The Hamiltonian revisited

I want to come back to something I mentioned in a previous post: when looking at that formula for those Uij amplitudes—which I’ll jot down once more:

Uij(t + Δt, t) = δij + ΔUij(t + Δt, t) = δij + Kij(t)·Δt ⇔ Uij(t + Δt, t) = δij − (i/ħ)·Hij(t)·Δt

—I noted that it resembles the general y(t + Δt) = y(t) + Δy = y(t) + (dy/dt)·Δt formula. So we can look at our Kij(t) function as being equal to the time derivative of the Uij(t + Δt, t) function. I want to re-visit that here, as it triggers a whole range of questions, which may or may not help to understand quantum math somewhat more intuitively.  Let’s quickly sum up what we’ve learned so far: it’s basically all about quantum-mechanical stuff that does not move in space. Hence, the x in our wavefunction ψ(x, t) is some fixed point in space and, therefore, our elementary wavefunction—which we wrote as:

ψ(x, t) = a·ei·θ a·ei·(ω·t − k∙x) = a·ei·[(E/ħ)·t − (p/ħ)∙x]

—reduces to ψ(t) = a·ei·ω·t = a·ei·[(E/ħ)·t.

Unlike what you might think, we’re not equating x with zero here. No. It’s the p = m·v factor that becomes zero, because our reference frame is that of the system that we’re looking at, so its velocity is zero: it doesn’t move in our reference frame. That immediately answers an obvious question: does our wavefunction look any different when choosing another reference frame? The answer is obviously: yes! It surely matters if the system moves or not, and it also matters how fast it moves, because it changes the energy and momentum values from E and p to some E’ and p’. However, we’ll not consider such complications here: that’s the realm of relativistic quantum mechanics. Let’s start with the simplest of situations.

A simple two-state system

One of the simplest examples of a quantum-mechanical system that does not move in space, is the textbook example of the ammonia molecule. The picture was as simple as the one below: an ammonia molecule consists of one nitrogen atom and three hydrogen atoms, and the nitrogen atom could be ‘up’ or ‘down’ with regard to the motion of the NH3 molecule around its axis of symmetry, as shown below.

Capture

It’s important to note that this ‘up’ or ‘down’ direction is, once again, defined with respect to the reference frame of the system itself. The motion of the molecule around its axis of symmetry is referred to as its spin—a term that’s used in a variety of contexts and, therefore, is annoyingly ambiguous. When we use the term ‘spin’ (up or down) to describe an electron state, for example, we’d associate it with the direction of its magnetic moment. Such magnetic moment arises from the fact that, for all practical purposes, we can think of an electron as a spinning electric charge. Now, while our ammonia molecule is electrically neutral, as a whole, the two states are actually associated with opposite electric dipole moments, as illustrated below. Hence, when we’d apply an electric field (denoted as ε) below, the two states are effectively associated with different energy levels, which we wrote as E0 ± εμ.

ammonia

But we’re getting ahead of ourselves here. Let’s revert to the system in free space, i.e. without an electromagnetic force field—or, what amounts to saying the same, without potential. Now, the ammonia molecule is a quantum-mechanical system, and so there is some amplitude for the nitrogen atom to tunnel through the plane of hydrogens. I told you before that this is the key to understanding quantum mechanics really: there is an energy barrier there and, classically, the nitrogen atom should not sneak across. But it does. It’s like it can borrow some energy – which we denote by A – to penetrate the energy barrier.

In quantum mechanics, the dynamics of this system are modeled using a set of two differential equations. These differential equations are really the equivalent of Newton’s classical Law of Motion (I am referring to the F = m·(dv/dt) = m·a equation here) in quantum mechanics, so I’ll have to explain them—which is not so easy as explaining Newton’s Law, because we’re talking complex-valued functions, but… Well… Let me first insert the solution of that set of differential equations:

graph

This graph shows how the probability of the nitrogen atom (or the ammonia molecule itself) being in state 1 (i.e. ‘up’) or, else, in state 2 (i.e. ‘down’), varies sinusoidally in time. Let me also give you the equations for the amplitudes to be in state 1 or 2 respectively:

  1. C1(t) = 〈 1 | ψ 〉 = (1/2)·e(i/ħ)·(E− A)·t + (1/2)·e(i/ħ)·(E+ A)·t = e(i/ħ)·E0·t·cos[(A/ħ)·t]
  2. C2(t) = 〈 2 | ψ 〉 = (1/2)·e(i/ħ)·(E− A)·t – (1/2)·e(i/ħ)·(E+ A)·t = i·e(i/ħ)·E0·t·sin[(A/ħ)·t]

So the P1(t) and P2(t) probabilities above are just the absolute square of these C1(t) and C2(t) functions. So as to help you understand what’s going on here, let me quickly insert the following technical remarks:

  • In case you wonder how we go from those exponentials to a simple sine and cosine factor, remember that the sum of complex conjugates, i.e eiθ eiθ reduces to 2·cosθ, while eiθ − eiθ reduces to 2·i·sinθ.
  • As for how to take the absolute square… Well… I shouldn’t be explaining that here, but you should be able to work that out remembering that (i) |a·b·c|2 = |a|2·|b|2·|c|2; (ii) |eiθ|2 = |e−iθ|= 12 = 1 (for any value of θ); and (iii) |i|2 = 1.
  • As for the periodicity of both probability functions, note that the period of the squared sine and cosine functions is equal to π. Hence, the argument of our sine and cosine function will be equal to 0, π, 2π, 3π etcetera if (A/ħ)·t = 0, π, 2π, 3π etcetera, i.e. if t = 0·ħ/A, π·ħ/A, 2π·ħ/A, 3π·ħ/A etcetera. So that’s why we measure time in units of ħ/A above.

The graph above is actually tricky to interpret, as it assumes that we know in what state the molecule starts out with at t = 0. This assumption is tricky because we usually do not know that: we have to make some observation which, curiously enough, will always yield one of the two states—nothing in-between. Or, else, we can use a state selector—an inhomogeneous electric field which will separate the ammonia molecules according to their state. It’s a weird thing really, and it summarizes all of the ‘craziness’ of quantum-mechanics: as long as we don’t measure anything – by applying that force field – our molecule is in some kind of abstract state, which mixes the two base states. But when we do make the measurement, always along some specific direction (which we usually take to be the z-direction in our reference frame), we’ll always find the molecule is either ‘up’ or, else, ‘down’. We never measure it as something in-between. Personally, I like to think the measurement apparatus – I am talking the electric field here – causes the nitrogen atom to sort of ‘snap into place’. However, physicists use more precise language here: they would say that the electric field does result in the two positions having very different energy levels (E0 + εμ and E0 – εμ, to be precise) and that, as a result, the amplitude for the nitrogen atom to flip back and forth has little effect. Now how do we model that?

