Quantum mechanics:
1900 (Planck): Max Planck proposed that light with frequency ν is emitted in quantized lumps of energy that come in integral multiples of the quantity,
E = hν = ¯hω (1)
where h ≈ 6.63 · 10−34 J ·s is Planck’s constant, and ¯h ≡ h/2π = 1.06 · 10−34 J ·s.
The frequency ν of light is generally very large (on the order of 1015 s−1 for the visible spectrum), but the smallness of h wins out, so the hν unit of energy is very small (at least on an everyday energy scale). The energy is therefore essentially continuous for most purposes. However, a puzzle in late 19th-century physics was the blackbody radiation problem. In a nutshell, the issue was that the classical (continuous) theory of light predicted that certain objects would radiate an infinite amount of energy, which of course can’t be correct. Planck’s hypothesis of quantized radiation not only got rid of the problem of the infinity, but also correctly predicted the shape of the power curve as a function of temperature.
The results that we derived for electromagnetic waves are still true. In particular, the energy flux is given by the Poynting vector And E = pc for a light. Planck’s hypothesis simply adds the information of how many lumps of energy a wave contains. Although strictly speaking, Planck initially thought that the quantization was only a function of the emission process and not inherent to the light itself.
1905 (Einstein): Albert Einstein stated that the quantization was in fact inherent to the light, and that the lumps can be interpreted as particles, which we now call “photons.” This proposal was a result of his work on the photoelectric effect, which deals with the absorption of light and the emission of electrons from a material.
We know that E = pc for a light wave. (This relation also follows from Einstein’s 1905 work on relativity, where he showed that E = pc for any massless particle, an example of which is a photon.) And we also know that ω = ck for a light wave. So Planck’s E = ¯hω relation becomes
E = ¯hω =⇒ pc = ¯h(ck) =⇒ p = ¯hk (2)
This result relates the momentum of a photon to the wavenumber of the wave it is associated with.
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1913 (Bohr): Niels Bohr stated that electrons in atoms have wavelike properties. This correctly explained a few things about hydrogen, in particular the quantized energy levels that were known.
1924 (de Broglie): Louis de Broglie proposed that all particles are associated with waves, where the frequency and wavenumber of the wave are given by the same relations we found above for photons, namely E = ¯hω and p = ¯hk. The larger E and p are, the larger ω and k are. Even for small E and p that are typical of a photon, ω and k are very large because ¯h is so small. So any everyday-sized particle with large (in comparison) energy and momentum values will have extremely large ω and k values. This (among other reasons) makes it virtually impossible to observe the wave nature of macroscopic amounts of matter.
This proposal (that E = ¯hω and p = ¯hk also hold for massive particles) was a big step, because many things that are true for photons are not true for massive (and nonrelativistic) particles. For example, E = pc (and hence ω = ck) holds only for massless particles (we’ll see below how ω and k are related for massive particles). But the proposal was a reasonable one to try. And it turned out to be correct, in view of the fact that the resulting predictions agree with experiments.
The fact that any particle has a wave associated with it leads to the so-called waveparticle duality. Are things particles, or waves, or both? Well, it depends what you’re doing with them. Sometimes things behave like waves, sometimes they behave like particles. A vaguely true statement is that things behave like waves until a measurement takes place, at which point they behave like particles. However, approximately one million things are left unaddressed in that sentence. The wave-particle duality is one of the things that few people, if any, understand about quantum mechanics.
1925 (Heisenberg): Werner Heisenberg formulated a version of quantum mechanics that made use of matrix mechanics. We won’t deal with this matrix formulation (it’s rather difficult), but instead with the following wave formulation due to Schrodinger (this is a waves book, after all).
1926 (Schrodinger): Erwin Schrodinger formulated a version of quantum mechanics that was based on waves. He wrote down a wave equation (the so-called Schrodinger equation) that governs how the waves evolve in space and time. We’ll deal with this equation in depth below. Even though the equation is correct, the correct interpretation of what the wave actually meant was still missing. Initially Schrodinger thought (incorrectly) that the wave represented the charge density.
