrary precision. I.e., there will be an inherent uncertainty in the measurements. Trying to be very accurate with one means you lose accuracy with they other all the more.
From Principle 8 you can show that if you have an eigenstate of the position observable Q, it will not be an eigenstate for P but will be a superposition for P.
So `collapsed’ states could still be superpositions! (Specifically, a collapsed state for Q will be a superposition, uncollapsed, state for P.)
That’s enough for now. There are of course other principles (and some of the above are interlinked), like the Schrodinger equation or the Dirac equation, which tell us what law the state ψ must obey, but I shall leave them out. The above should give an idea of the fundamental principles on which the theory is based.
This has been a wonderful discovery back in the 1950s that gave Radio Astronomy a good push forward. It also helped in mapping out our Milky Way galaxy (which we really can’t see very well!).
It arose from a feature of quantum field theory, specifically from the hyperfine structure of hydrogen. (I’ll try to explain.)
You know that the hydrogen atom consists of a single proton at its central nucleus and a single electron moving around it somehow in certain specific quantized orbits. It cannot just circle around in any orbit.
That was one of Niels Bohr’s major contributions to our understanding of the atom. In fact this year we’re celebrating the 100th anniversary of his model of the atom (his major papers written in 1913). Some articles in the June issue of Nature magazine are in honor of Bohr’s work.
Normally the electron circles in the lowest orbit associated with the lowest energy state – usually called the ground state (the one with n = 1).
It is known that protons and electrons are particles that have “spin”. (That’s why they are sometimes also called ‘fermions’.) It’s as if they behave like spinning tops. (The Earth and Milky Way are spinning too!)
The spin can be in one direction (say ‘up’) or in the other direction (we label as ‘down’). (These labels of where ‘up’ and ‘down’ are depends on the coordinates we choose, but let’s now worry about that.)
When scientists looked at the spectrum of hydrogen more closely they saw that even while the electron can be in the same ground state – and with definite smallest energy – it can have slightly different energies that are very very close to one another. That’s what is meant by “hyperfine structure” — meaning that the usual energy levels of hydrogen are basically correct except that there are ever so slight deviations from the normal energy levels.
It was discovered by means of quantum field theory that this difference in ground state energies arise when the electron and proton switch between spinning in the same direction to spinning in opposite directions (or vice versa).
When they spin in the same direction the hydrogen atom has slightly more energy than when they are spinning in opposite direction.
And the difference between them?
The difference in these energies corresponds to an electromagnetic wave corresponding to about 21 cm wavelength. And that falls in the radio band of the electromagnetic spectrum.
So when the hydrogen atom shows an emission or absorption spectrum in that wavelength level it means that the electron and proton have switched between having parallel spins to having opposite spins. When the switch happens you see an electromagnetic ray either emitted or absorbed.
It does not happen too often, but when you have a huge number of hydrogen atoms — as you would in hydrogen clouds in our galaxy — it will invariably happen and can be measured.
Now it’s a really nice thing that our galaxy contains several hydrogen clouds. So by measuring the Doppler shift in the spectrum of hydrogen — at the 21 cm line! — you can measure the velocities of these clouds in relation to our location near the sun.
These velocity distributions are used together with other techniques to map out the hydrogen clouds in order to map out and locate the spiral arms they fall into.
That work (lots of hard work!) showed astronomers that our Milky Way does indeed have arms, just as we would see in some other galaxies, such as in the picture shown here of NGC 1232.
The one UNKNOWN about the structure of our Milky Way is that we don’t know whether it has 2 or 4 arms.
 University Astronomy, by Pasachoff and Kutner.
 Astronomy (The Evolving Universe), by Michael Zeilik.
(These are excellent sources, by the way.)
The first time I heard of August Kekule’s dream/vision was from my dear mother! (My mom is a geologist who obviously had to know a lot of chemistry.) I am referring to Kekule’s vision while gazing at a fireplace which somehow prompted him onto the idea for the structure of the benzene molecule C6H6. And then I heard that the story is suspect maybe even a myth cooked up by unscientific minds. Now I have learned that Kekule himself recounted that story which was translated into English and published in the Journal of Chemical Education (Volume 35, No. 1, Jan. 1958, pp 21-23, translator: Theodor Benfey). Here is an excerpt from that paper relevant to the story where Kekule talks about his discovery.
I was sitting writing at my textbook but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by repeated visions of the kind, could now distinguish larger structures of manifold conformation: long rows, sometimes more closely fitted together all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis. Let us learn to dream, gentlemen, then perhaps we shall find the truth.
And to those who don’t think The truth will be given. They’ll have it without effort.
But let us beware of publishing our dreams till they have been tested by the making understanding.
Countless spores of the inner life fill the universe, but only in a few rare beings do they find the soil for their development; in them the idea, whose origin is known to no men, comes to life in creative action. (J. Von Liebig)
I believe it is unnecessary to rule out or ridicule dreams, trances, visions in the pursuit of scientific truth. Because, after all, they still have to be tested and examined in our sober existence (as Kekule already alluded). I see them as extensions of thinking and contemplation, and surely there is nothing wrong with these.
Long ago (late 1980s) I attended a lecture by Einstein biographer I. Bernard Cohen. (Cohen actually interviewed Einstein and published it in the Scientific American in the 1955 issue.)