The Hamiltonian equations

I shouldn’t be using the term above, as it usually refers to a set of differential equations describing classical systems. However, I’ll also use it for the quantum-mechanical analog, which amounts to the following for our simple two-state example above:

Hamiltonian maser

Don’t panic. We’ll explain. The equations above are all the same but use different formats: the first block writes them as a set of equations, while the second uses the matrix notation, which involves the use of that rather infamous Hamiltonian matrix, which we denote by H = [Hij]. Now, we’ve postponed a lot of technical stuff, so… Well… We can’t avoid it any longer. Let’s look at those Hamiltonian coefficients Hij first. Where do they come from?

You’ll remember we thought of time as some kind of apparatus, with particles entering in some initial state φ and coming out in some final state χ. Both are to be described in terms of our base states. To be precise, we associated the (complex) coefficients C1 and C2 with |φ〉 and D1 and D2 with |χ〉. However, the χ state is a final state, so we have to write it as 〈χ| = |χ〉† (read: chi dagger). The dagger symbol tells us we need to take the conjugate transpose of |χ〉, so the column vector becomes a row vector, and its coefficients are the complex conjugate of D1 and D2, which we denote as D1* and D2*. We combined this with Dirac’s bra-ket notation for the amplitude to go from one base state to another, as a function in time (or a function of time, I should say):

Uij(t + Δt, t) = 〈i|U(t + Δt, t)|j〉

This allowed us to write the following matrix equation:

U coefficients

To see what it means, you should write it all out:

〈χ|U(t + Δt, t)|φ〉 = D1*·(U11(t + Δt, t)·C1 + U12(t + Δt, t)·C2) + D2*·(U21(t + Δt, t)·C1 + U22(t + Δt, t)·C2)

= D1*·U11(t + Δt, t)·C+ D1*·U12(t + Δt, t)·C+ D2*·U21(t + Δt, t)·C+ D2*·U22(t + Δt, t)·C2

It’s a horrendous expression, but it’s a complex-valued amplitude or, quite simply, a complex number. So this is not nonsensical. We can now take the next step, and that’s to go from those Uij amplitudes to the Hij amplitudes of the Hamiltonian matrix. The key is to consider the following: if Δt goes to zero, nothing happens, so we write: Uij = 〈i|U|j〉 → 〈i|j〉 = δij for Δt → 0, with δij = 1 if i = j, and δij = 0 if i ≠ j. We then assume that, for small t, those Uij amplitudes should differ from δij (i.e. from 1 or 0) by amounts that are proportional to Δt. So we write:

Uij(t + Δt, t) = δij + ΔUij(t + Δt, t) = δij + Kij(t)·Δt

We then equated those Kij(t) factors with − (i/ħ)·Hij(t), and we were done: Uij(t + Δt, t) = δij − (i/ħ)·Hij(t)·Δt. […] Well… I show you how we get those differential equations in a moment. Let’s pause here for a while to see what’s going on really. You’ll probably remember how one can mathematically ‘construct’ the complex exponential eiθ by using the linear approximation eiε = 1 + iε near θ = 0 and for infinitesimally small values of ε. In case you forgot, we basically used the definition of the derivative of the real exponential eε for ε going to zero:

FormulaSo we’ve got something similar here for U11(t + Δt, t) = 1 − i·[H11(t)/ħ]·Δt and U22(t + Δt, t) = 1 − i·[H22(t)/ħ]·Δt. Just replace the ε in eiε = 1 + iε by ε = − (E0/ħ)·Δt. Indeed, we know that H11 = H22 = E0, and E0/ħ is, of course, just the energy measured in (reduced) Planck units, i.e. in its natural unit. Hence, if our ammonia molecule is in one of the two base states, we start at θ = 0 and then we just start moving on the unit circle, clockwise, because of the minus sign in eiθ. Let’s write it out:

U11(t + Δt, t) = 1 − i·[H11(t)/ħ]·Δt = 1 − i·[E0/ħ]·Δt and

U22(t + Δt, t) = 1 − i·[H22(t)/ħ]·Δt = 1 − i·[E0/ħ]·Δt

But what about U12 and U21? Is there a similar interpretation? Let’s write those equations down and think about them:

U12(t + Δt, t) = 0 − i·[H12(t)/ħ]·Δt = 0 + i·[A/ħ]·Δt and

U21(t + Δt, t) = 0 − i·[H21(t)/ħ]·Δt = 0 + i·[A/ħ]·Δt

We can visualize this as follows:

circle

Let’s remind ourselves of the definition of the derivative of a function by looking at the illustration below:izvodThe f(x0) value in this illustration corresponds to the Uij(t, t), obviously. So now things make somewhat more sense: U11(t, t) = U11(t, t) = 1, obviously, and U12(t, t) = U21(t, t) = 0. We then add the ΔUij(t + Δt, t) to Uij(t, t). Hence, we can, and probably should, think of those Kij(t) coefficients as the derivative of the Uij(t, t) functions with respect to time. So we can write something like this:

H and U

These derivatives are pure imaginary numbers. That does not mean that the Uij(t + Δt, t) functions are purely imaginary: U11(t + Δt, t) and U22(t + Δt, t) can be approximated by 1 − i·[E0/ħ]·Δt for small Δt, so they do have a real part. In contrast, U12(t + Δt, t) and U21(t + Δt, t) are, effectively, purely imaginary (for small Δt, that is).

I can’t help thinking these formulas reflect a deep and beautiful geometry, but its meaning escapes me so far. 😦 When everything is said and done, none of the reflections above makes things somewhat more intuitive: these wavefunctions remain as mysterious as ever.

I keep staring at those P1(t) and P2(t) functions, and the C1(t) and C2(t) functions that ‘generate’ them, so to speak. They’re not independent, obviously. In fact, they’re exactly the same, except for a phase difference, which corresponds to the phase difference between the sine and cosine. So it’s all one reality, really: all can be described in one single functional form, so to speak. I hope things become more obvious as I move forward. :-/

Post scriptum: I promised I’d show you how to get those differential equations but… Well… I’ve done that in other posts, so I’ll refer you to one of those. Sorry for not repeating myself. 🙂

The Hamiltonian for a two-state system: the ammonia example

Ammonia, i.e. NH3, is a colorless gas with a strong smell. Its serves as a precursor in the production of fertilizer, but we also know it as a cleaning product, ammonium hydroxide, which is NH3 dissolved in water. It has a lot of other uses too. For example, its use in this post, is to illustrate a two-state system. 🙂 We’ll apply everything we learned in our previous posts and, as I  mentioned when finishing the last of those rather mathematical pieces, I think the example really feels like a reward after all of the tough work on all of those abstract concepts – like that Hamiltonian matrix indeed – so I hope you enjoy it. So… Here we go!