1926 (Born): Max Born correctly interpreted Schrodinger’s wave as a probability amplitude. By “amplitude” we mean that the wave must be squared to obtain the desired probability. More precisely, since the wave (as we’ll see) is in general complex, we need to square its absolute value. This yields the probability of finding a particle at a given location (assuming that the wave is written as a function of x).
This probability isn’t a consequence of ignorance, as is the case with virtually every other example of probability you’re familiar with. For example, in a coin toss, if you know everything about the initial motion of the coin (velocity, angular velocity), along with all external influences (air currents, nature of the floor it lands on, etc.), then you can predict which side will land facing up. Quantum mechanical probabilities aren’t like this. They aren’t a consequence of missing information. The probabilities are truly random, and there is no further information (so-called “hidden variables”) that will make things unrandom. The topic of hidden variables includes various theorems (such as Bell’s theorem) and experimental results that you will learn about in a quantum mechanics course.
1926 (Dirac): Paul Dirac showed that Heisenberg’s and Schrodinger’s versions of quantum mechanics were equivalent, in that they could both be derived from a more general version of quantum mechanics.
The Schrodinger equation
we’ll give a “derivation” of the Schrodinger equation. Our starting point will be the classical nonrelativistic expression for the energy of a particle, which is the sum of the kinetic and potential energies. We’ll assume as usual that the potential is a function of only x. We have
E = K + V = 1 /2 mv2 + V (x) = p2 /2m + V (x). (3)
We’ll now invoke de Broglie’s claim that all particles can be represented as waves with frequency ω and wavenumber k, and that E = ¯hω and p = ¯hk. This turns the expression for the energy into
¯hω = ¯h2 k2 /2m + V (x). (4)
A wave with frequency ω and wavenumber k can be written as usual as, ψ(x, t) = Aei(kx−ωt) (the convention is to put a minus sign in front of the ωt). In 3-D we would have ψ(r, t) = Aei(k·r−ωt) , but let’s just deal with 1-D. We now note that
∂ψ /∂t = −iωψ =⇒ ωψ = i ∂ψ /∂t , and ∂ 2ψ /∂x2 = −k2ψ =⇒ k2ψ = − ∂ 2ψ /∂x2 . (5)
If we multiply the energy equation in Eq. (4) by ψ, and then plug in these relations, we obtain
¯h(ωψ) = ¯h2 /2m .(k 2ψ) + V (x)ψ =⇒ i¯h ∂ψ /∂t = −¯h2 /2m · ∂ 2ψ /∂x2 + V ψ (6)
This is the time-dependent Schrodinger equation. If we put the x and t arguments back in, the equation takes the form,
i¯h ∂ψ(x, t) /∂t = −¯h2 /2m · ∂ 2ψ(x, t) /∂x2 + V (x)ψ(x, t). (7)
In 3-D, the x dependence turns into dependence on all three coordinates (x, y, z), and the ∂ 2ψ/∂x2 term becomes ∇2ψ (the sum of the second derivatives). Remember that Born’s (correct) interpretation of ψ(x) is that |ψ(x)|2 gives the probability of finding the particle at position x.
Having successfully produced the time-dependent Schrodinger equation, we should ask: Did the above reasoning actually prove that the Schrodinger equation is valid? No, it didn’t, for three reasons.
1. The reasoning is based on de Broglie’s assumption that there is a wave associated with every particle, and also on the assumption that the ω and k of the wave are related to E and p via Planck’s constant in Eqs. (1) and (2). We had to accept these assumptions on faith.
2. Said in a different way, it is impossible to actually prove anything in physics. All we can do is make an educated guess at a theory, and then do experiments to try to show that the theory is consistent with the real world. The more experiments we do, the more comfortable we are that the theory is a good one. But we can never be absolutely sure that we have the correct theory. In fact, odds are that it’s simply the limiting case of a more correct theory.
3.