In his lecture, Cohen described Einstein’s view of scientific discovery as a sort of ‘leap’ from experiences to theory. That theory is not logically deduced from experiences but that theory is “jumped at” — or “swooped” is the word Cohen used, I think — thru the imagination or intuition based on our experiences (which of course would/could include experiments). This reminds one of the known Einstein quote that “imagination is more important that knowledge.”
In his book Ideas and Opinions, Albert Einstein said:
“Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and end in it. Propositions arrived at by purely logical means are completely empty as regards reality. Because Galileo saw this, and particularly because he drummed it into the scientific world, he is the father of modern physics—indeed, of modern science altogether.”
(See Part V of his “Ideas and Opinions” in the section entitled “On the Method of Theoretical Physics.”)
In a related passage from the same section, Einstein noted:
“If, then, it is true that the axiomatic foundation of theoretical physics cannot be extracted from experience but must be freely invented, may we ever hope to find the right way? Furthermore, does this right way exist anywhere other than in our illusions? May we hope to be guided safely by experience at all, if there exist theories (such as classical mechanics) which to a large extent do justice to experience, without comprehending the matter in a deep way?
To these questions, I answer with complete confidence, that, in my opinion, the right way exists, and that we are capable of finding it. Our experience hitherto justifies us in trusting that nature is the realization of the simplest that is mathematically conceivable. I am convinced that purely mathematical construction enables us to find those concepts and those lawlike connections between them that provide the key to the understanding of natural phenomena. Useful mathematical concepts may well be suggested by experience, but in no way can they be derived from it. Experience naturally remains the sole criterion of the usefulness of a mathematical construction for physics. But the actual creative principle lies in mathematics. Thus, in a certain sense, I take it to be true that pure thought can grasp the real, as the ancients had dreamed.”
Note his reference to theory as being ‘freely invented’ (and even ‘illusion’) which echo the role of intuition and imagination in the scientific development of theory (but which are probably not completely divorced from experience either!).
The last two quotes above incidentally can be found online in Standford’s Encyclopedia of Philosophy: Einstein’s Philosophy of Science
The volume of a sphere of radius R in 2 dimensions is just the area of a circle which is π R2. (The symbol π is Pi which is 3.1415….)
The volume of a sphere of radius R in 3 dimensions is (4/3) π R3.
The volume of a sphere of radius R in 4 dimensions is (1/2) π2 R4.
Hold everything! How’d you get that? With a little calculus!
Ok, you see a sort of pattern here. The volume of a sphere in n dimensions has a nice form: it is some constant C(n) (involving π and some fractions) times the radius R raised to the dimension n:
V(n,R) = C(n) Rn —————– (1)
where I wrote V(n,R) for the volume of a sphere in n dimensions of radius R (as a function of these two variables).
Now, how do you get the constants C(n)?
With a bit of calculus and some ‘telescoping’ as we say in Math.
(In the cases n = 2 and n = 3, we can see that C(2) = π, and C(3) = 4π/3.)
First, let’s do the calculus. The volume of a sphere in n dimensions can be obtained by integrating the volume of a sphere in n-1 dimensions like this using the form (1):
V(n,R) = ∫ V(n-1,r) dz = ∫ C(n-1) rn-1 dz
where here r2 + z2 = R2 and your integral goes from -R to R (or you can take 2 times the integral from 0 to R). You solve the latter for r, plug it into the last integral, and compute it using the trig substitution z = R sin θ. When you do, you get
V(n,R) = 2C(n-1) Rn ∫ cosnθ dθ.
The cosine integral here goes from 0 to π/2, and it can be expressed in terms of the Gamma function Γ, so it becomes
V(n,R) = 2C(n-1) Rn π1/2 Γ((n+1)/2) / Γ((n+2)/2).
Now compare this with the form for V in equation (1), you see that the R’s cancel and you have C(n) expressed in terms of C(n-1). After telescoping and simplifying you eventually get the volume of a sphere in n dimensions to be:
V(n,R) = πn/2 Rn / Γ((n+2)/2).
Now plug in n equals 4 dimensions and you have what we said above. (Note: Γ(3) = 2 — in fact, for positive integers N the Gamma function has simple values given by Γ(N) = (N-1)!, using the factorial notation.)
How about the volume of a sphere in 5 dimensions? It works out to
V(5,R) = (8/15) π2 R5.
One last thing that’s neat: if you take the derivative of the Volume with respect to the radius R, you get its Surface Area! How crazy is that?!!
Someone on twitter asked if a product of four consecutive odd positive numbers can be a perfect square.
My answer: No.
For example, 3 x 5 x 7 x 9 = 945 which is not a perfect square. (30^2 = 900 and 31^2 = 961.) Similarly, 7 x 9 x 11 x 13 = 9009, again is not a perfect square.
Here is my proof. Write the four consecutive positive odd numbers as:
2x+1, 2x+3, 2x+5, 2x+7
where x is a positive integer.
The middle two numbers 2x+3, 2x+5 cannot both be perfect squares since their difference is 2 — the difference between two positive consecutive perfect squares is at least 3. Let’s suppose it is 2x+3 that is not a perfect square. (The argument can still be adapted if it was 2x+5.) So 2x+3 is divisible by an odd prime p that has an odd power in its prime factorization. This prime p cannot divide the other three factors 2x+1, 2x+5, 2x+7, or else it would divide their differences from 2x+3, which are 2 or 4 (and p is odd). Therefore the prime p appears in the prime factorization of the product
with only an odd power, hence this product cannot be a perfect square. QED