The geometry of the NH3 molecule can be described by thinking of it as a trigonal pyramid, with the nitrogen atom (N) at its apex, and the three hydrogen atoms (H) at the base, as illustrated below. [Feynman’s illustration is slightly misleading, though, because it may give the impression that the hydrogen atoms are bonded together somehow. That’s not the case: the hydrogen atoms share their electron with the nitrogen, thereby completing the outer shell of both atoms. This is referred to as a covalent bond. You may want to look it up, but it is of no particular relevance to what follows here.]

Capture

Here, we will only worry about the spin of the molecule about its axis of symmetry, as shown above, which is either in one direction or in the other, obviously. So we’ll discuss the molecule as a two-state system. So we don’t care about its translational (i.e. linear) momentum, its internal vibrations, or whatever else that might be going on. It is one of those situations illustrating that the spin vector, i.e. the vector representing angular momentum, is an axial vector: the first state, which is denoted by | 1 〉 is not the mirror image of state | 2 〉. In fact, there is a more sophisticated version of the illustration above, which usefully reminds us of the physics involved.

dipoleIt should be noted, however, that we don’t need to specify what the energy barrier really consists of: moving the center of mass obviously requires some energy, but it is likely that a ‘flip’ also involves overcoming some electrostatic forces, as shown by the reversal of the electric dipole moment in the illustration above. In fact, the illustration may confuse you, because we’re usually thinking about some net electric charge that’s spinning, and so the angular momentum results in a magnetic dipole moment, that’s either ‘up’ or ‘down’, and it’s usually also denoted by the very same μ symbol that’s used below. As I explained in my post on angular momentum and the magnetic moment, it’s related to the angular momentum J through the so-called g-number. In the illustration above, however, the μ symbol is used to denote an electric dipole moment, so that’s different. Don’t rack your brain over it: just accept there’s an energy barrier, and it requires energy to get through it. Don’t worry about its details!

Indeed, in quantum mechanics, we abstract away from such nitty-gritty, and so we just say that we have base states | i 〉 here, with i equal to 1 or 2. One or the other. Now, in our post on quantum math, we introduced what Feynman only half-jokingly refers to as the Great Law of Quantum Physics: | = ∑ | i 〉〈 i | over all base states i. It basically means that we should always describe our initial and end states in terms of base states. Applying that principle to the state of our ammonia molecule, which we’ll denote by | ψ 〉, we can write:

f1

You may – in fact, you should – mechanically apply that | = ∑ | i 〉〈 i | substitution to | ψ 〉 to get what you get here, but you should also think about what you’re writing. It’s not an easy thing to interpret, but it may help you to think of the similarity of the formula above with the description of a vector in terms of its base vectors, which we write as A = Ax·e+ Ay·e2 + Az·e3. Just substitute the Acoefficients for Ci and the ebase vectors for the | i 〉 base states, and you may understand this formula somewhat better. It also explains why the | ψ 〉 state is often referred to as the | ψ 〉 state vector: unlike our  A = ∑ Ai·esum of base vectors, our | 1 〉 C1 + | 2 〉 Csum does not have any geometrical interpretation but… Well… Not all ‘vectors’ in math have a geometric interpretation, and so this is a case in point.

It may also help you to think of the time-dependency. Indeed, this formula makes a lot more sense when realizing that the state of our ammonia molecule, and those coefficients Ci, depend on time, so we write: ψ = ψ(t) and C= Ci(t). Hence, if we would know, for sure, that our molecule is always in state | 1 〉, then C1 = 1 and C2 = 0, and we’d write: | ψ 〉 = | 1 〉 = | 1 〉 1 + | 2 〉 0. [I am always tempted to insert a little dot (·), and change the order of the factors, so as to show we’re talking some kind of product indeed – so I am tempted to write | ψ 〉 = C1·| 1 〉 C1 + C2·| 2 〉 C2, but I note that’s not done conventionally, so I won’t do it either.]  

Why this time dependency? It’s because we’ll allow for the possibility of the nitrogen to push its way through the pyramid – through the three hydrogens, really – and flip to the other side. It’s unlikely, because it requires a lot of energy to get half-way through (we’ve got what we referred to as an energy barrier here), but it may happen and, as we’ll see shortly, it results in us having to think of the the ammonia molecule as having two separate energy levels, rather than just one. We’ll denote those energy levels as E0 ± A. However, I am getting ahead of myself here, so let me get back to the main story.

To fully understand the story, you should really read my previous post on the Hamiltonian, which explains how those Ci coefficients, as a function of time, can be determined. They’re determined by a set of differential equations (i.e. equations involving a function and the derivative of that function) which we wrote as:

H6

 If we have two base states only – which is the case here – then this set of equations is:

set - two-base

Two equations and two functions – C= C1(t) and C= C2(t) – so we should be able to solve this thing, right? Well… No. We don’t know those Hij coefficients. As I explained in my previous post, they also evolve in time, so we should write them as Hij(t) instead of Hij tout court, and so it messes the whole thing up. We have two equations and six functions really. There is no way we can solve this! So how do we get out of this mess?

Well… By trial and error, I guess. 🙂 Let us just assume the molecule would behave nicely—which we know it doesn’t, but so let’s push the ‘classical’ analysis as far as we can, so we might get some clues as to how to solve this problem. In fact, our analysis isn’t ‘classical’ at all, because we’re still talking amplitudes here! However, you’ll agree the ‘simple’ solution would be that our ammonia molecule doesn’t ‘tunnel’. It just stays in the same spin direction forever. Then H12 and H21 must be zero (think of the U12(t + Δt, t) and U21(t + Δt, t) functions) and H11 and H22 are equal to… Well… I’d love to say they’re equal to 1 but… Well… You should go through my previous posts: these Hamiltonian coefficients are related to probabilities but… Well… Same-same but different, as they say in Asia. 🙂 They’re amplitudes, which are things you use to calculate probabilities. But calculating probabilities involve normalization and other stuff, like allowing for interference of amplitudes, and so… Well… To make a long story short, if our ammonia molecule would stay in the same spin direction forever, then H11 and H22  are not one but some constant. In any case, the point is that they would not change in time (so H11(t) = H11  and H22(t ) = H22), and, therefore, our two equations would reduce to:

S1

So the coefficients are now proper coefficients, in the sense that they’ve got some definite value, and so we have two equations and two functions only now, and so we can solve this. Indeed, remembering all of the stuff we wrote on the magic of exponential functions (more in particular, remembering that d[ex]/dx), we can understand the proposed solution:

S2

As Feynman notes: “These are just the amplitudes for stationary states with the energies E= H11 and E= H22.” Now let’s think about that. Indeed, I find the term ‘stationary’ state quite confusing, as it’s ill-defined. In this context, it basically means that we have a wavefunction that is determined by (i) a definite (i.e. unambiguous, or precise) energy level and (ii) that there is no spatial variation. Let me refer you to my post on the basics of quantum math here. We often use a sort of ‘Platonic’ example of the wavefunction indeed:

a·ei·θ ei·(ω·t − k ∙x) = a·e(i/ħ)·(E·t − px)

So that’s a wavefunction assuming the particle we’re looking at has some well-defined energy E and some equally well-defined momentum p. Now, that’s kind of ‘Platonic’ indeed, because it’s more like an idea, rather than something real. Indeed, a wavefunction like that means that the particle is everywhere and nowhere, really—because its wavefunction is spread out all of over space. Of course, we may think of the ‘space’ as some kind of confined space, like a box, and then we can think of this particle as being ‘somewhere’ in that box, and then we look at the temporal variation of this function only – which is what we’re doing now: we don’t consider the space variable x at all. So then the equation reduces to a·e–(i/ħ)·(E·t), and so… Well… Yes. We do find that our Hamiltonian coefficient Hii is like the energy of the | i 〉 state of our NH3 molecule, so we write: H11 = E1, and H22 = E2, and the ‘wavefunctions’ of our Cand Ccoefficients can be written as:

  • Ca·e(i/ħ)·(H11·t) a·e(i/ħ)·(E1·t), with H11 = E1, and
  • C= a·e(i/ħ)·(H22·t) a·e(i/ħ)·(E2·t), with H22 = E2.

But can we interpret Cand  Cas proper amplitudes? They are just coefficients in these equations, aren’t they? Well… Yes and no. From what we wrote in previous posts, you should remember that these Ccoefficients are equal to 〈 i | ψ 〉, so they are the amplitude to find our ammonia molecule in one state or the other.

Back to Feynman now. He adds, logically but brilliantly:

We note, however, that for the ammonia molecule the two states |1〉 and |2〉 have a definite symmetry. If nature is at all reasonable, the matrix elements H11 and H22 must be equal. We’ll call them both E0, because they correspond to the energy the states would have if H11 and H22 were zero.”

So our Cand Camplitudes then reduce to:

  • C〈 1 | ψ 〉 = a·e(i/ħ)·(E0·t)
  • C=〈 2 | ψ 〉 = a·e(i/ħ)·(E0·t)

We can now take the absolute square of both to find the probability for the molecule to be in state 1 or in state 2:

  • |〈 1 | ψ 〉|= |a·e(i/ħ)·(E0·t)|a
  • |〈 2 | ψ 〉|= |a·e(i/ħ)·(E0·t)|a

Now, the probabilities have to add up to 1, so a+ a= 1 and, therefore, the probability to be in either in state 1 or state 2 is 0.5, which is what we’d expect.

Note: At this point, it is probably good to get back to our | ψ 〉 = | 1 〉 C1 + | 2 〉 Cequation, so as to try to understand what it really says. Substituting the a·e(i/ħ)·(E0·t) expression for C1 and C2 yields:

| ψ 〉 = | 1 〉 a·e(i/ħ)·(E0·t) + | 2 〉 a·e(i/ħ)·(E0·t) = [| 1 〉 + | 2 〉] a·e(i/ħ)·(E0·t)

Now, what is this saying, really? In our previous post, we explained this is an ‘open’ equation, so it actually doesn’t mean all that much: we need to ‘close’ or ‘complete’ it by adding a ‘bra’, i.e. a state like 〈 χ |, so we get a 〈 χ | ψ〉 type of amplitude that we can actually do something with. Now, in this case, our final 〈 χ | state is either 〈 1 | or 〈 2 |, so we write:

  • 〈 1 | ψ 〉 = [〈 1 | 1 〉 + 〈 1 | 2 〉]·a·e(i/ħ)·(E0·t) = [1 + 0]·a·e(i/ħ)·(E0·t)· = a·e(i/ħ)·(E0·t)
  • 〈 2 | ψ 〉 = [〈 2 | 1 〉 + 〈 2 | 2 〉]·a·e(i/ħ)·(E0·t) = [0 + 1]·a·e(i/ħ)·(E0·t)· = a·e(i/ħ)·(E0·t)

Note that I finally added the multiplication dot (·) because we’re talking proper amplitudes now and, therefore, we’ve got a proper product too: we multiply one complex number with another. We can now take the absolute square of both to find the probability for the molecule to be in state 1 or in state 2:

  • |〈 1 | ψ 〉|= |a·e(i/ħ)·(E0·t)|a
  • |〈 2 | ψ 〉|= |a·e(i/ħ)·(E0·t)|a

Unsurprisingly, we find the same thing: these probabilities have to add up to 1, so a+ a= 1 and, therefore, the probability to be in state 1 or state 2 is 0.5. So the notation and the logic behind makes perfect sense. But let me get back to the lesson now.

The point is: the true meaning of a ‘stationary’ state here, is that we have non-fluctuating probabilities. So they are and remain equal to some constant, i.e. 1/2 in this case. This implies that the state of the molecule does not change: there is no way to go from state 1 to state 2 and vice versa. Indeed, if we know the molecule is in state 1, it will stay in that state. [Think about what normalization of probabilities means when we’re looking at one state only.]

You should note that these non-varying probabilities are related to the fact that the amplitudes have a non-varying magnitude. The phase of these amplitudes varies in time, of course, but their magnitude is and remains aalways. The amplitude is not being ‘enveloped’ by another curve, so to speak.

OK. That should be clear enough. Sorry I spent so much time on this, but this stuff on ‘stationary’ states comes back again and again and so I just wanted to clear that up as much as I can. Let’s get back to the story.

So we know that, what we’re describing above, is not what ammonia does really. As Feynman puts it: “The equations [i.e. the Cand Cequations above] don’t tell us what what ammonia really does. It turns out that it is possible for the nitrogen to push its way through the three hydrogens and flip to the other side. It is quite difficult; to get half-way through requires a lot of energy. How can it get through if it hasn’t got enough energy? There is some amplitude that it will penetrate the energy barrier. It is possible in quantum mechanics to sneak quickly across a region which is illegal energetically. There is, therefore, some [small] amplitude that a molecule which starts in |1〉 will get to the state |2. The coefficients H12 and H21 are not really zero.”