The Schrodinger equation actually isn’t valid, so there’s certainly no
way that we proved it. Consistent with the above point concerning
limiting cases, the quantum theory based on Schrodinger’s equation is
just a limiting theory of a more correct one, which happens to be
quantum field theory (which unifies quantum mechanics with special
relativity). This in turn must be a limiting theory of yet another more
correct one, because it doesn’t incorporate gravity. Eventually, there
will be one theory that covers everything (although this point can be
debated), but we’re definitely not there yet.
Due to the “i” that appears in Eq. (6), ψ(x) is complex. And in contrast with waves in classical mechanics, the entire complex function now matters in quantum mechanics. We won’t be taking the real part in the end. Up to this point in the book, the use of complex functions was simply a matter of convenience, because it is easier to work with exponentials than trig functions. Only the real part mattered (or imaginary part – take your pick, but not both). But in quantum mechanics the whole complex wavefunction is relevant. However, the theory is structured in such a way that anything you might want to measure (position, momentum, energy, etc.) will always turn out to be a real quantity. This is a necessary feature of any valid theory, of course, because you’re not going to go out and measure a distance of 2 + 5i meters, or pay an electrical bill of 17 + 6i kilowatt-hours.
As mentioned, there is an endless number of difficult questions about quantum mechanics that can be discussed. But in this short introduction to the subject, let’s just accept Schrodinger’s equation as valid, and see where it takes us.
Solving the equation
If we put aside the profound implications of the Schrodinger equation and regard it as simply a mathematical equation, then it’s just another wave equation. We already know the solution, of course, because we used the function ψ(x, t) = Aei(kx−ωt) to produce Eqs. (5) and (6) in the first place. But let’s pretend that we don’t know this, and let’s solve the Schrodinger equation as if we were given it out of the blue. As always, we’ll guess an exponential solution. If we first look at exponential behavior in the time coordinate, our guess is ψ(x, t) = e −iωtf(x) (the minus sign here is convention). Plugging this into Eq. (7) and canceling the e −iωt yields,
¯hωf(x) = − ¯h2 /2m. ∂2f(x) /∂x2 + V (x)f(x). (8)
But from Eq. (1), we have ¯hω = E. And we’ll now replace f(x) with ψ(x). This might cause a little confusion, since we’ve already used ψ to denote the entire wavefunction ψ(x, t). However, it is general convention to also use the letter ψ to denote the spatial part. So we now have
E ψ(x) = − ¯h2 /2m. ∂2ψ(x) /∂x2 + V (x)ψ(x) (9)
This is called the time-independent Schrodinger equation. This equation is more restrictive than the original time-dependent Schrodinger equation, because it assumes that the particle/wave has a definite energy (that is, a definite ω). In general, a particle can be in a state that is the superposition of states with various definite energies, just like the motion of a string can be the superposition of various normal modes with definite ω’s. The same reasoning applies here as with all the other waves we’ve discussed: From Fourier analysis and from the linearity of the Schrodinger equation, we can build up any general wavefunction from ones with specific energies. Because of this, it suffices to consider the time-independent Schrodinger equation. The solutions for that equation form a basis for all possible solutions. Continuing with our standard strategy of guessing exponentials, we’ll let ψ(x) = Aeikx. Plugging this into Eq. (9) and canceling the e ikx gives (going back to the ¯hω instead of E)
¯hω = − ¯h2 /2m (−k2 ) + V (x) =⇒ ¯hω = ¯h2 k2 /2m + V (x). (10)
This is simply Eq. (4), so we’ve ended up back where we started, as expected. However, our goal here was to show how the Schrodinger equation can be solved from scratch, without knowing where it came from.
Eq. (10) is (sort of) a dispersion relation. If V (x) is a constant C in a given region, then the relation between ω and k (namely ω = ¯hk2/2m + C) is independent of x, so we have a nice sinusoidal wavefunction (or exponential, if k is imaginary). However, if V (x) isn’t constant, then the wavefunction isn’t characterized by a unique wavenumber. So a function of the form e ikx doesn’t work as a solution for ψ(x). (A Fourier superposition can certainly work, since any function can be expressed that way, but a single e ikx by itself doesn’t work.) This is similar to the case where the density of a string isn’t constant. We don’t obtain sinusoidal waves there either.