He adds: “Again, by symmetry, they should both be the same—at least in magnitude. In fact, we already know that, in general, Hij must be equal to the complex conjugate of Hji.”

His next step, then, is to interpreted as either a stroke of genius or, else, as unexplained. 🙂 He invokes the symmetry of the situation to boldly state that H12 is some real negative number, which he denotes as −A, which – because it’s a real number (so the imaginary part is zero) – must be equal to its complex conjugate H21. So then Feynman does this fantastic jump in logic. First, he keeps using the E0 value for H11 and H22, motivating that as follows: “If nature is at all reasonable, the matrix elements H11 and H22 must be equal, and we’ll call them both E0, because they correspond to the energy the states would have if H11 and H22 were zero.” Second, he uses that minus A value for H12 and H21. In short, the two equations and six functions are now reduced to:

equations

Solving these equations is rather boring. Feynman does it as follows:

solution

Now, what does these equations actually mean? It depends on those a and b coefficients. Looking at the solutions, the most obvious question to ask is: what if a or b are zero? If b is zero, then the second terms in both equations is zero, and so C1 and C2 are exactly the same: two amplitudes with the same temporal frequency ω = (E− A)/ħ. If a is zero, then C1 and C2 are the same too, but with opposite sign: two amplitudes with the same temporal frequency ω = (E+ A)/ħ. Squaring them – in both cases (i.e. for a = 0 or b = 0) – yields, once again, an equal and constant probability for the spin of the ammonia molecule to in the ‘up’ or ‘down’ or ‘down’. To be precise, we We can now take the absolute square of both to find the probability for the molecule to be in state 1 or in state 2:

  • For b = 0: |〈 1 | ψ 〉|= |(a/2)·e(i/ħ)·(E− A)·t|a2/4 = |〈 2 | ψ 〉|
  • For a = 0: |〈 1 | ψ 〉|=|(b/2)·e(i/ħ)·(E+ A)·t|= b2/4 = |〈 2 | ψ 〉|(the minus sign in front of b/2 is squared away)

So we get two stationary states now. Why two instead of one? Well… You need to use your imagination a bit here. They actually reflect each other: they’re the same as the one stationary state we found when assuming our nitrogen atom could not ‘flip’ from one position to the other. It’s just that the introduction of that possibility now results in a sort of ‘doublet’ of energy levels. But so we shouldn’t waste our time on this, as we want to analyze the general case, for which the probabilities to be in state 1 or state 2 do vary in time. So that’s when a and b are non-zero.

To analyze it all, we may want to start with equating t to zero. We then get:

C1

This leads us to conclude that a = b = 1, so our equations for C1(t) and C2(t) can now be written as:

C2

Remembering our rules for adding and subtracting complex conjugates (eiθ + e–iθ = 2cosθ and eiθ − e–iθ = 2sinθ), we can re-write this as:

C3

Now these amplitudes are much more interesting. Their temporal variation is defined by Ebut, on top of that, we have an envelope here: the cos(A·t/ħ) and sin(A·t/ħ) factor respectively. So their magnitude is no longer time-independent: both the phase as well as the amplitude now vary with time. What’s going on here becomes quite obvious when calculating and plotting the associated probabilities, which are

  • |C1(t)|= cos2(A·t/ħ), and
  • |C2(t)|= sin2(A·t/ħ)

respectively (note that the absolute square of i is equal to 1, not −1). The graph of these functions is depicted below.

graph

As Feynman puts it: “The probability sloshes back and forth.” Indeed, the way to think about this is that, if our ammonia molecule is in state 1, then it will not stay in that state. In fact, one can be sure the nitrogen atom is going to flip at some point in time, with the probabilities being defined by that fluctuating probability density function above. Indeed, as time goes by, the probability to be in state 2 increases, until it will effectively be in state 2. And then the cycle reverses.

Our | ψ 〉 = | 1 〉 C1 + | 2 〉 Cequation is a lot more interesting now, as we do have a proper mix of pure states now: we never really know in what state our molecule will be, as we have these ‘oscillating’ probabilities now, which we should interpret carefully.

The point to note is that the a = 0 and b = 0 solutions came with precise temporal frequencies: (E− A)/ħ and (E0 + A)/ħ respectively, which correspond to two separate energy levels: E− A and E0 + A respectively, with |A| = H12 = H21. So everything is related to everything once again: allowing the nitrogen atom to push its way through the three hydrogens, so as to flip to the other side, thereby breaking the energy barrier, is equivalent to associating two energy levels to the ammonia molecule as a whole, thereby introducing some uncertainty, or indefiniteness as to its energy, and that, in turn, gives us the amplitudes and probabilities that we’ve just calculated.

Note that the probabilities “sloshing back and forth”, or “dumping into each other” – as Feynman puts it – is the result of the varying magnitudes of our amplitudes, going up and down and, therefore, their absolute square varies too.

So… Well… That’s it as an introduction to a two-state system. There’s more to come. Ammonia is used in the ammonia maser. Now that is something that’s interesting to analyze—both from a classical as well as from a quantum-mechanical perspective. Feynman devotes a full chapter to it, so I’d say… Well… Have a look. 🙂

Post scriptum: I must assume this analysis of the NH3 molecule, with the nitrogen ‘flipping’ across the hydrogens, triggers a lot of questions, so let me try to answer some. Let me first insert the illustration once more, so you don’t have to scroll up:

dipole

The first thing that you should note is that the ‘flip’ involves a change in the center of mass position. So that requires energy, which is why we associate two different energy levels with the molecule: E+ A and E− A. However, as mentioned above, we don’t care about the nitty-gritty here: the energy barrier is likely to combine a number of factors, including electrostatic forces, as evidenced by the flip in the electric dipole moment, which is what the μ symbol here represents! Just note that the two energy levels are separated by an amount that’s equal to 2·A, rather than A and that, once again, it becomes obvious now why Feynman would prefer the Hamiltonian to be called the ‘energy matrix’, as its coefficients do represent specific energy levels, or differences between them! Now, that assumption yielded the following wavefunctions for C= 〈 1 | ψ 〉 and C= 〈 2 | ψ 〉:

  • C= 〈 1 | ψ 〉 = (1/2)·e(i/ħ)·(E− A)·t + (1/2)·e(i/ħ)·(E+ A)·t
  • C= 〈 2 | ψ 〉 = (1/2)·e(i/ħ)·(E− A)·t – (1/2)·e(i/ħ)·(E+ A)·t

Both are composite waves. To be precise, they are the sum of two component waves with a temporal frequency equal to ω= (E− A)/ħ and ω= (E+ A)/ħ respectively. [As for the minus sign in front of the second term in the wave equation for C2, −1 = e±iπ, so + (1/2)·e(i/ħ)·(E+ A)·t and – (1/2)·e(i/ħ)·(E+ A)·t are the same wavefunction: they only differ because their relative phase is shifted by ±π.]

Now, writing things this way, rather than in terms of probabilities, makes it clear that the two base states of the molecule themselves are associated with two different energy levels, so it is not like one state has more energy than the other. It’s just that the possibility of going from one state to the other requires an uncertainty about the energy, which is reflected by the energy doublet  E± A in the wavefunction of the base states. Now, if the wavefunction of the base states incorporates that energy doublet, then it is obvious that the state of the ammonia molecule, at any point in time, will also incorporate that energy doublet.

This triggers the following remark: what’s the uncertainty really? Is it an uncertainty in the energy, or is it an uncertainty in the wavefunction? I mean: we have a function relating the energy to a frequency. Introducing some uncertainty about the energy is mathematically equivalent to introducing uncertainty about the frequency. Think of it: two energy levels implies two frequencies, and vice versa. More in general, introducing n energy levels, or some continuous range of energy levels ΔE, amounts to saying that our wave function doesn’t have a specific frequency: it now has n frequencies, or a range of frequencies Δω = ΔE/ħ. Of course, the answer is: the uncertainty is in both, so it’s in the frequency and in the energy and both are related through the wavefunction. So… In a way, we’re chasing our own tail.

Having said that, the energy may be uncertain, but it is real. It’s there, as evidenced by the fact that the ammonia molecule behaves like an atomic oscillator: we can excite it in exactly the same way as we can excite an electron inside an atom, i.e. by shining light on it. The only difference is the photon energies: to cause a transition in an atom, we use photons in the optical or ultraviolet range, and they give us the same radiation back. To cause a transition in an ammonia molecule, we only need photons with energies in the microwave range. Here, I should quickly remind you of the frequencies and energies involved. visible light is radiation in the 400–800 terahertz range and, using the E = h·f equation, we can calculate the associated energies of a photon as 1.6 to 3.2 eV. Microwave radiation – as produced in your microwave oven – is typically in the range of 1 to 2.5 gigahertz, and the associated photon energy is 4 to 10 millionths of an eV. Having illustrated the difference in terms of the energies involved, I should add that masers and lasers are based on the same physical principle: LASER and MASER stand for Light/Micro-wave Amplification by Stimulated Emission of Radiation, respectively.

So… How shall I phrase this? There’s uncertainty, but the way we are modeling that uncertainty matters. So yes, the uncertainty in the frequency of our wavefunction and the uncertainty in the energy are mathematically equivalent, but the wavefunction has a meaning that goes much beyond that. [You may want to reflect on that yourself.]

Finally, another question you may have is why would Feynman take minus A (i.e. −A) for H12 and H21. Frankly, my first thought on this was that it should have something to do with the original equation for these Hamiltonian coefficients, which also has a minus sign: Uij(t + Δt, t) = δij + Kij(t)·Δt = δij − (i/ħ)·Hij(t)·Δt. For i ≠ j, this reduces to:

Uij(t + Δt, t) = + Kij(t)·Δt = − (i/ħ)·Hij(t)·Δt

However, the answer is: it really doesn’t matter. One could write: H12 and H21 = +A, and we’d find the same equations. We’d just switch the indices 1 and 2, and the coefficients a and b. But we get the same solutions. You can figure that out yourself. Have fun with it !

Oh ! And please do let me know if some of the stuff above would trigger other questions. I am not sure if I’ll be able to answer them, but I’ll surely try, and good question always help to ensure we sort of ‘get’ this stuff in a more intuitive way. Indeed, when everything is said and done, the goal of this blog is not simply re-produce stuff, but to truly ‘get’ it, as good as we can. 🙂

Quantum math: the Hamiltonian

After all of the ‘rules’ and ‘laws’ we’ve introduced in our previous post, you might think we’re done but, of course, we aren’t. Things change. As Feynman puts it: “One convenient, delightful ‘apparatus’ to consider is merely a wait of a few minutes; During the delay, various things could be going on—external forces applied or other shenanigans—so that something is happening. At the end of the delay, the amplitude to find the thing in some state χ is no longer exactly the same as it would have been without the delay.”

In short, the picture we presented in the previous posts was a static one. Time was frozen. In reality, time passes, and so we now need to look at how amplitudes change over time. That’s where the Hamiltonian kicks in. So let’s have a look at that now.

[If you happen to understand the Hamiltonian already, you may want to have a look at how we apply it to a real situation: we’ll explain the basics involving state transitions of the ammonia molecule, which are a prerequisite to understanding how a maser works, which is not unlike a laser. But that’s for later. First we need to get the basics.]

Using Dirac’s bra-ket notation, which we introduced in the previous posts, we can write the amplitude to find a ‘thing’ – i.e. a particle, for example, or some system, of particles or other things – in some state χ at the time t = t2, when it was in some state φ state at the time t = t1 as follows:

H1

Don’t be scared of this thing. If you’re unfamiliar with the notation, just check out my previous posts: we’re just replacing A by U, and the only thing that we’ve modified is that the amplitudes to go from φ to χ now depend on t1 and t2. Of course, we’ll describe all states in terms of base states, so we have to choose some representation and expand this expression, so we write: 

H2

I’ve explained the point a couple of time already, but let me note it once more: in quantum physics, we always measure some (vector) quantity – like angular momentum, or spin – in some direction, let’s say the z-direction, or the x-direction, or whatever direction really. Now we can do that in classical mechanics too, of course, and then we find the component of that vector quantity (vector quantities are defined by their magnitude and, importantly, their direction). However, in classical mechanics, we know the components in the x-, y- and z-direction will unambiguously determine that vector quantity. In quantum physics, it doesn’t work that way. The magnitude is never all in one direction only, so we can always some of it in some other direction. (see my post on transformations, or on quantum math in general). So there is an ambiguity in quantum physics has no parallel in classical mechanics. So the concept of a component of a vector needs to be carefully interpreted. There’s nothing definite there, like in classical mechanics: all we have is amplitudes, and all we can do is calculate probabilities, i.e. expected values based on those amplitudes.

In any case, I can’t keep repeating this, so let me move on. In regard to that 〈 χ | U | φ 〉 expression, I should, perhaps, add a few remarks. First, why U instead of A? The answer: no special reason, but it’s true that the use of U reminds us of energy, like potential energy, for example. We might as well have used W. The point is: energy and momentum do appear in the argument of our wavefunctions, and so we might as well remind ourselves of that by choosing symbols like W or U here. Second, we may, of course, want to choose our time scale such that t1 = 0. However, it’s fine to develop the more general case. Third, it’s probably good to remind ourselves we can think of matrices to model it all. More in particular, if we have three base states, say ‘plus‘, ‘zero, or ‘minus‘, and denoting 〈 i | φ 〉 and 〈 i | χ 〉 as Ci and Di respectively (so 〈 χ | i 〉 = 〈 i | χ 〉* = Di*), then we can re-write the expanded expression above as:

Matrix U

Fourth, you may have heard of the S-matrix, which is also known as the scattering matrix—which explains the S in front but it’s actually a more general thing. Feynman defines the S-matrix as the U(t1, t2) matrix for t→ −∞ and t→ +∞, so as some kind of limiting case of U. That’s true in the sense that the S-matrix is used to relate initial and final states, indeed. However, the relation between the S-matrix and the so-called evolution operators U is slightly more complex than he wants us to believe. I can’t say too much about this now, so I’ll just refer you to the Wikipedia article on that, as I have to move on.

The key to the analysis is to break things up once more. More in particular, one should appreciate that we could look at three successive points in time, t1, t2, t3, and write U(t1, t3) as:

U(t3, t1) = U(t3, t2)·U(t2, t1)

It’s just like adding another apparatus in series, so it’s just like what did in our previous post, when we wrote:

B1

So we just put a | bar between B and A and wrote it all out. That | bar is really like a factor 1 in multiplication but – let me caution you – you really need to watch the order of the various factors in your product, and read symbols in the right order, which is often from right to left, like in Hebrew or Arab, rather than from left to right. In that regard, you should note that we wrote U(t3, t1) rather than U(t1, t3): you need to keep your wits about you here! So as to make sure we can all appreciate that point, let me show you what that U(t3, t1) = U(t3, t2)·U(t2, t1) actually says by spelling it out if we have two base states only (like ‘up‘ or ‘down‘, which I’ll note as ‘+’ and ‘−’ again) :

Matrix U2

So now you appreciate why we try to simplify our notation as much as we can! But let me get back to the lesson. To explain the Hamiltonian, which we need to describe how states change over time, Feynman embarks on a rather spectacular differential analysis. Now, we’ve done such exercises before, so don’t be too afraid. He substitutes t1 for t tout court, and tfor t + Δt, with Δt the infinitesimal you know from Δy = (dy/dx)·Δx, with the derivative dy/dx being defined as the Δy/Δx ratio for Δx → 0. So we write U(t2, t1) = U(t + Δt, t). Now, we also explained the idea of an operator in our previous post. It came up when we’re being creative, and so we dropped the 〈 χ | state from the 〈 χ | A | φ〉 expression and just wrote:

C1

If you ‘get’ that, you’ll also understand what I am writing now:

chi1

This is quite abstract, however. It is an ‘open’ equation, really: one needs to ‘complete’ it with a ‘bra’, i.e. a state like 〈 χ |, so as to give a 〈 χ | ψ〉 = 〈 χ | A | φ〉 type of amplitude that actually means something. What we’re saying is that our operator (or our ‘apparatus’ if it helps you to think that way) does not mean all that much as long as we don’t measure what comes out, so we have to choose some set of base states, i.e. a representation, which allows us to describe the final state, which we write as 〈 χ |. In fact, what we’re interested in is the following amplitudes:

chi2

So now we’re in business, really. 🙂 If we can find those amplitudes, for each of our base states i, we know what’s going on. Of course, we’ll want to express our ψ(t) state in terms of our base states too, so the expression we should be thinking of is:

chi3

Phew! That looks rather unwieldy, doesn’t it? You’re right. It does. So let’s simplify. We can do the following substitutions:

  • 〈 i | ψ(t + Δt)〉 = Ci(t + Δt) or, more generally, 〈 j | ψ(t)〉 = Cj(t)
  • 〈 i | U(t2, t1) | j〉 = Uij(t2, t1) or, more specifically, 〈 i | U(t + Δt, t) | j〉 = Uij(t + Δt, t)

H3

As Feynman notes, that’s how the dynamics of quantum mechanics really look like. But, of course, we do need something in terms of derivatives rather than in terms of differentials. That’s where the Δy = (dy/dx)·Δx equation comes in. The analysis looks kinda dicey because it’s like doing some kind of first-order linear approximation of things – rather than an exact kinda thing – but that’s how it is. Let me remind you of the following formula: if we write our function y as y = f(x), and we’re evaluating the function near some point a, then our Δy = (dy/dx)·Δx equation can be used to write:

y = f(x) ≈ f(a) + f'(a)·(x − a) = f(a) + (dy/dx)·Δx

To remind yourself of how this works, you can complete the drawing below with the actual y = f(x) as opposed to the f(a) + Δy approximation, remembering that the (dy/dx) derivative gives you the slope of the tangent to the curve, but it’s all kids’ stuff really and so we shouldn’t waste too much spacetime on this. 🙂

300px-TangentGraphic2

The point is: our Uij(t + Δt, t) is a function too, not only of time, but also of i and j. It’s just a rather special function, because we know that, for Δt → 0, Uij will be equal to 1 if i = (in plain language: if Δt → 0 goes to zero, nothing happens and we’re just in state i), and equal to 0 if i = j. That’s just as per the definition of our base states. Indeed, remember the first ‘rule’ of quantum math:

〈 i | j〉 = 〈 j | i〉 = δij, with δij = δji is equal to 1 if i = j, and zero if i ≠ j

So we can write our f(x) ≈ f(a) + (dy/dx)·Δx expression for Uij as:

H4

So Kij is also some kind of derivative and the Kronecker delta, i.e. δij, serves as the reference point around which we’re evaluating UijHowever, that’s about as far as the comparison goes. We need to remind ourselves that we’re talking complex-valued amplitudes here. In that regard, it’s probably also good to remind ourselves once more that we need to watch the order of stuff: Uij = 〈 i | U | j〉, so that’s the amplitude to go from base state to base state i, rather than the other way around. Of course, we have the 〈 χ | φ 〉 = 〈 φ | χ 〉* rule, but we still need to see how that plays out with an expression like 〈 i | U(t + Δt, t) | j〉. So, in short, we should be careful here! 

Having said that, we can actually play a bit with that expression, and so that’s what we’re going to do now. The first thing we’ll do is to write Kij as a function of time indeed:

Kij = Kij(t)

So we don’t have that Δt in the argument. It’s just like dy/dx = f'(x): a derivative is a derivative—a function which we derive from some other function. However, we’ll do something weird now: just like any function, we can multiply or divide it by some constant, so we can write something like G(x) = F(x), which is equivalent to saying that F(x) = G(x)/c. I know that sound silly but it is how is, and we can also do it with complex-valued functions: we can define some other function by multiplying or dividing by some complex-valued constant, like a + b·i, or ξ or whatever other constant. Just note we’re no longer talking the base state but the imaginary unit i. So it’s all done so as to confuse you even more. 🙂

So let’s take −i/ħ as our constant and re-write our Kij(t) function as −itimes some other function, which we’ll denote by Hij(t), so Kij(t) = –(i/ħ)·Hij(t). You guess it, of course: Hij(t) is the infamous Hamiltonian, and it’s written the way it’s written both for historical as well as for practical reasons, which you’ll soon discover. Of course, we’re talking one coefficient only and we’ll have nine if we have three base states i and j, or four if we have only two. So we’ve got a n-by-n matrix once more. As for its name… Well… As Feynman notes: “How Hamilton, who worked in the 1830s, got his name on a quantum mechanical matrix is a tale of history. It would be much better called the energy matrix, for reasons that will become apparent as we work with it.”

OK. So we’ll just have to acknowledge that and move on. Our Uij(t + Δt, t) = δij + Kij(t)·Δt expression becomes:

 Uij(t + Δt, t) = δij –(i/ħ)·Hij(t)·Δt

[Isn’t it great you actually start to understand those Chinese-looking formulas? :-)] We’re not there yet, however. In fact, we’ve still got quite a bit of ground to cover. We now need to take that other monster:

H3

So let’s substitute now, so we get:

H5

We can get this in the form we want to get – so that’s the form you’ll find in textbooks 🙂 – by noting that the ∑δij·Cj(t) sum, taking over all is, quite simply, equal to Ci(t). [Think about the indexes here: we’re looking at some i, and so it’s only the j that’s taking on whatever value it can possibly have.] So we can move that to the other side, which gives us Ci(t + Δt) – Ci(t). We can then divide both sides of our expression by Δt, which gives us an expression like [f(x + Δx) – f(x)]/Δx = Δy//Δx, which is actually the definition of the derivative for Δx going to zero. Now, that allows us to re-write the whole thing in terms of a proper derivative, rather than having to work with this rather unwieldy differential stuff. So, if we substitute [Ci(t + Δt) – Ci(t)]/Δx for d[Ci(t)]/dt, and then also move –(i/ħ) to the left-hand side, remembering that 1/i = –i (and, hence, [–(i/ħ)]−1 = i/ħ), we get the formula in the shape we wanted it in:

H6

Done ! Of course, this is a set of differential equations and… Well… Yes. Yet another set of differential equations. 🙂 It seems like we can’t solve anything without involving differential equations in physics, isn’t it? But… Well… I guess that’s the way it is. So, before we turn to some example, let’s note a few things.

First, we know that a particle, or a system, must be in some state at any point of time. That’s equivalent to stating that the sum of the probabilities |Ci(t)|= |〈 i | ψ(t)〉|is some constant. In fact, we’d like to say it’s equal to one, but then we haven’t normalized anything here. You can fiddle with the formulas but it’s probably easier to just acknowledge that, if we’d measure anything – think of the angular momentum along the z-direction, or some other direction, if you’d want an example – then we’ll find it’s either ‘up’ or ‘down’ for a spin-1/2 particle, or ‘plus’, ‘zero’, or ‘minus’ for a spin-1 particle.

Now, we know that the complex conjugate of a sum is equal to the sum of the complex conjugates: [∑ z]* = ∑ zi*, and that the complex conjugate of a product is the product of the complex conjugates, so we have [∑ ziz]* = ∑ zi*zj*. Now, some fiddling with the formulas above should allow you to prove that Hij = Hij*, and the associated matrix is usually referred to as the Hermitian or conjugate transpose. If if the original Hamiltonian matrix is denoted as H, then its conjugate transpose will be denoted by H*, H or even H(so the in the superscript stands for Hermitian, instead of Hamiltonean). So… Yes. There’s competing notations around. 🙂

The simplest situation, of course, is when the Hamiltonian do not depend on time. In that case, we’re back in the static case, and all Hij coefficients are just constants. For a system with two base states, we’d have the following set of equations:

set - two-base

This set of two equations can be easily solved by remembering the solution for one equation only. Indeed, if we assume there’s only base state – which is like saying: the particle is at rest somewhere (yes: it’s that stupid!) – our set of equations reduces to only one:

one equation

This is a differential equation which is easily solved to give:

solution

[As for being ‘easily solved’, just remember the exponential function is its own derivative and, therefore, d[a·e–(i/ħ)Hijt]/dt = a·d[e–(i/ħ)Hijt]/dt = –a·(i/ħ)·Hij·e–(i/ħ)Hijt, which gives you the differential equation, so… Well… That’s the solution.]

This should, of course, remind you of the equation that inspired Louis de Broglie to write down his now famous matter-wave equation (see my post on the basics of quantum math):

a·ei·θ ei·(ω·t − k ∙x) = a·e(i/ħ)·(E·t − px)

Indeed, if we look at the temporal variation of this function only – so we don’t consider the space variable x – then this equation reduces to a·e–(i/ħ)·(E·t), and so find that our Hamiltonian coefficient H11 is equal to the energy of our particle, so we write: H11 = E, which, of course, explains why Feynman thinks the Hamiltonian matrix should be referred to as the energy matrix. As he puts it: “The Hamiltonian is the generalization of the energy for more complex situations.”

Now, I’ll conclude this post by giving you the answer to Feynman’s remark on why the Irish 19th century mathematician William Rowan Hamilton should be associated with the Hamiltonian. The truth is: the term ‘Hamiltonian matrix’ may also refer to a more general notion. Let me copy Wikipedia here: “In mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix

J= \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}

and In is the n-by-n identity matrix. In other words, A is Hamiltonian if and only if (JA)T = JA where ()T denotes the transpose. So… That’s the answer. 🙂 And there’s another reason too: Hamilton invented the quaternions and… Well… I’ll leave it to you to check out what these have got to do with quantum physics. 🙂

[…] Oh ! And what about the maser example? Well… I am a bit tired now, so I’ll just refer you to Feynman’s exposé on it. It’s not that difficult if you understood all of the above. In fact, it’s actually quite straightforward, and so I really recommend you work your way through the example, as it will give you a much better ‘feel’ for the quantum-mechanical framework we’ve developed so far. In fact, walking through the whole thing is like a kind of ‘reward’ for having worked so hard on the more abstract stuff in this and my previous posts. So… Yes. Just go for it! 🙂 [And, just in case you don’t want to go for it, I did write a little introduction to in the following post. :-)]